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mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
(hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
z ∈ adjoin R ({B.gen} : Set L) | case neg.hz
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (Submodule.span R {p})
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : NoZeroSMulDivisors R L
x✝ : L[X] := Polynomial.map (algebraMap R L) P
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
a✝ : 0 ∈ range (Q.natDegree + 1)
⊢ p ∣ Q.coeff 0 | exact dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt hp hBint hQ hzint hei | no goals | fa4f10e8cbdbb4c7 |
Array.SatisfiesM_anyM | Mathlib/.lake/packages/batteries/Batteries/Data/Array/Monadic.lean | theorem SatisfiesM_anyM [Monad m] [LawfulMonad m] (p : α → m Bool) (as : Array α) (start stop)
(hstart : start ≤ min stop as.size) (tru : Prop) (fal : Nat → Prop) (h0 : fal start)
(hp : ∀ i : Fin as.size, i.1 < stop → fal i.1 →
SatisfiesM (bif · then tru else fal (i + 1)) (p as[i])) :
SatisfiesM
(fun res => bif res then tru else fal (min stop as.size))
(anyM p as start stop) | case isFalse
m : Type → Type u_1
α : Type u_2
inst✝¹ : Monad m
inst✝ : LawfulMonad m
p : α → m Bool
as : Array α
start stop✝ : Nat
hstart : start ≤ min stop✝ as.size
tru : Prop
fal : Nat → Prop
h0✝ : fal start
hp✝ : ∀ (i : Fin as.size), ↑i < stop✝ → fal ↑i → SatisfiesM (fun x => bif x then tru else fal (↑i + 1)) (p as[i])
stop j : Nat
hj' : j ≤ stop
hstop : stop ≤ as.size
h0 : fal j
hp : ∀ (i : Fin as.size), ↑i < stop → fal ↑i → SatisfiesM (fun x => bif x then tru else fal (↑i + 1)) (p as[i])
h✝ : ¬j < stop
⊢ SatisfiesM (fun res => bif res then tru else fal stop) (pure false) | next hj => exact .pure <| Nat.le_antisymm hj' (Nat.ge_of_not_lt hj) ▸ h0 | no goals | d562cc739db72f2f |
rootsOfUnity.integer_power_of_ringEquiv' | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | theorem rootsOfUnity.integer_power_of_ringEquiv' (g : L ≃+* L) :
∃ m : ℤ, ∀ t ∈ rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) | L : Type u
inst✝² : CommRing L
inst✝¹ : IsDomain L
n : ℕ
inst✝ : NeZero n
g : L ≃+* L
⊢ ∃ m, ∀ t ∈ rootsOfUnity n L, g ↑t = ↑(t ^ m) | simpa using rootsOfUnity.integer_power_of_ringEquiv n g | no goals | 3ffe73c8d1d46269 |
List.findIdx?_eq_fst_find?_zipIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem findIdx?_eq_fst_find?_zipIdx {xs : List α} {p : α → Bool} :
xs.findIdx? p = (xs.zipIdx.find? fun ⟨x, _⟩ => p x).map (·.2) | case nil
α : Type u_1
p : α → Bool
⊢ findIdx? p [] =
Option.map (fun x => x.snd)
(find?
(fun x =>
match x with
| (x, snd) => p x)
[].zipIdx) | simp | no goals | b0e05c550ded97a0 |
AffineSubspace.setOf_wSameSide_eq_image2 | Mathlib/Analysis/Convex/Side.lean | theorem setOf_wSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{ y | s.WSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ici 0) s | case h.mp.intro.intro.inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
h : y -ᵥ p₂ = 0
⊢ ∃ a, 0 ≤ a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y | rw [vsub_eq_zero_iff_eq] at h | case h.mp.intro.intro.inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
h : y = p₂
⊢ ∃ a, 0 ≤ a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y | 0ddcbc2292b9cdb2 |
CategoryTheory.extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types | Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean | lemma extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types [FinitaryPreExtensive C]
{F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
(h : Presheaf.IsLocallySurjective (extensiveTopology C) f) {X : C} :
Function.Surjective (f.app (op X)) | case intro.intro.intro.intro.intro.a
C : Type u_1
inst✝³ : Category.{u_4, u_1} C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone :=
(Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) :=
(Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) :=
ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
⊢ (ConcreteCategory.hom (G.map (π a).op)) (f.app (op X) (i.hom y)) = (ConcreteCategory.hom (G.map (π a).op)) x | rw [← (h' a).choose_spec] | case intro.intro.intro.intro.intro.a
C : Type u_1
inst✝³ : Category.{u_4, u_1} C
inst✝² : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ Type w
f : F ⟶ G
inst✝¹ : PreservesFiniteProducts F
inst✝ : PreservesFiniteProducts G
X : C
x : G.obj (op X)
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), (Presheaf.imageSieve f x).arrows (π a)
y : (a : α) → F.obj (op (Y a)) := fun a => Exists.choose ⋯
x✝ : Fintype α := Fintype.ofFinite α
ht : IsLimit (Types.productLimitCone fun a => F.obj (op (Y a))).cone :=
(Types.productLimitCone fun a => F.obj (op (Y a))).isLimit
ht' : IsLimit (Cone.whisker (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))) :=
(Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse (Cocone.op (Cofan.mk X π))).symm h.some.op
i : ((a : α) → F.obj (op (Y a))) ≅ F.obj (op X) :=
ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht') (Discrete.natIso fun x => Iso.refl (F.obj (op (Y x.as))))
a : α
⊢ (ConcreteCategory.hom (G.map (π a).op)) (f.app (op X) (i.hom y)) =
(ConcreteCategory.hom (f.app (op (Y a)))) (Exists.choose ⋯) | f246db99c628b9b2 |
FirstOrder.Language.Theory.models_formula_iff_onTheory_models_equivSentence | Mathlib/ModelTheory/Satisfiability.lean | theorem models_formula_iff_onTheory_models_equivSentence {φ : L.Formula α} :
T ⊨ᵇ φ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ Formula.equivSentence φ | case refine_1
L : Language
T : L.Theory
α : Type w
φ : L.Formula α
h : T ⊨ᵇ φ
M : ((L.lhomWithConstants α).onTheory T).ModelType
this : L.Structure ↑M := (L.lhomWithConstants α).reduct ↑M
⊢ ↑M ⊨ Formula.equivSentence φ | have : (L.lhomWithConstants α).IsExpansionOn M := LHom.isExpansionOn_reduct _ _ | case refine_1
L : Language
T : L.Theory
α : Type w
φ : L.Formula α
h : T ⊨ᵇ φ
M : ((L.lhomWithConstants α).onTheory T).ModelType
this✝ : L.Structure ↑M := (L.lhomWithConstants α).reduct ↑M
this : (L.lhomWithConstants α).IsExpansionOn ↑M
⊢ ↑M ⊨ Formula.equivSentence φ | 68157140d9be2023 |
ProbabilityTheory.Kernel.compProd_preimage_fst | Mathlib/Probability/Kernel/Composition/CompProd.lean | lemma compProd_preimage_fst {s : Set β} (hs : MeasurableSet s) (κ : Kernel α β)
(η : Kernel (α × β) γ) [IsSFiniteKernel κ] [IsMarkovKernel η] (x : α) :
(κ ⊗ₖ η) x (Prod.fst ⁻¹' s) = κ x s | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set β
hs : MeasurableSet s
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsMarkovKernel η
x : α
⊢ ∫⁻ (b : β), (η (x, b)) {c | b ∈ s} ∂κ x = (κ x) s | have : ∀ b : β, η (x, b) {_c | b ∈ s} = s.indicator (fun _ ↦ 1) b := by
intro b
by_cases hb : b ∈ s <;> simp [hb] | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set β
hs : MeasurableSet s
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsMarkovKernel η
x : α
this : ∀ (b : β), (η (x, b)) {_c | b ∈ s} = s.indicator (fun x => 1) b
⊢ ∫⁻ (b : β), (η (x, b)) {c | b ∈ s} ∂κ x = (κ x) s | 36d443ec521ea3bb |
Nat.Partrec.Code.encode_lt_pair | Mathlib/Computability/PartrecCode.lean | theorem encode_lt_pair (cf cg) :
encode cf < encode (pair cf cg) ∧ encode cg < encode (pair cf cg) | cf cg : Code
⊢ encode cf < encode (cf.pair cg) ∧ encode cg < encode (cf.pair cg) | simp only [encodeCode_eq, encodeCode] | cf cg : Code
⊢ cf.encodeCode < 2 * (2 * Nat.pair cf.encodeCode cg.encodeCode) + 4 ∧
cg.encodeCode < 2 * (2 * Nat.pair cf.encodeCode cg.encodeCode) + 4 | 5e05412ef5ddc786 |
IntermediateField.sup_toSubalgebra_of_isAlgebraic_right | Mathlib/FieldTheory/IntermediateField/Adjoin/Algebra.lean | theorem sup_toSubalgebra_of_isAlgebraic_right [Algebra.IsAlgebraic K E2] :
(E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra | K : Type u_3
L : Type u_4
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
E1 E2 : IntermediateField K L
inst✝ : Algebra.IsAlgebraic K ↥E2
this : (adjoin ↥E1 ↑E2).toSubalgebra = Algebra.adjoin ↥E1 ↑E2
⊢ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra | apply_fun Subalgebra.restrictScalars K at this | K : Type u_3
L : Type u_4
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
E1 E2 : IntermediateField K L
inst✝ : Algebra.IsAlgebraic K ↥E2
this :
Subalgebra.restrictScalars K (adjoin ↥E1 ↑E2).toSubalgebra = Subalgebra.restrictScalars K (Algebra.adjoin ↥E1 ↑E2)
⊢ (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra | 6f61e299237242c9 |
Setoid.IsPartition.sUnion_eq_univ | Mathlib/Data/Setoid/Partition.lean | theorem IsPartition.sUnion_eq_univ {c : Set (Set α)} (hc : IsPartition c) : ⋃₀ c = Set.univ :=
Set.eq_univ_of_forall fun x =>
Set.mem_sUnion.2 <|
let ⟨t, ht⟩ := hc.2 x
⟨t, by
simp only [existsUnique_iff_exists] at ht
tauto⟩
| α : Type u_1
c : Set (Set α)
hc : IsPartition c
x : α
t : Set α
ht : (t ∈ c ∧ x ∈ t) ∧ ∀ (y : Set α), y ∈ c ∧ x ∈ y → y = t
⊢ t ∈ c ∧ x ∈ t | tauto | no goals | 6df30b3690c4e8b9 |
exteriorPower.alternatingMapLinearEquiv_symm_map | Mathlib/LinearAlgebra/ExteriorPower/Basic.lean | @[simp] lemma alternatingMapLinearEquiv_symm_map (f : M →ₗ[R] N) :
alternatingMapLinearEquiv.symm (map n f) = (ιMulti R n).compLinearMap f | R : Type u
inst✝⁴ : CommRing R
n : ℕ
M : Type u_1
N : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
⊢ alternatingMapLinearEquiv.symm (map n f) = (ιMulti R n).compLinearMap f | simp only [map, LinearEquiv.symm_apply_apply] | no goals | da62e5af6f98c521 |
CategoryTheory.Meq.congr_apply | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | lemma congr_apply {X} {P : Cᵒᵖ ⥤ D} {S : J.Cover X} (x : Meq P S) {Y}
{f g : Y ⟶ X} (h : f = g) (hf : S f) :
x ⟨_, _, hf⟩ = x ⟨_, g, by simpa only [← h] using hf⟩ | C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
X : C
P : Cᵒᵖ ⥤ D
S : J.Cover X
x : Meq P S
Y : C
f g : Y ⟶ X
h : f = g
hf : (↑S).arrows f
⊢ ↑x { Y := Y, f := f, hf := hf } = ↑x { Y := Y, f := g, hf := ⋯ } | subst h | C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
X : C
P : Cᵒᵖ ⥤ D
S : J.Cover X
x : Meq P S
Y : C
f : Y ⟶ X
hf : (↑S).arrows f
⊢ ↑x { Y := Y, f := f, hf := hf } = ↑x { Y := Y, f := f, hf := ⋯ } | aaa88f681fa58bcc |
Std.Tactic.BVDecide.BVExpr.bitblast.blastZeroExtend.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ZeroExtend.lean | theorem go_get_aux (aig : AIG α) (w : Nat) (input : AIG.RefVec aig w) (newWidth curr : Nat)
(hcurr : curr ≤ newWidth) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr) (hfoo),
(go aig w input newWidth curr hcurr s).vec.get idx (by omega)
=
(s.get idx hidx).cast hfoo | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
w : Nat
input : aig.RefVec w
newWidth curr : Nat
hcurr : curr ≤ newWidth
s : aig.RefVec curr
⊢ ∀ (idx : Nat) (hidx : idx < curr) (hfoo : aig.decls.size ≤ (go aig w input newWidth curr hcurr s).aig.decls.size),
(go aig w input newWidth curr hcurr s).vec.get idx ⋯ = (s.get idx hidx).cast hfoo | intro idx hidx | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
w : Nat
input : aig.RefVec w
newWidth curr : Nat
hcurr : curr ≤ newWidth
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
⊢ ∀ (hfoo : aig.decls.size ≤ (go aig w input newWidth curr hcurr s).aig.decls.size),
(go aig w input newWidth curr hcurr s).vec.get idx ⋯ = (s.get idx hidx).cast hfoo | aa7bc21c8b54f977 |
Equiv.Perm.ofSubtype_swap_eq | Mathlib/GroupTheory/Perm/Support.lean | theorem ofSubtype_swap_eq {p : α → Prop} [DecidablePred p] (x y : Subtype p) :
ofSubtype (Equiv.swap x y) = Equiv.swap ↑x ↑y :=
Equiv.ext fun z => by
by_cases hz : p z
· rw [swap_apply_def, ofSubtype_apply_of_mem _ hz]
split_ifs with hzx hzy
· simp_rw [hzx, Subtype.coe_eta, swap_apply_left]
· simp_rw [hzy, Subtype.coe_eta, swap_apply_right]
· rw [swap_apply_of_ne_of_ne] <;>
simp [Subtype.ext_iff, *]
· rw [ofSubtype_apply_of_not_mem _ hz, swap_apply_of_ne_of_ne]
· intro h
apply hz
rw [h]
exact Subtype.prop x
intro h
apply hz
rw [h]
exact Subtype.prop y
| case neg.a
α : Type u_1
inst✝¹ : DecidableEq α
p : α → Prop
inst✝ : DecidablePred p
x y : Subtype p
z : α
hz : ¬p z
⊢ z ≠ ↑x | intro h | case neg.a
α : Type u_1
inst✝¹ : DecidableEq α
p : α → Prop
inst✝ : DecidablePred p
x y : Subtype p
z : α
hz : ¬p z
h : z = ↑x
⊢ False | 7c967a4cc58d1491 |
ProbabilityTheory.Kernel.indep_limsup_atTop_self | Mathlib/Probability/Independence/ZeroOne.lean | theorem Kernel.indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
Indep (limsup s atTop) (limsup s atTop) κ μα | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ Indep (limsup s atTop) (limsup s atTop) κ μα | refine indep_limsup_self h_le h_indep ?_ ?_ hnsp ?_ | case refine_1
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ ∀ (t : Set ι), BddAbove t → tᶜ ∈ atTop
case refine_2
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ Directed (fun x1 x2 => x1 ≤ x2) ns
case refine_3
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
hnsp : ∀ (i : ι), BddAbove (ns i)
⊢ ∀ (n : ι), ∃ a, n ∈ ns a | 24b40639abdf5ed0 |
VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | lemma hasFTaylorSeriesUpTo_fourierIntegral {N : WithTop ℕ∞}
(hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖) μ)
(h'f : AEStronglyMeasurable f μ) :
HasFTaylorSeriesUpTo N (fourierIntegral 𝐞 μ L.toLinearMap₂ f)
(fun w n ↦ fourierIntegral 𝐞 μ L.toLinearMap₂ (fun v ↦ fourierPowSMulRight L f v n) w) | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : WithTop ℕ∞
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ
h'f : AEStronglyMeasurable f μ
n : ℕ
hn : ↑n < N
w : W
I₁ : Integrable (fun v => fourierPowSMulRight L f v n) μ
I₂ : Integrable (fun v => ‖v‖ * ‖fourierPowSMulRight L f v n‖) μ
I₃ : Integrable (fun v => fourierPowSMulRight L f v (n + 1)) μ
⊢ Integrable (fun v => fourierSMulRight L (fun v => fourierPowSMulRight L f v n) v) μ | apply (I₂.const_mul ((2 * π * ‖L‖))).mono' (h'f.fourierPowSMulRight L n).fourierSMulRight | E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
N : WithTop ℕ∞
hf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ
h'f : AEStronglyMeasurable f μ
n : ℕ
hn : ↑n < N
w : W
I₁ : Integrable (fun v => fourierPowSMulRight L f v n) μ
I₂ : Integrable (fun v => ‖v‖ * ‖fourierPowSMulRight L f v n‖) μ
I₃ : Integrable (fun v => fourierPowSMulRight L f v (n + 1)) μ
⊢ ∀ᵐ (a : V) ∂μ,
‖fourierSMulRight L (fun v => fourierPowSMulRight L f v n) a‖ ≤ 2 * π * ‖L‖ * (‖a‖ * ‖fourierPowSMulRight L f a n‖) | de7ff764a7066172 |
List.length_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem length_map (as : List α) (f : α → β) : (as.map f).length = as.length | case cons
α : Type u_1
β : Type u_2
f : α → β
head✝ : α
as : List α
ih : (map f as).length = as.length
⊢ (map f (head✝ :: as)).length = (head✝ :: as).length | simp [List.map, ih] | no goals | e236928c127beadd |
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.e_s
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
⊢ IsEmpty (Fin 0) | infer_instance | no goals | 4597273329d964a3 |
MonomialOrder.degree_add_le | Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean | theorem degree_add_le {f g : MvPolynomial σ R} :
m.toSyn (m.degree (f + g)) ≤ m.toSyn (m.degree f) ⊔ m.toSyn (m.degree g) | case pos.h
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
b : σ →₀ ℕ
hb : b ∈ (f + g).support
hf : b ∈ f.support
⊢ m.toSyn b ≤ m.toSyn (m.degree f) | exact m.le_degree hf | no goals | 3ccc5da830d2ee29 |
Fin.insertNth_binop | Mathlib/Data/Fin/Tuple/Basic.lean | theorem insertNth_binop (op : ∀ j, α j → α j → α j) (i : Fin (n + 1)) (x y : α i)
(p q : ∀ j, α (i.succAbove j)) :
(i.insertNth (op i x y) fun j ↦ op _ (p j) (q j)) = fun j ↦
op j (i.insertNth x p j) (i.insertNth y q j) :=
insertNth_eq_iff.2 <| by unfold removeNth; simp
| n : ℕ
α : Fin (n + 1) → Sort u_1
op : (j : Fin (n + 1)) → α j → α j → α j
i : Fin (n + 1)
x y : α i
p q : (j : Fin n) → α (i.succAbove j)
⊢ op i x y = op i (i.insertNth x p i) (i.insertNth y q i) ∧
(fun j => op (i.succAbove j) (p j) (q j)) = i.removeNth fun j => op j (i.insertNth x p j) (i.insertNth y q j) | unfold removeNth | n : ℕ
α : Fin (n + 1) → Sort u_1
op : (j : Fin (n + 1)) → α j → α j → α j
i : Fin (n + 1)
x y : α i
p q : (j : Fin n) → α (i.succAbove j)
⊢ op i x y = op i (i.insertNth x p i) (i.insertNth y q i) ∧
(fun j => op (i.succAbove j) (p j) (q j)) = fun i_1 =>
(fun j => op j (i.insertNth x p j) (i.insertNth y q j)) (i.succAbove i_1) | 0a8a748d549c1480 |
HurwitzKernelBounds.isBigO_atTop_F_int_zero_sub | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | lemma isBigO_atTop_F_int_zero_sub (a : UnitAddCircle) : ∃ p, 0 < p ∧
(fun t ↦ F_int 0 a t - (if a = 0 then 1 else 0)) =O[atTop] fun t ↦ exp (-p * t) | a : UnitAddCircle
⊢ ∃ p, 0 < p ∧ (fun t => F_int 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t) | obtain ⟨a, ha, rfl⟩ := a.eq_coe_Ico | case intro.intro
a : ℝ
ha : a ∈ Ico 0 1
⊢ ∃ p, 0 < p ∧ (fun t => F_int 0 (↑a) t - if ↑a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t) | 216d869dd795975f |
UniformSpace.Completion.mul_hatInv_cancel | Mathlib/Topology/Algebra/UniformField.lean | theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 | K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
⊢ x * x.hatInv = 1 | haveI : T1Space (hat K) := T2Space.t1Space | K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : TopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : UniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
⊢ x * x.hatInv = 1 | 64ff41b42847f96a |
ArithmeticFunction.vonMangoldt.summable_F'' | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | private lemma summable_F'' : Summable F'' | hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
⊢ Summable ArithmeticFunction.vonMangoldt.F'' | suffices Summable fun (pk : Nat.Primes × ℕ) ↦ (pk.1 : ℝ)⁻¹ ^ (pk.2 + 3 / 2 : ℝ) by
refine (Summable.mul_left 2 this).of_nonneg_of_le (fun pk ↦ ?_) (fun pk ↦ F''_le pk.1 pk.2)
simp only [F'', Function.comp_apply, F', F₀, Prod.map_fst, id_eq, Prod.map_snd, Nat.cast_pow]
have := vonMangoldt_nonneg (n := (pk.1 : ℕ) ^ (pk.2 + 2))
positivity | hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹
hp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1
⊢ Summable fun pk => (↑↑pk.1)⁻¹ ^ (↑pk.2 + 3 / 2) | c675abf56f959c99 |
Even.pow_nonneg | Mathlib/Algebra/Order/Ring/Basic.lean | lemma Even.pow_nonneg (hn : Even n) (a : R) : 0 ≤ a ^ n | case intro
R : Type u_3
inst✝¹ : LinearOrderedSemiring R
inst✝ : ExistsAddOfLE R
a : R
k : ℕ
⊢ 0 ≤ a ^ (k + k) | rw [pow_add] | case intro
R : Type u_3
inst✝¹ : LinearOrderedSemiring R
inst✝ : ExistsAddOfLE R
a : R
k : ℕ
⊢ 0 ≤ a ^ k * a ^ k | edbb2af1e09ebb9e |
MeasureTheory.eLpNorm_one_le_of_le | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem eLpNorm_one_le_of_le {r : ℝ≥0} (hfint : Integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ)
(hf : ∀ᵐ ω ∂μ, f ω ≤ r) : eLpNorm f 1 μ ≤ 2 * μ Set.univ * r | case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : r = 0
⊢ eLpNorm f 1 μ ≤ 2 * μ univ * ↑r | suffices f =ᵐ[μ] 0 by
rw [eLpNorm_congr_ae this, eLpNorm_zero, hr, ENNReal.coe_zero, mul_zero] | case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : r = 0
⊢ f =ᶠ[ae μ] 0 | f01ab8867e396603 |
CommGroup.exists_apply_ne_one_aux | Mathlib/GroupTheory/FiniteAbelian/Duality.lean | private
lemma exists_apply_ne_one_aux
(H : ∀ n : ℕ, n ∣ Monoid.exponent G → ∀ a : ZMod n, a ≠ 0 →
∃ φ : Multiplicative (ZMod n) →* M, φ (.ofAdd a) ≠ 1)
{a : G} (ha : a ≠ 1) :
∃ φ : G →* M, φ a ≠ 1 | G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
H : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1
a : G
ha : a ≠ 1
ι : Type
w✝ : Fintype ι
n : ι → ℕ
left✝ : ∀ (i : ι), 1 < n i
h : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h.some
⊢ ∃ i, e a i ≠ 1 | contrapose! ha | G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
H : ∀ (n : ℕ), n ∣ Monoid.exponent G → ∀ (a : ZMod n), a ≠ 0 → ∃ φ, φ (Multiplicative.ofAdd a) ≠ 1
a : G
ι : Type
w✝ : Fintype ι
n : ι → ℕ
left✝ : ∀ (i : ι), 1 < n i
h : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h.some
ha : ∀ (i : ι), e a i = 1
⊢ a = 1 | ca48e086b93104fd |
exteriorPower.pairingDual_apply_apply_eq_one_zero | Mathlib/LinearAlgebra/ExteriorPower/Pairing.lean | lemma pairingDual_apply_apply_eq_one_zero (a b : Fin n ↪o ι) (h : a ≠ b) :
pairingDual R M n (ιMulti _ _ (f ∘ a)) (ιMulti _ _ (x ∘ b)) = 0 | case h
R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
σ : Equiv.Perm (Fin n)
x✝ : σ ∈ Finset.univ
h' : ¬∏ x_1 : Fin n, (f (a x_1)) (x (b (σ x_1))) = 0
i : Fin n
⊢ a i = (⇑b ∘ ⇑σ) i | by_contra hi | case h
R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
σ : Equiv.Perm (Fin n)
x✝ : σ ∈ Finset.univ
h' : ¬∏ x_1 : Fin n, (f (a x_1)) (x (b (σ x_1))) = 0
i : Fin n
hi : ¬a i = (⇑b ∘ ⇑σ) i
⊢ False | fd3ea1bd40177425 |
WeierstrassCurve.natDegree_coeff_ΨSq_ofNat | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | private lemma natDegree_coeff_ΨSq_ofNat (n : ℕ) :
(W.ΨSq n).natDegree ≤ n ^ 2 - 1 ∧ (W.ΨSq n).coeff (n ^ 2 - 1) = (n ^ 2 : ℤ) | R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
h : ∀ {n : ℕ},
(W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
(W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1) ^ 2 - 1 = 2 * WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
⊢ (↑n + 1) ^ 2 = (if Even (n + 1) then (↑n + 1) / 2 else ↑n + 1) ^ 2 * if Even (n + 1) then 4 else 1 | split_ifs <;> ring1 | no goals | 96420254b4130cb9 |
MeasureTheory.Measure.withDensity_rnDeriv_eq | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) :
ν.withDensity (rnDeriv μ ν) = μ | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
h : μ ≪ ν
h_sing : μ.singularPart ν ⟂ₘ ν
⊢ (μ.singularPart ν) Set.univ = 0 | rw [← measure_add_measure_compl h_sing.measurableSet_nullSet] | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
h : μ ≪ ν
h_sing : μ.singularPart ν ⟂ₘ ν
⊢ (μ.singularPart ν) h_sing.nullSet + (μ.singularPart ν) h_sing.nullSetᶜ = 0 | e394f2342b33f5a1 |
AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff | Mathlib/AlgebraicGeometry/AffineScheme.lean | theorem basicOpen_union_eq_self_iff (s : Set Γ(X, U)) :
⨆ f : s, X.basicOpen (f : Γ(X, U)) = U ↔ Ideal.span s = ⊤ | X : Scheme
U : X.Opens
hU : IsAffineOpen U
s : Set ↑Γ(X, U)
⊢ ⋃ i, (PrimeSpectrum.basicOpen ↑i).carrier = Set.univ ↔ Ideal.span s = ⊤ | simp only [Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl] | X : Scheme
U : X.Opens
hU : IsAffineOpen U
s : Set ↑Γ(X, U)
⊢ ⋃ i, (PrimeSpectrum.zeroLocus {↑i})ᶜ = Set.univ ↔ Ideal.span s = ⊤ | e342cbef83e9eeeb |
exists_sum_eq_one_iff_pairwise_coprime | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
(∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) = 1) ↔
Pairwise (IsCoprime on fun i : t ↦ s i) | case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (if x = a then u * ∏ j ∈ insert a t \ {x}, s j else v * μ x * ∏ j ∈ insert a t \ {x}, s j) =
v * ((μ x * ∏ j ∈ t \ {x}, s j) * s a) | rw [mul_assoc, if_neg fun ha : x = a ↦ hat (ha.casesOn hx)] | case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ v * (μ x * ∏ j ∈ insert a t \ {x}, s j) = v * ((μ x * ∏ j ∈ t \ {x}, s j) * s a) | 8a28cdbc75233b6c |
MvPolynomial.eval₂_assoc | Mathlib/Algebra/MvPolynomial/Eval.lean | theorem eval₂_assoc (q : S₂ → MvPolynomial σ R) (p : MvPolynomial S₂ R) :
eval₂ f (fun t => eval₂ f g (q t)) p = eval₂ f g (eval₂ C q p) | case e_f.a
R : Type u
S₁ : Type v
S₂ : Type w
σ : Type u_1
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
f : R →+* S₁
g : σ → S₁
q : S₂ → MvPolynomial σ R
p : MvPolynomial S₂ R
a : R
⊢ f a = ((eval₂Hom f g).comp C) a | simp | no goals | 2eea5a3597162dd7 |
TensorPower.cast_eq_cast | Mathlib/LinearAlgebra/TensorPower/Basic.lean | theorem cast_eq_cast {i j} (h : i = j) :
⇑(cast R M h) = _root_.cast (congrArg (fun i => ⨂[R]^i M) h) | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i : ℕ
⊢ ⇑(cast R M ⋯) = _root_.cast ⋯ | rw [cast_refl] | R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i : ℕ
⊢ ⇑(LinearEquiv.refl R (⨂[R]^i M)) = _root_.cast ⋯ | 0321341678e9c761 |
Mon_.mul_associator | Mathlib/CategoryTheory/Monoidal/Mon_.lean | theorem mul_associator {M N P : Mon_ C} :
(tensorμ (M.X ⊗ N.X) P.X (M.X ⊗ N.X) P.X ≫
(tensorμ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensorμ M.X (N.X ⊗ P.X) M.X (N.X ⊗ P.X) ≫
(M.mul ⊗ tensorμ N.X P.X N.X P.X ≫ (N.mul ⊗ P.mul)) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ (tensorμ (M.X ⊗ N.X) P.X (M.X ⊗ N.X) P.X ≫ (tensorμ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensorμ M.X (N.X ⊗ P.X) M.X (N.X ⊗ P.X) ≫ (M.mul ⊗ tensorμ N.X P.X N.X P.X ≫ (N.mul ⊗ P.mul)) | simp only [tensor_obj, prodMonoidal_tensorObj, Category.assoc] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ tensorμ (M.X ⊗ N.X) P.X (M.X ⊗ N.X) P.X ≫ (tensorμ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensorμ M.X (N.X ⊗ P.X) M.X (N.X ⊗ P.X) ≫ (M.mul ⊗ tensorμ N.X P.X N.X P.X ≫ (N.mul ⊗ P.mul)) | c740555ca5cf3587 |
Std.DHashMap.Internal.List.getKey_insertListConst_of_contains_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getKey_insertListConst_of_contains_eq_false [BEq α] [EquivBEq α]
{l : List ((_ : α) × β)} {toInsert : List (α × β)} {k : α}
(not_contains : (toInsert.map Prod.fst).contains k = false)
{h} :
getKey k (insertListConst l toInsert) h =
getKey k l (containsKey_of_containsKey_insertListConst h not_contains) | α : Type u
β : Type v
inst✝¹ : BEq α
inst✝ : EquivBEq α
l : List ((_ : α) × β)
toInsert : List (α × β)
k : α
not_contains : (List.map Prod.fst toInsert).contains k = false
h : containsKey k (insertListConst l toInsert) = true
⊢ getKey k (insertListConst l toInsert) h = getKey k l ⋯ | rw [← Option.some_inj, ← getKey?_eq_some_getKey,
getKey?_insertListConst_of_contains_eq_false not_contains, getKey?_eq_some_getKey] | no goals | dd2cab53f1d3098e |
surjective_of_isSwap_of_isPretransitive | Mathlib/GroupTheory/Perm/ClosureSwap.lean | theorem surjective_of_isSwap_of_isPretransitive [Finite α] (S : Set G)
(hS1 : ∀ σ ∈ S, Perm.IsSwap (MulAction.toPermHom G α σ)) (hS2 : Subgroup.closure S = ⊤)
[h : MulAction.IsPretransitive G α] : Function.Surjective (MulAction.toPermHom G α) | G : Type u_1
α : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : DecidableEq α
inst✝ : Finite α
S : Set G
hS1 : ∀ σ ∈ S, ((toPermHom G α) σ).IsSwap
hS2 : closure S = ⊤
h : IsPretransitive G α
this : IsPretransitive (↥(toPermHom G α).range) α
⊢ (toPermHom G α).range = ⊤ | rw [MonoidHom.range_eq_map, ← hS2, MonoidHom.map_closure] at this ⊢ | G : Type u_1
α : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
inst✝¹ : DecidableEq α
inst✝ : Finite α
S : Set G
hS1 : ∀ σ ∈ S, ((toPermHom G α) σ).IsSwap
hS2 : closure S = ⊤
h : IsPretransitive G α
this : IsPretransitive (↥(closure (⇑(toPermHom G α) '' S))) α
⊢ closure (⇑(toPermHom G α) '' S) = ⊤ | 666fb2f6f05ce63a |
List.perm_lookmap | Mathlib/Data/List/Lookmap.lean | theorem perm_lookmap (f : α → Option α) {l₁ l₂ : List α}
(H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁) (p : l₁ ~ l₂) :
lookmap f l₁ ~ lookmap f l₂ | α : Type u_1
f : α → Option α
l₁ l₂ : List α
H : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) l₁
p : l₁ ~ l₂
⊢ lookmap f l₁ ~ lookmap f l₂ | induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ p₁ _ IH₁ IH₂ | case nil
α : Type u_1
f : α → Option α
l₁ l₂ : List α
H : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) []
⊢ lookmap f [] ~ lookmap f []
case cons
α : Type u_1
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) (a :: l₁)
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)
case swap
α : Type u_1
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) (b :: a :: l)
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)
case trans
α : Type u_1
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ (c : α), c ∈ f a → ∀ (d : α), d ∈ f b → a = b ∧ c = d) l₁
⊢ lookmap f l₁ ~ lookmap f l₃ | 50ba40462e6c2ee0 |
image_subset_closure_compl_image_compl_of_isOpen | Mathlib/Topology/ExtremallyDisconnected.lean | /-- Lemma 2.1 in [Gleason, *Projective topological spaces*][gleason1958]:
if $\rho$ is a continuous surjection from a topological space $E$ to a topological space $A$
satisfying the "Zorn subset condition", then $\rho(G)$ is contained in
the closure of $A \setminus \rho(E \setminus G)$ for any open set $G$ of $E$. -/
lemma image_subset_closure_compl_image_compl_of_isOpen {ρ : E → A} (ρ_cont : Continuous ρ)
(ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ)
{G : Set E} (hG : IsOpen G) : ρ '' G ⊆ closure ((ρ '' Gᶜ)ᶜ) | case neg
A E : Type u
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
N : Set A
N_open : IsOpen N
hN : a ∈ N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty | rcases (G.mem_image ρ a).mp ha with ⟨e, he, rfl⟩ | case neg.intro.intro
A E : Type u
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty | 7e7582b54cb8739a |
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 | case refine_1
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 ⊔ Cardinal.lift.{v, u} #ι < c.cof | apply max_lt _ hc' | 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 | 1ab519ba3d97b754 |
List.erase_eq_eraseIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem erase_eq_eraseIdx (l : List α) (a : α) :
l.erase a = match l.idxOf? a with
| none => l
| some i => l.eraseIdx i | case cons
α : Type u_1
inst✝ : BEq α
a x : α
xs : List α
ih :
xs.erase a =
match idxOf? a xs with
| none => xs
| some i => xs.eraseIdx i
⊢ (if (x == a) = true then xs else x :: xs.erase a) =
match if (x == a) = true then some 0 else Option.map (fun x => x + 1) (idxOf? a xs) with
| none => x :: xs
| some i => (x :: xs).eraseIdx i | split | case cons.isTrue
α : Type u_1
inst✝ : BEq α
a x : α
xs : List α
ih :
xs.erase a =
match idxOf? a xs with
| none => xs
| some i => xs.eraseIdx i
h✝ : (x == a) = true
⊢ xs =
match some 0 with
| none => x :: xs
| some i => (x :: xs).eraseIdx i
case cons.isFalse
α : Type u_1
inst✝ : BEq α
a x : α
xs : List α
ih :
xs.erase a =
match idxOf? a xs with
| none => xs
| some i => xs.eraseIdx i
h✝ : ¬(x == a) = true
⊢ x :: xs.erase a =
match Option.map (fun x => x + 1) (idxOf? a xs) with
| none => x :: xs
| some i => (x :: xs).eraseIdx i | 75458e5765fb9afd |
Std.Tactic.BVDecide.LRAT.Internal.DefaultClause.contains_iff | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Clause.lean | theorem contains_iff :
∀ (c : DefaultClause n) (l : Literal (PosFin n)), contains c l = true ↔ l ∈ toList c | n : Nat
c : DefaultClause n
l : Literal (PosFin n)
⊢ c.contains l = true ↔ l ∈ c.toList | simp only [contains, List.contains] | n : Nat
c : DefaultClause n
l : Literal (PosFin n)
⊢ List.elem l c.clause = true ↔ l ∈ c.toList | 9a231d639ed11448 |
VectorField.DifferentiableWithinAt.pullbackWithin | Mathlib/Analysis/Calculus/VectorField.lean | lemma DifferentiableWithinAt.pullbackWithin {f : E → F} {V : F → F} {s : Set E} {t : Set F} {x : E}
(hV : DifferentiableWithinAt 𝕜 V t (f x))
(hf : ContDiffWithinAt 𝕜 2 f s x) (hf' : (fderivWithin 𝕜 f s x).IsInvertible)
(hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (hst : MapsTo f s t) :
DifferentiableWithinAt 𝕜 (pullbackWithin 𝕜 f V s) s x | case hc
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
f : E → F
V : F → F
s : Set E
t : Set F
x : E
hV : DifferentiableWithinAt 𝕜 V t (f x)
hf : ContDiffWithinAt 𝕜 2 f s x
hf' : (fderivWithin 𝕜 f s x).IsInvertible
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
hst : MapsTo f s t
M : E → E ≃L[𝕜] F
M_symm_smooth : ContDiffWithinAt 𝕜 1 (fun y => ↑(M y).symm) s x
hM : ∀ᶠ (y : E) in 𝓝[s] x, ↑(M y) = fderivWithin 𝕜 f s y
⊢ DifferentiableWithinAt 𝕜 (fun y => ↑(M y).symm) s x | exact M_symm_smooth.differentiableWithinAt le_rfl | no goals | 9464e97a9d907f07 |
Array.zipWith_comm_of_comm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Zip.lean | theorem zipWith_comm_of_comm (f : α → α → β) (comm : ∀ x y : α, f x y = f y x) (l l' : Array α) :
zipWith f l l' = zipWith f l' l | α : Type u_1
β : Type u_2
f : α → α → β
comm : ∀ (x y : α), f x y = f y x
l l' : Array α
⊢ zipWith (fun b a => f a b) l' l = zipWith f l' l | simp only [comm] | no goals | cdf1c835b4c70ca1 |
Ideal.finprod_count | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem finprod_count (I : Ideal R) (hI : I ≠ 0) : (Associates.mk v.asIdeal).count
(Associates.mk (∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I)).factors =
(Associates.mk v.asIdeal).count (Associates.mk I).factors | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠
0
hv : Irreducible (Associates.mk v.asIdeal)
h_dvd :
Associates.mk (v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ∣
Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)
h_not_dvd :
¬Associates.mk (v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1)) ∣
Associates.mk
(∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)
⊢ (Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
(Associates.mk v.asIdeal).count (Associates.mk I).factors | simp only [Associates.dvd_eq_le] at h_dvd h_not_dvd | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠
0
hv : Irreducible (Associates.mk v.asIdeal)
h_dvd :
Associates.mk (v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≤
Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)
h_not_dvd :
¬Associates.mk (v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1)) ≤
Associates.mk
(∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)
⊢ (Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
(Associates.mk v.asIdeal).count (Associates.mk I).factors | 3755a7f07538b12d |
Nat.pos_of_lt_mul_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem pos_of_lt_mul_right {a b c : Nat} (h : a < b * c) : 0 < b | a b c : Nat
h : a < b * c
⊢ 0 < b * c | omega | no goals | d9ecb2226b6aa123 |
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) | f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
⊢ 1 < 2 | norm_num | no goals | 69674f7264744f71 |
Relation.reflTransGen_iff_eq | Mathlib/Logic/Relation.lean | theorem reflTransGen_iff_eq (h : ∀ b, ¬r a b) : ReflTransGen r a b ↔ b = a | α : Type u_1
r : α → α → Prop
a b : α
h : ∀ (b : α), ¬r a b
⊢ (a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b) ↔ b = a | simp [h, eq_comm] | no goals | 9fd6a0ab909b1619 |
Int.sum_div | Mathlib/Algebra/BigOperators/Ring/Finset.lean | protected lemma sum_div (hf : ∀ i ∈ s, n ∣ f i) : (∑ i ∈ s, f i) / n = ∑ i ∈ s, f i / n | case inl
ι : Type u_7
s : Finset ι
f : ι → ℤ
hf : ∀ i ∈ s, 0 ∣ f i
⊢ (∑ i ∈ s, f i) / 0 = ∑ i ∈ s, f i / 0 | simp | no goals | 82058ce44f02b8ca |
Finset.Iic_erase | Mathlib/Order/Interval/Finset/Basic.lean | theorem Iic_erase [DecidableEq α] (b : α) : (Iic b).erase b = Iio b | α : Type u_2
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrderBot α
inst✝ : DecidableEq α
b : α
⊢ (Iic b).erase b = Iio b | ext | case h
α : Type u_2
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrderBot α
inst✝ : DecidableEq α
b a✝ : α
⊢ a✝ ∈ (Iic b).erase b ↔ a✝ ∈ Iio b | b3668786661fd204 |
CategoryTheory.Functor.isZero_rightDerived_obj_injective_succ | Mathlib/CategoryTheory/Abelian/RightDerived.lean | /-- The higher derived functors vanish on injective objects. -/
lemma Functor.isZero_rightDerived_obj_injective_succ
(F : C ⥤ D) [F.Additive] (n : ℕ) (X : C) [Injective X] :
IsZero ((F.rightDerived (n+1)).obj X) | C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
n : ℕ
X : C
inst✝ : Injective X
⊢ IsZero
((HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) (n + 1)).obj
((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (InjectiveResolution.self X).cocomplex)) | erw [← HomologicalComplex.exactAt_iff_isZero_homology] | C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
n : ℕ
X : C
inst✝ : Injective X
⊢ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (InjectiveResolution.self X).cocomplex).ExactAt (n + 1) | 865b83e734765103 |
MeasureTheory.SimpleFunc.lintegral_restrict_iUnion_of_directed | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem lintegral_restrict_iUnion_of_directed {ι : Type*} [Countable ι]
(f : α →ₛ ℝ≥0∞) {s : ι → Set α} (hd : Directed (· ⊆ ·) s) (μ : Measure α) :
f.lintegral (μ.restrict (⋃ i, s i)) = ⨆ i, f.lintegral (μ.restrict (s i)) | α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 => x1 ⊆ x2) s
μ : Measure α
⊢ ∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (⇑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (⇑f ⁻¹' {x}) | refine finsetSum_iSup fun i j ↦ (hd i j).imp fun k ⟨hik, hjk⟩ ↦ fun a ↦ ?_ | α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 => x1 ⊆ x2) s
μ : Measure α
i j k : ι
x✝ : (fun x1 x2 => x1 ⊆ x2) (s i) (s k) ∧ (fun x1 x2 => x1 ⊆ x2) (s j) (s k)
hik : (fun x1 x2 => x1 ⊆ x2) (s i) (s k)
hjk : (fun x1 x2 => x1 ⊆ x2) (s j) (s k)
a : ℝ≥0∞
⊢ a * (μ.restrict (s i)) (⇑f ⁻¹' {a}) ≤ a * (μ.restrict (s k)) (⇑f ⁻¹' {a}) ∧
a * (μ.restrict (s j)) (⇑f ⁻¹' {a}) ≤ a * (μ.restrict (s k)) (⇑f ⁻¹' {a}) | 271c53c8279667c1 |
Filter.mem_prod_iff_right | Mathlib/Order/Filter/Prod.lean | theorem mem_prod_iff_right {s : Set (α × β)} :
s ∈ f ×ˢ g ↔ ∃ t ∈ g, ∀ᶠ x in f, ∀ y ∈ t, (x, y) ∈ s | α : Type u_1
β : Type u_2
f : Filter α
g : Filter β
s : Set (α × β)
⊢ s ∈ f ×ˢ g ↔ ∃ t ∈ g, ∀ᶠ (x : α) in f, ∀ y ∈ t, (x, y) ∈ s | rw [prod_comm, mem_map, mem_prod_iff_left] | α : Type u_1
β : Type u_2
f : Filter α
g : Filter β
s : Set (α × β)
⊢ (∃ t ∈ g, ∀ᶠ (y : α) in f, ∀ x ∈ t, (x, y) ∈ (fun p => (p.2, p.1)) ⁻¹' s) ↔
∃ t ∈ g, ∀ᶠ (x : α) in f, ∀ y ∈ t, (x, y) ∈ s | 82d8167a05c091e6 |
Finset.mem_finsupp_iff | Mathlib/Data/Finset/Finsupp.lean | theorem mem_finsupp_iff {t : ι → Finset α} :
f ∈ s.finsupp t ↔ f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i | case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f | refine fun h => ⟨fun i _ => f i, mem_pi.2 h.2, ?_⟩ | case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι → Finset α
h : f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i
⊢ ({ toFun := indicator s, inj' := ⋯ } fun i x => f i) = f | d85aaa6cf73479f2 |
tendstoUniformly_tsum_of_cofinite_eventually | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | theorem tendstoUniformly_tsum_of_cofinite_eventually {ι : Type*} {f : ι → β → F} {u : ι → ℝ}
(hu : Summable u) (hfu : ∀ᶠ (n : ι) in cofinite, ∀ x : β, ‖f n x‖ ≤ u n) :
TendstoUniformly (fun t x => ∑ n ∈ t, f n x) (fun x => ∑' n, f n x) atTop | β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
ι : Type u_4
f : ι → β → F
u : ι → ℝ
hu : Summable u
hfu : ∀ᶠ (n : ι) in cofinite, ∀ (x : β), ‖f n x‖ ≤ u n
⊢ ∀ᶠ (n : ι) in cofinite, ∀ x ∈ univ, ‖f n x‖ ≤ u n | simpa using hfu | no goals | 9e3b2f25945fc21b |
Int.gcd_one | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Gcd.lean | theorem gcd_one {a : Int} : gcd a 1 = 1 | a : Int
⊢ a.gcd 1 = 1 | simp [gcd] | no goals | 543a072c1db4fa9b |
Order.height_enat | Mathlib/Order/KrullDimension.lean | @[simp]
lemma height_enat (n : ℕ∞) : height n = n | n : ℕ∞
⊢ height n = n | cases n with
| top => simp only [← WithBot.coe_eq_coe, height_top_eq_krullDim, krullDim_enat, WithBot.coe_top]
| coe n => exact (height_coe_withTop _).trans (height_nat _) | no goals | 76c06ba17cf6871f |
Matrix.vecAlt0_vecAppend | Mathlib/Data/Fin/VecNotation.lean | theorem vecAlt0_vecAppend (v : Fin n → α) :
vecAlt0 rfl (vecAppend rfl v v) = v ∘ (fun n ↦ n + n) | α : Type u
n : ℕ
v : Fin n → α
⊢ vecAlt0 ⋯ (vecAppend ⋯ v v) = v ∘ fun n_1 => n_1 + n_1 | ext i | case h
α : Type u
n : ℕ
v : Fin n → α
i : Fin n
⊢ vecAlt0 ⋯ (vecAppend ⋯ v v) i = (v ∘ fun n_1 => n_1 + n_1) i | 4c85d9edab2b8fd9 |
List.not_of_mem_foldl_argAux | Mathlib/Data/List/MinMax.lean | theorem not_of_mem_foldl_argAux (hr₀ : Irreflexive r) (hr₁ : Transitive r) :
∀ {a m : α} {o : Option α}, a ∈ l → m ∈ foldl (argAux r) o l → ¬r a m | case append_singleton.some
α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b m : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
ho : (if r a c then some a else some c) = some m
hf : foldl (argAux r) o tl = some c
⊢ ¬r b m | split_ifs at ho with hac <;> rcases mem_append.1 hb with h | h <;>
injection ho with ho <;> subst ho | case pos.inl
α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : r a c
h : b ∈ tl
⊢ ¬r b a
case pos.inr
α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : r a c
h : b ∈ [a]
⊢ ¬r b a
case neg.inl
α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : ¬r a c
h : b ∈ tl
⊢ ¬r b c
case neg.inr
α : Type u_1
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : ¬r a c
h : b ∈ [a]
⊢ ¬r b c | fd32a85061587928 |
Ideal.primeHeight_add_one_le_of_lt | Mathlib/RingTheory/Ideal/Height.lean | lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J) :
I.primeHeight + 1 ≤ J.primeHeight | R : Type u_1
inst✝² : CommRing R
I J : Ideal R
inst✝¹ : I.IsPrime
inst✝ : J.IsPrime
h : I < J
⊢ I.primeHeight + 1 ≤ J.primeHeight | unfold primeHeight | R : Type u_1
inst✝² : CommRing R
I J : Ideal R
inst✝¹ : I.IsPrime
inst✝ : J.IsPrime
h : I < J
⊢ Order.height { asIdeal := I, isPrime := inst✝¹ } + 1 ≤ Order.height { asIdeal := J, isPrime := inst✝ } | bbaacd39c8a6cbfe |
Equiv.Perm.IsCycle.isConj_iff | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.isConj_iff (hσ : IsCycle σ) (hτ : IsCycle τ) :
IsConj σ τ ↔ #σ.support = #τ.support where
mp h | case intro.refine_1
α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ : σ.IsCycle
π : Perm α
hτ : (π * σ * π⁻¹).IsCycle
h : IsConj σ (π * σ * π⁻¹)
x✝ : α
ha : x✝ ∈ σ.support
⊢ (fun a x => π a) x✝ ha ∈ (π * σ * π⁻¹).support | simp [mem_support.1 ha] | no goals | d8d050eb8efa4f7b |
AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom | Mathlib/AlgebraicGeometry/Scheme.lean | theorem Scheme.Spec_map_presheaf_map_eqToHom {X : Scheme} {U V : X.Opens} (h : U = V) (W) :
(Spec.map (X.presheaf.map (eqToHom h).op)).app W = eqToHom (by cases h; dsimp; simp) | case refl
X : Scheme
U : X.Opens
this : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 (Scheme.Spec.obj (op Γ(X, U)))
W : (Spec Γ(X, U)).Opens
⊢ Hom.app (Spec.map (X.presheaf.map (eqToHom ⋯).op)) W = eqToHom ⋯ | refine (Scheme.congr_app this _).trans ?_ | case refl
X : Scheme
U : X.Opens
this : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 (Scheme.Spec.obj (op Γ(X, U)))
W : (Spec Γ(X, U)).Opens
⊢ Hom.app (𝟙 (Scheme.Spec.obj (op Γ(X, U)))) W ≫ (Scheme.Spec.obj (op Γ(X, U))).presheaf.map (eqToHom ⋯).op = eqToHom ⋯ | 6819d9d83e818d30 |
List.mem_inits | Mathlib/Data/List/Infix.lean | theorem mem_inits : ∀ s t : List α, s ∈ inits t ↔ s <+: t
| s, [] =>
suffices s = nil ↔ s <+: nil by simpa only [inits, mem_singleton]
⟨fun h => h.symm ▸ prefix_rfl, eq_nil_of_prefix_nil⟩
| s, a :: t =>
suffices (s = nil ∨ ∃ l ∈ inits t, a :: l = s) ↔ s <+: a :: t by simpa
⟨fun o =>
match s, o with
| _, Or.inl rfl => ⟨_, rfl⟩
| s, Or.inr ⟨r, hr, hs⟩ => by
let ⟨s, ht⟩ := (mem_inits _ _).1 hr
rw [← hs, ← ht]; exact ⟨s, rfl⟩,
fun mi =>
match s, mi with
| [], ⟨_, rfl⟩ => Or.inl rfl
| b :: s, ⟨r, hr⟩ =>
(List.noConfusion hr) fun ba (st : s ++ r = t) =>
Or.inr <| by rw [ba]; exact ⟨_, (mem_inits _ _).2 ⟨_, st⟩, rfl⟩⟩
| α : Type u_1
s✝ : List α
a : α
t : List α
mi : s✝ <+: a :: t
b : α
s r : List α
hr : b :: s ++ r = a :: t
ba : b = a
st : s ++ r = t
⊢ ∃ l, l ∈ t.inits ∧ a :: l = a :: s | exact ⟨_, (mem_inits _ _).2 ⟨_, st⟩, rfl⟩ | no goals | bd51da11fa244872 |
affineCombination_mem_affineSpan | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem affineCombination_mem_affineSpan [Nontrivial k] {s : Finset ι} {w : ι → k}
(h : ∑ i ∈ s, w i = 1) (p : ι → P) :
s.affineCombination k p w ∈ affineSpan k (Set.range p) | 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
s : Finset ι
w : ι → k
h : ∑ i ∈ s, w i = 1
p : ι → P
hnz : ∑ i ∈ s, w i ≠ 0
i1 : ι
hi1 : i1 ∈ s
w1 : ι → k := Function.update (Function.const ι 0) i1 1
hw1 : ∑ i ∈ s, w1 i = 1
hw1s : (Finset.affineCombination k s p) w1 = p i1
⊢ (Finset.affineCombination k s p) w -ᵥ p i1 ∈ (affineSpan k (Set.range p)).direction | rw [direction_affineSpan, ← hw1s, Finset.affineCombination_vsub] | 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
s : Finset ι
w : ι → k
h : ∑ i ∈ s, w i = 1
p : ι → P
hnz : ∑ i ∈ s, w i ≠ 0
i1 : ι
hi1 : i1 ∈ s
w1 : ι → k := Function.update (Function.const ι 0) i1 1
hw1 : ∑ i ∈ s, w1 i = 1
hw1s : (Finset.affineCombination k s p) w1 = p i1
⊢ (s.weightedVSub p) (w - w1) ∈ vectorSpan k (Set.range p) | b0078337a6cd7107 |
Filter.mul_top_of_one_le | Mathlib/Order/Filter/Pointwise.lean | theorem mul_top_of_one_le (hf : 1 ≤ f) : f * ⊤ = ⊤ | α : Type u_2
inst✝ : Monoid α
f : Filter α
hf : 1 ≤ f
⊢ f * ⊤ = ⊤ | refine top_le_iff.1 fun s => ?_ | α : Type u_2
inst✝ : Monoid α
f : Filter α
hf : 1 ≤ f
s : Set α
⊢ s ∈ f * ⊤ → s ∈ ⊤ | 9240cc9c89569e01 |
PowerSeries.degree_trunc_lt | Mathlib/RingTheory/PowerSeries/Trunc.lean | theorem degree_trunc_lt (f : R⟦X⟧) (n) : (trunc n f).degree < n | case neg
R : Type u_1
inst✝ : Semiring R
f : R⟦X⟧
n m✝ : ℕ
a✝ : n ≤ m✝
h : ¬m✝ < n
⊢ 0 = 0 | rfl | no goals | 3a93048cc0d1e880 |
Nat.findGreatest_eq_iff | Mathlib/Data/Nat/Find.lean | lemma findGreatest_eq_iff :
Nat.findGreatest P k = m ↔ m ≤ k ∧ (m ≠ 0 → P m) ∧ ∀ ⦃n⦄, m < n → n ≤ k → ¬P n | P : ℕ → Prop
inst✝ : DecidablePred P
k : ℕ
ihk : ∀ {m : ℕ}, findGreatest P k = m ↔ m ≤ k ∧ (m ≠ 0 → P m) ∧ ∀ ⦃n : ℕ⦄, m < n → n ≤ k → ¬P n
hk : P (k + 1)
n : ℕ
hlt : k + 1 < n
hle : n ≤ k + 1
⊢ ¬P n | omega | no goals | 9905ebac396451ee |
Transcendental.of_aeval | Mathlib/RingTheory/Algebraic/Basic.lean | theorem Transcendental.of_aeval {r : A} {f : R[X]}
(H : Transcendental R (Polynomial.aeval r f)) : Transcendental R f | R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : A
f : R[X]
H : ∀ (p : R[X]), (Polynomial.aeval ((Polynomial.aeval r) f)) p = 0 → p = 0
⊢ ∀ (p : R[X]), (Polynomial.aeval f) p = 0 → p = 0 | intro p hp | R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : A
f : R[X]
H : ∀ (p : R[X]), (Polynomial.aeval ((Polynomial.aeval r) f)) p = 0 → p = 0
p : R[X]
hp : (Polynomial.aeval f) p = 0
⊢ p = 0 | f95d2ba8e4b6f764 |
List.getElem?_zip_eq_some | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean | theorem getElem?_zip_eq_some {l₁ : List α} {l₂ : List β} {z : α × β} {i : Nat} :
(zip l₁ l₂)[i]? = some z ↔ l₁[i]? = some z.1 ∧ l₂[i]? = some z.2 | case mk
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
i : Nat
fst✝ : α
snd✝ : β
⊢ (l₁.zip l₂)[i]? = some (fst✝, snd✝) ↔ l₁[i]? = some (fst✝, snd✝).fst ∧ l₂[i]? = some (fst✝, snd✝).snd | rw [zip, getElem?_zipWith_eq_some] | case mk
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
i : Nat
fst✝ : α
snd✝ : β
⊢ (∃ x y, l₁[i]? = some x ∧ l₂[i]? = some y ∧ (x, y) = (fst✝, snd✝)) ↔
l₁[i]? = some (fst✝, snd✝).fst ∧ l₂[i]? = some (fst✝, snd✝).snd | ff8f01ddc18035f4 |
ContinuousOn.if | Mathlib/Topology/ContinuousOn.lean | theorem ContinuousOn.if {p : α → Prop} [∀ a, Decidable (p a)]
(hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
(hf : ContinuousOn f <| s ∩ closure { a | p a })
(hg : ContinuousOn g <| s ∩ closure { a | ¬p a }) :
ContinuousOn (fun a => if p a then f a else g a) s | case hpf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s : Set α
p : α → Prop
inst✝ : (a : α) → Decidable (p a)
hp : ∀ a ∈ s ∩ frontier {a | p a}, f a = g a
hf : ContinuousOn f (s ∩ closure {a | p a})
hg : ContinuousOn g (s ∩ closure {a | ¬p a})
a : α
ha : a ∈ s ∩ frontier {a | p a}
⊢ Tendsto f (𝓝[s ∩ closure {a | p a}] a) (𝓝 (f a)) | exact hf a ⟨ha.1, ha.2.1⟩ | no goals | 342fb5057472ed38 |
Int.neg_add_emod_self | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem neg_add_emod_self (a b : Int) : (-a + b) % a = b % a | a b : Int
⊢ (-a + b) % a = b % a | rw [Int.add_comm, add_neg_emod_self] | no goals | fe3f1c2653fd2218 |
MeasureTheory.exists_not_mem_null_le_laverage | Mathlib/MeasureTheory/Integral/Average.lean | theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
∃ x, x ∉ N ∧ f x ≤ ⨍⁻ a, f a ∂μ | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : AEMeasurable f μ
hN : μ N = 0
this : 0 < μ ({x | f x ≤ ⨍⁻ (a : α), f a ∂μ} \ N)
⊢ ∃ x ∉ N, f x ≤ ⨍⁻ (a : α), f a ∂μ | obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne' | case intro.intro
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : AEMeasurable f μ
hN : μ N = 0
this : 0 < μ ({x | f x ≤ ⨍⁻ (a : α), f a ∂μ} \ N)
x : α
hx : x ∈ {x | f x ≤ ⨍⁻ (a : α), f a ∂μ}
hxN : x ∉ N
⊢ ∃ x ∉ N, f x ≤ ⨍⁻ (a : α), f a ∂μ | b5d2c1ec1f07f375 |
Orientation.oangle_smul_add_right_eq_zero_or_eq_pi_iff | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_smul_add_right_eq_zero_or_eq_pi_iff {x y : V} (r : ℝ) :
o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = π ↔
o.oangle x y = 0 ∨ o.oangle x y = π | case refine_1
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
h : ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0) | rcases h with ⟨m, h, hm⟩ | case refine_1.intro.intro
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
m : Fin (Nat.succ 0).succ → ℝ
h : m 0 • ![x, r • x + y] 0 + m 1 • ![x, r • x + y] 1 = 0
hm : m 0 ≠ 0 ∨ m 1 ≠ 0
⊢ ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0) | 8163cb9f535da18e |
MvPolynomial.quotient_mk_comp_C_isIntegral_of_isJacobsonRing | Mathlib/RingTheory/Jacobson/Ring.lean | theorem quotient_mk_comp_C_isIntegral_of_isJacobsonRing {R : Type*} [CommRing R] [IsJacobsonRing R]
(P : Ideal (MvPolynomial (Fin n) R)) [hP : P.IsMaximal] :
RingHom.IsIntegral (RingHom.comp (Ideal.Quotient.mk P) (MvPolynomial.C)) | n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
P : Ideal (MvPolynomial (Fin n) R)
hP : P.IsMaximal
⊢ (algebraMap R (MvPolynomial (Fin n) R ⧸ P)).IsIntegral | apply quotient_mk_comp_C_isIntegral_of_isJacobsonRing' | case hP
n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsJacobsonRing R
P : Ideal (MvPolynomial (Fin n) R)
hP : P.IsMaximal
⊢ P.IsMaximal | 18215d953b985915 |
ContinuousMap.polynomial_comp_attachBound_mem | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | theorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
f : ↥A
g : ℝ[X]
⊢ (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound ∈ A | rw [polynomial_comp_attachBound] | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
f : ↥A
g : ℝ[X]
⊢ ↑((Polynomial.aeval f) g) ∈ A | 6fc1f65f86aa5c1c |
CategoryTheory.Pretriangulated.isIso₂_of_isIso₁₃ | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | lemma isIso₂_of_isIso₁₃ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)
(hT' : T' ∈ distTriang C) (h₁ : IsIso φ.hom₁) (h₃ : IsIso φ.hom₃) : IsIso φ.hom₂ | case intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
h : A ⟶ T'.invRotate.obj₁
k : A ⟶ T.invRotate.obj₁
hf : (k ≫ T.invRotate.mor₁) ≫ T.mor₁ ≫ φ.hom₂ = 0
hh : (k ≫ T.invRotate.mor₁) ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
⊢ (k ≫ T.invRotate.mor₁) ≫ T.mor₁ = 0 | erw [assoc, comp_distTriang_mor_zero₁₂ _ (inv_rot_of_distTriang _ hT), comp_zero] | no goals | 8a538ff25cabfee4 |
HasStrictDerivAt.div | Mathlib/Analysis/Calculus/Deriv/Inv.lean | theorem HasStrictDerivAt.div (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x)
(hx : d x ≠ 0) : HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x | case h.e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') * (d x * d x ^ 2) = (c' * d x ^ 2 + -(c x * d' * d x)) * d x ^ 2 | ring | no goals | a430038d012aeadb |
MeasureTheory.setToFun_smul | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
(hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
(f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f | case pos
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hf : ¬Integrable f μ
hr : c = 0
⊢ setToFun μ T hT (0 • f) = 0 • setToFun μ T hT f | simp | no goals | c0e0423fc4390b6f |
Nat.diag_induction | Mathlib/Data/Nat/Init.lean | theorem diag_induction (P : ℕ → ℕ → Prop) (ha : ∀ a, P (a + 1) (a + 1)) (hb : ∀ b, P 0 (b + 1))
(hd : ∀ a b, a < b → P (a + 1) b → P a (b + 1) → P (a + 1) (b + 1)) : ∀ a b, a < b → P a b
| 0, _ + 1, _ => hb _
| a + 1, b + 1, h => by
apply hd _ _ (Nat.add_lt_add_iff_right.1 h)
· have this : a + 1 = b ∨ a + 1 < b | P : ℕ → ℕ → Prop
ha : ∀ (a : ℕ), P (a + 1) (a + 1)
hb : ∀ (b : ℕ), P 0 (b + 1)
hd : ∀ (a b : ℕ), a < b → P (a + 1) b → P a (b + 1) → P (a + 1) (b + 1)
a b : ℕ
h : a + 1 < b + 1
⊢ a + 1 = b ∨ a + 1 < b | omega | no goals | 42f141b2edc38622 |
Int.mul_eq_zero | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem mul_eq_zero {a b : Int} : a * b = 0 ↔ a = 0 ∨ b = 0 | a b : Int
h : a * b = 0
x✝¹ : Int
x✝ : x✝¹ * ofNat 0 = 0
⊢ x✝¹ = 0 ∨ ofNat 0 = 0 | simp | no goals | 6f0602ae658aec2d |
Batteries.UnionFind.parent_push | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | theorem parent_push {m : UnionFind} : m.push.parent a = m.parent a | a : Nat
m : UnionFind
⊢ m.push.parent a = m.parent a | simp [parent] | no goals | 3a7b5e17d1394d4f |
Multiset.rel_add_left | Mathlib/Data/Multiset/AddSub.lean | theorem rel_add_left {as₀ as₁} :
∀ {bs}, Rel r (as₀ + as₁) bs ↔ ∃ bs₀ bs₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁ :=
@(Multiset.induction_on as₀ (by simp) fun a s ih bs ↦ by
simp only [ih, cons_add, rel_cons_left]
constructor
· intro h
rcases h with ⟨b, bs', hab, h, rfl⟩
rcases h with ⟨bs₀, bs₁, h₀, h₁, rfl⟩
exact ⟨b ::ₘ bs₀, bs₁, ⟨b, bs₀, hab, h₀, rfl⟩, h₁, by simp⟩
· intro h
rcases h with ⟨bs₀, bs₁, h, h₁, rfl⟩
rcases h with ⟨b, bs, hab, h₀, rfl⟩
exact ⟨b, bs + bs₁, hab, ⟨bs, bs₁, h₀, h₁, rfl⟩, by simp⟩)
| case mp
α : Type u_1
β : Type v
r : α → β → Prop
as₀ as₁ : Multiset α
a : α
s : Multiset α
ih : ∀ {bs : Multiset β}, Rel r (s + as₁) bs ↔ ∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁
bs : Multiset β
h : ∃ b bs', r a b ∧ (∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs' = bs₀ + bs₁) ∧ bs = b ::ₘ bs'
⊢ ∃ bs₀ bs₁, (∃ b bs', r a b ∧ Rel r s bs' ∧ bs₀ = b ::ₘ bs') ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁ | rcases h with ⟨b, bs', hab, h, rfl⟩ | case mp.intro.intro.intro.intro
α : Type u_1
β : Type v
r : α → β → Prop
as₀ as₁ : Multiset α
a : α
s : Multiset α
ih : ∀ {bs : Multiset β}, Rel r (s + as₁) bs ↔ ∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁
b : β
bs' : Multiset β
hab : r a b
h : ∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs' = bs₀ + bs₁
⊢ ∃ bs₀ bs₁, (∃ b bs', r a b ∧ Rel r s bs' ∧ bs₀ = b ::ₘ bs') ∧ Rel r as₁ bs₁ ∧ b ::ₘ bs' = bs₀ + bs₁ | 528c1d60fea179d9 |
MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_eq_lintegral_pos_part_sub_lintegral_neg_part {f : α → ℝ} (hf : Integrable f μ) :
∫ a, f a ∂μ =
ENNReal.toReal (∫⁻ a, .ofReal (f a) ∂μ) - ENNReal.toReal (∫⁻ a, .ofReal (-f a) ∂μ) | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
eq₂ : (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal = ‖Lp.negPart f₁‖
⊢ ∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal - (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal | rw [eq₁, eq₂, integral, dif_pos, dif_pos] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
eq₂ : (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal = ‖Lp.negPart f₁‖
⊢ L1.integral (Integrable.toL1 (fun a => f a) ?hc) = ‖Lp.posPart f₁‖ - ‖Lp.negPart f₁‖
case hc
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
eq₂ : (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal = ‖Lp.negPart f₁‖
⊢ Integrable (fun a => f a) μ
case hc
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
f₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf
eq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖
eq₂ : (∫⁻ (a : α), ENNReal.ofReal (-f a) ∂μ).toReal = ‖Lp.negPart f₁‖
⊢ CompleteSpace ℝ | c38c71e9a9c235bc |
Matroid.Indep.insert_isBasis_iff_mem_closure | Mathlib/Data/Matroid/Closure.lean | lemma Indep.insert_isBasis_iff_mem_closure (hI : M.Indep I) :
M.IsBasis I (insert e I) ↔ e ∈ M.closure I | α : Type u_2
M : Matroid α
e : α
I : Set α
hI : M.Indep I
⊢ M.IsBasis I (insert e I) ↔ e ∈ M.closure I | rw [hI.closure_eq_setOf_isBasis_insert, mem_setOf] | no goals | fc1bbc2380a27a84 |
Valuation.map_sum_lt | Mathlib/RingTheory/Valuation/Basic.lean | theorem map_sum_lt {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ 0)
(hf : ∀ i ∈ s, v (f i) < g) : v (∑ i ∈ s, f i) < g | R : Type u_3
Γ₀ : Type u_4
inst✝¹ : Ring R
inst✝ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
ι : Type u_7
s✝ : Finset ι
f : ι → R
g : Γ₀
hg : g ≠ 0
hf✝ : ∀ i ∈ s✝, v (f i) < g
a : ι
s : Finset ι
has : a ∉ s
ih : (∀ i ∈ s, v (f i) < g) → v (∑ i ∈ s, f i) < g
hf : v (f a) < g ∧ ∀ x ∈ s, v (f x) < g
⊢ v (f a + ∑ x ∈ s, f x) < g | exact v.map_add_lt hf.1 (ih hf.2) | no goals | ecfbbabc73c096f9 |
ProbabilityTheory.mgf_sum_of_identDistrib | Mathlib/Probability/Moments/Basic.lean | theorem mgf_sum_of_identDistrib
{X : ι → Ω → ℝ}
{s : Finset ι} {j : ι}
(h_meas : ∀ i, Measurable (X i))
(h_indep : iIndepFun (fun _ => inferInstance) X μ)
(hident : ∀ i ∈ s, ∀ j ∈ s, IdentDistrib (X i) (X j) μ μ)
(hj : j ∈ s) (t : ℝ) : mgf (∑ i ∈ s, X i) μ t = mgf (X j) μ t ^ #s | Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ι → Ω → ℝ
s : Finset ι
j : ι
h_meas : ∀ (i : ι), Measurable (X i)
h_indep : iIndepFun (fun x => inferInstance) X μ
hident : ∀ i ∈ s, ∀ j ∈ s, IdentDistrib (X i) (X j) μ μ
hj : j ∈ s
t : ℝ
⊢ mgf (∑ i ∈ s, X i) μ t = mgf (X j) μ t ^ #s | rw [h_indep.mgf_sum h_meas] | Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ι → Ω → ℝ
s : Finset ι
j : ι
h_meas : ∀ (i : ι), Measurable (X i)
h_indep : iIndepFun (fun x => inferInstance) X μ
hident : ∀ i ∈ s, ∀ j ∈ s, IdentDistrib (X i) (X j) μ μ
hj : j ∈ s
t : ℝ
⊢ ∏ i ∈ s, mgf (X i) μ t = mgf (X j) μ t ^ #s | a251942609d925ba |
PMF.map_apply | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | theorem map_apply : (map f p) b = ∑' a, if b = f a then p a else 0 | α : Type u_1
β : Type u_2
f : α → β
p : PMF α
b : β
⊢ (map f p) b = ∑' (a : α), if b = f a then p a else 0 | simp [map] | no goals | 8eba2aee49430f17 |
measurableSet_of_differentiableAt_of_isComplete_with_param | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem measurableSet_of_differentiableAt_of_isComplete_with_param
(hf : Continuous f.uncurry) {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
MeasurableSet {p : α × E | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
f : α → E → F
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝ : ℕ
⊢ MeasurableSet
(⋃ i,
⋂ i_1,
⋂ (_ : i_1 ≥ i), ⋂ i_2, ⋂ (_ : i_2 ≥ i), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i_1) ((1 / 2) ^ i_2) ((1 / 2) ^ x✝)}) | refine MeasurableSet.iUnion (fun _ ↦ ?_) | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : LocallyCompactSpace E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
α : Type u_4
inst✝⁴ : TopologicalSpace α
f : α → E → F
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
hf : Continuous (Function.uncurry f)
K : Set (E →L[𝕜] F)
hK : IsComplete K
this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}
x✝¹ x✝ : ℕ
⊢ MeasurableSet
(⋂ i,
⋂ (_ : i ≥ x✝), ⋂ i_1, ⋂ (_ : i_1 ≥ x✝), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i) ((1 / 2) ^ i_1) ((1 / 2) ^ x✝¹)}) | 77d1b3ef6979d5fd |
Ordnode.Valid'.rotateL | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
(H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ | α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : rl.size < ratio * rr.size
rr0 : rr.size > 0
this : BalancedSz l.size rl.size ∧ BalancedSz (l.size + rl.size + 1) rr.size
⊢ Valid' o₁ (l.node3L x rl rx rr) o₂ | exact hl.node3L hr.left hr.right this.1 this.2 | no goals | 0ebe7132c42c2866 |
Nat.bitIndices_sorted | Mathlib/Data/Nat/BitIndices.lean | theorem bitIndices_sorted {n : ℕ} : n.bitIndices.Sorted (· < ·) | n : ℕ
⊢ Sorted (fun x1 x2 => x1 < x2) n.bitIndices | induction' n using binaryRec with b n hs | case z
⊢ Sorted (fun x1 x2 => x1 < x2) (bitIndices 0)
case f
b : Bool
n : ℕ
hs : Sorted (fun x1 x2 => x1 < x2) n.bitIndices
⊢ Sorted (fun x1 x2 => x1 < x2) (bit b n).bitIndices | af76496aedb3f73c |
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case h.a₃
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : E.a₄ = 0
ha₆ : E.a₆ ≠ 0
ha₄' : E'.a₄ = 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ (E.variableChange { u := Units.mk0 u hu0, r := 0, s := 0, t := 0 }).a₃ = E'.a₃ | simp | no goals | 5599f8482dc6841e |
BitVec.getLsbD_ge | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false | w : Nat
x : BitVec w
i : Nat
ge : w ≤ i
⊢ x.getLsbD i = false | let ⟨x, x_lt⟩ := x | w : Nat
x✝ : BitVec w
i : Nat
ge : w ≤ i
x : Nat
x_lt : x < 2 ^ w
⊢ { toFin := ⟨x, x_lt⟩ }.getLsbD i = false | 9350744387bd6450 |
ContDiffWithinAt.eventually | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffWithinAt.eventually (h : ContDiffWithinAt 𝕜 n f s x) (hn : n ≠ ∞) :
∀ᶠ y in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ∞
⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y | rcases h.contDiffOn le_rfl (by simp [hn]) with ⟨u, hu, _, hd⟩ | case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
left✝ : u ⊆ insert x s
hd : ContDiffOn 𝕜 n f u
⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y | 8e6eeba4d8f14914 |
CategoryTheory.GradedObject.mapBifunctor_triangle | Mathlib/CategoryTheory/GradedObject/Unitor.lean | lemma mapBifunctor_triangle
(triangle : ∀ (X₁ : C₁) (X₃ : C₃), ((associator.hom.app X₁).app X₂).app X₃ ≫
(G.obj X₁).map (e₂.hom.app X₃) = (G.map (e₁.hom.app X₁)).app X₃) :
(mapBifunctorAssociator associator τ.ρ₁₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃).hom ≫
mapBifunctorMapMap G π (𝟙 X₁) (mapBifunctorLeftUnitor F₂ X₂ e₂ τ.p₂₃ τ.h₃ X₃).hom =
mapBifunctorMapMap G π (mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ τ.h₁ X₁).hom (𝟙 X₃) | case h.h.e_a.e_a.e_a.e_a
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D : Type u_4
I₁ : Type u_5
I₂ : Type u_6
I₃ : Type u_7
J : Type u_8
inst✝¹⁵ : Category.{u_12, u_1} C₁
inst✝¹⁴ : Category.{u_11, u_2} C₂
inst✝¹³ : Category.{u_10, u_3} C₃
inst✝¹² : Category.{u_9, u_4} D
inst✝¹¹ : Zero I₂
inst✝¹⁰ : DecidableEq I₂
inst✝⁹ : HasInitial C₂
F₁ : C₁ ⥤ C₂ ⥤ C₁
F₂ : C₂ ⥤ C₃ ⥤ C₃
G : C₁ ⥤ C₃ ⥤ D
associator : bifunctorComp₁₂ F₁ G ≅ bifunctorComp₂₃ G F₂
X₂ : C₂
e₁ : F₁.flip.obj X₂ ≅ 𝟭 C₁
e₂ : F₂.obj X₂ ≅ 𝟭 C₃
inst✝⁸ : ∀ (X₁ : C₁), PreservesColimit (Functor.empty C₂) (F₁.obj X₁)
inst✝⁷ : ∀ (X₃ : C₃), PreservesColimit (Functor.empty C₂) (F₂.flip.obj X₃)
r : I₁ × I₂ × I₃ → J
π : I₁ × I₃ → J
τ : TriangleIndexData r π
X₁ : GradedObject I₁ C₁
X₃ : GradedObject I₃ C₃
inst✝⁶ : (((mapBifunctor F₁ I₁ I₂).obj X₁).obj ((single₀ I₂).obj X₂)).HasMap τ.p₁₂
inst✝⁵ : (((mapBifunctor G I₁ I₃).obj (mapBifunctorMapObj F₁ τ.p₁₂ X₁ ((single₀ I₂).obj X₂))).obj X₃).HasMap π
inst✝⁴ : (((mapBifunctor F₂ I₂ I₃).obj ((single₀ I₂).obj X₂)).obj X₃).HasMap τ.p₂₃
inst✝³ : (((mapBifunctor G I₁ I₃).obj X₁).obj (mapBifunctorMapObj F₂ τ.p₂₃ ((single₀ I₂).obj X₂) X₃)).HasMap π
inst✝² : HasGoodTrifunctor₁₂Obj F₁ G τ.ρ₁₂ X₁ ((single₀ I₂).obj X₂) X₃
inst✝¹ : HasGoodTrifunctor₂₃Obj G F₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃
inst✝ : (((mapBifunctor G I₁ I₃).obj X₁).obj X₃).HasMap π
triangle :
∀ (X₁ : C₁) (X₃ : C₃),
((associator.hom.app X₁).app X₂).app X₃ ≫ (G.obj X₁).map (e₂.hom.app X₃) = (G.map (e₁.hom.app X₁)).app X₃
j : J
i₁ : I₁
i₃ : I₃
hj : π (i₁, i₃) = j
⊢ (G.map ((F₁.obj (X₁ i₁)).map (singleObjApplyIso 0 X₂).hom)).app (X₃ i₃) ≫
((associator.hom.app (X₁ i₁)).app X₂).app (X₃ i₃) =
((associator.hom.app (X₁ i₁)).app ((single₀ I₂).obj X₂ 0)).app (X₃ i₃) ≫
(G.obj (X₁ i₁)).map ((F₂.map (singleObjApplyIso 0 X₂).hom).app (X₃ i₃)) | apply NatTrans.naturality_app (associator.hom.app (X₁ i₁)) | no goals | 59516bb5bca5f0f8 |
Complex.HadamardThreeLines.mem_verticalClosedStrip_of_scale_id_mem_verticalClosedStrip | Mathlib/Analysis/Complex/Hadamard.lean | /-- If z is on the closed strip `re ⁻¹' [l, u]`, then `(z - l) / (u - l)` is on the closed strip
`re ⁻¹' [0, 1]`. -/
lemma mem_verticalClosedStrip_of_scale_id_mem_verticalClosedStrip {z : ℂ} {l u : ℝ} (hul : l < u)
(hz : z ∈ verticalClosedStrip l u) : z / (u - l) - l / (u - l) ∈ verticalClosedStrip 0 1 | z : ℂ
l u : ℝ
hul : l < u
hz : z ∈ re ⁻¹' Icc l u
⊢ u - l ≠ 0 | linarith | no goals | 1ee8b1bc34ef4d29 |
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
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 n)) (ys 0) (List.range n)
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop | have hsucc (n : ℕ) : xs (n + 1) = c (xs n) (ys n) := by simp [xs, List.range_succ] | case intro
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 n)) (ys 0) (List.range n)
hsucc : ∀ (n : ℕ), xs (n + 1) = c (xs n) (ys n)
⊢ ∃ xs, Monotone xs ∧ Tendsto xs atTop atTop | cfd85fba236c098c |
BoxIntegral.integralSum_neg | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem integralSum_neg (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (-f) vol π = -integralSum f vol π | ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π : TaggedPrepartition I
⊢ integralSum (-f) vol π = -integralSum f vol π | simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib] | no goals | 25debb8c308b0182 |
tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
(hf2 : HasCompactSupport f) :
Tendsto (fun w : V => ∫ v : V, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) | case intro.intro.intro.intro.intro.intro.refine_1.hf
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B : ℝ≥0
hB_pos : 0 < B
hB_vol : volume A ≤ ↑B
δ : ℝ
hδ1 : δ > 0
hδ2 : ∀ ⦃a b : V⦄, dist a b < δ → dist (f a) (f b) < ε / ↑B
w : V
hw_bd : 1 / 2 + 1 / (2 * δ) ≤ ‖w‖
hw_ne : w ≠ 0
hw'_nm : ‖i w‖ = 1 / (2 * ‖w‖)
this : ‖1 / 2‖ = 2⁻¹
bdA : ∀ v ∈ A, ‖‖f v - f (v + i w)‖‖ ≤ ε / ↑B
⊢ Continuous fun x => ‖f x - f (x + i w)‖ | exact continuous_norm.comp <| hf1.sub <| hf1.comp <| continuous_id'.add continuous_const | no goals | e1b4f22b1d3b4e6d |
Vector.take_mk | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem take_mk (a : Array α) (h : a.size = n) (m) :
(Vector.mk a h).take m = Vector.mk (a.take m) (by simp [h]) := rfl
| α : Type ?u.17462
n : Nat
a : Array α
h : a.size = n
m : Nat
⊢ (a.take m).size = min m n | simp [h] | no goals | 1572c49faa65a5c2 |
eventually_residual_liouville | Mathlib/NumberTheory/Transcendental/Liouville/Residual.lean | theorem eventually_residual_liouville : ∀ᶠ x in residual ℝ, Liouville x | case h.e'_4.h.e'_3
r : ℚ
n : ℕ
⊢ ↑r.num * 2 / (↑r.den * 2) = ↑r | norm_cast | case h.e'_4.h.e'_3
r : ℚ
n : ℕ
⊢ Rat.divInt (r.num * 2) ↑(r.den * 2) = r | e53ba645462ad542 |
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