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 |
|---|---|---|---|---|---|---|
Ideal.quotientInfToPiQuotient_surj | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | lemma quotientInfToPiQuotient_surj {I : ι → Ideal R}
(hI : Pairwise (IsCoprime on I)) : Surjective (quotientInfToPiQuotient I) | case h.h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) =... | rw [quotientInfToPiQuotient_mk', map_sum, Fintype.sum_eq_single i] | case h.h
R : Type u_2
inst✝¹ : CommRing R
ι : Type u_3
inst✝ : Finite ι
I : ι → Ideal R
hI : Pairwise (IsCoprime on I)
val✝ : Fintype ι
g : (i : ι) → R ⧸ I i
f : ι → R
hf : ∀ (i : ι), (Quotient.mk (I i)) (f i) = g i
e : ι → R
he : ∀ (i : ι), (Quotient.mk (I i)) (e i) = 1 ∧ ∀ (j : ι), j ≠ i → (Quotient.mk (I j)) (e i) =... | df184cf9a63521b9 |
Order.isLUB_Iio_iff_isSuccPrelimit | Mathlib/Order/SuccPred/Limit.lean | theorem isLUB_Iio_iff_isSuccPrelimit : IsLUB (Iio a) a ↔ IsSuccPrelimit a | α : Type u_1
inst✝ : LinearOrder α
b : α
ha : IsLUB (Iic b) b
hb : b ⋖ b
⊢ False | cases hb.lt.false | no goals | 789b5fa9a40f35ce |
CoxeterSystem.length_alternatingWord | Mathlib/GroupTheory/Coxeter/Basic.lean | theorem length_alternatingWord (i i' : B) (m : ℕ) :
List.length (alternatingWord i i' m) = m | B : Type u_1
i i' : B
m : ℕ
⊢ (alternatingWord i i' m).length = m | induction' m with m ih generalizing i i' | case zero
B : Type u_1
i i' : B
⊢ (alternatingWord i i' 0).length = 0
case succ
B : Type u_1
m : ℕ
ih : ∀ (i i' : B), (alternatingWord i i' m).length = m
i i' : B
⊢ (alternatingWord i i' (m + 1)).length = m + 1 | cca0d9605e2516f0 |
LinearMap.exists_leftInverse_of_injective | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | theorem LinearMap.exists_leftInverse_of_injective (f : V →ₗ[K] V') (hf_inj : LinearMap.ker f = ⊥) :
∃ g : V' →ₗ[K] V, g.comp f = LinearMap.id | K : Type u_3
V : Type u_4
V' : Type u_5
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
f : V →ₗ[K] V'
hf_inj : ker f = ⊥
B : Set V := Basis.ofVectorSpaceIndex K V
hB : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V
hB₀ : LinearInd... | exact leftInverse_invFun (LinearMap.ker_eq_bot.1 hf_inj) _ | no goals | 3d5853a3c9ced8e8 |
Nat.image_Ico_mod | Mathlib/Order/Interval/Finset/Nat.lean | theorem image_Ico_mod (n a : ℕ) : (Ico n (n + a)).image (· % a) = range a | case inr.h.mpr.inl.refine_1
n a : ℕ
ha : a ≠ 0
i : ℕ
hia : i < a
hn : n % a + a * (n / a) = n
hi : i < n % a
⊢ n ≤ i + a * (n / a + 1) | rw [add_comm (n / a), Nat.mul_add, mul_one, ← add_assoc] | case inr.h.mpr.inl.refine_1
n a : ℕ
ha : a ≠ 0
i : ℕ
hia : i < a
hn : n % a + a * (n / a) = n
hi : i < n % a
⊢ n ≤ i + a + a * (n / a) | 20b926d60f75b07c |
Multiset.transGen_oneStep_of_isDershowitzMannaLT | Mathlib/Data/Multiset/DershowitzManna.lean | private lemma transGen_oneStep_of_isDershowitzMannaLT :
IsDershowitzMannaLT M N → TransGen OneStep M N | α : Type u_1
inst✝ : Preorder α
z : α
M N X Y : Multiset α
hM : M = X + Y
ih :
∀ {M N : Multiset α} (X Y : Multiset α),
0 ≠ ∅ → M = X + Y → N = X + 0 → (∀ y ∈ Y, ∃ z ∈ 0, y < z) → TransGen Multiset.OneStep M N
hZ : z ::ₘ 0 ≠ ∅
hN : N = X + z ::ₘ 0
hYZ : ∀ y ∈ Y, ∃ z_1 ∈ z ::ₘ 0, y < z_1
⊢ ∀ y ∈ Y, y < z | simpa using hYZ | no goals | dd05ea058a96ae5b |
ProbabilityTheory.iIndepSet_iff_meas_biInter | Mathlib/Probability/Independence/Basic.lean | theorem iIndepSet_iff_meas_biInter {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
iIndepSet f μ ↔ ∀ s, μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) | Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : ι → Set Ω
hf : ∀ (i : ι), MeasurableSet (f i)
⊢ iIndepSet f μ ↔ ∀ (s : Finset ι), μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) | simp_rw [iIndepSet, Kernel.iIndepSet_iff_meas_biInter hf, ae_dirac_eq, Filter.eventually_pure,
Kernel.const_apply] | no goals | fb306e4d8a20aea5 |
Polynomial.coeff_preHilbertPoly_self | Mathlib/RingTheory/Polynomial/HilbertPoly.lean | lemma coeff_preHilbertPoly_self [CharZero F] (d k : ℕ) :
(preHilbertPoly F d k).coeff d = (d ! : F)⁻¹ | F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
d k : ℕ
hne : ↑d ! ≠ 0
heq : d = ((ascPochhammer F d).comp (X - C ↑k + 1)).natDegree
⊢ ((↑d !)⁻¹ • (ascPochhammer F d).comp (X - C ↑k + 1)).coeff d = (↑d !)⁻¹ | nth_rw 3 [heq] | F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
d k : ℕ
hne : ↑d ! ≠ 0
heq : d = ((ascPochhammer F d).comp (X - C ↑k + 1)).natDegree
⊢ ((↑d !)⁻¹ • (ascPochhammer F d).comp (X - C ↑k + 1)).coeff ((ascPochhammer F d).comp (X - C ↑k + 1)).natDegree =
(↑d !)⁻¹ | 103b1d6d649a6ef2 |
Equiv.optionCongr_swap | Mathlib/GroupTheory/Perm/Option.lean | theorem Equiv.optionCongr_swap {α : Type*} [DecidableEq α] (x y : α) :
optionCongr (swap x y) = swap (some x) (some y) | case neg
α : Type u_1
inst✝ : DecidableEq α
x y i a✝ : α
hx : ¬i = x
⊢ a✝ ∈ (optionCongr (swap x y)) (some i) ↔ a✝ ∈ (swap (some x) (some y)) (some i) | by_cases hy : i = y <;> simp [hx, hy, swap_apply_of_ne_of_ne] | no goals | f153f5aacb7303b6 |
Dynamics.IsDynCoverOf.iterate_le_pow | Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean | /-- From a dynamical cover `s` with entourage `U` and time `m`, we construct covers with entourage
`U ○ U` and any multiple `m * n` of `m` with controlled cardinality. This lemma is the first step
in a submultiplicative-like property of `coverMincard`, with consequences such as explicit bounds
for the topological entro... | case inr.intro.inr.refine_2
X : Type u_1
T : X → X
F : Set X
F_inv : MapsTo T F F
U : Set (X × X)
U_symm : SymmetricRel U
m n : ℕ
s : Finset X
h : IsDynCoverOf T F U m ↑s
x✝ : Nonempty X
s_nemp : (↑s).Nonempty
x : X
x_F : x ∈ F
m_pos : m > 0
dyncover : (Fin n → { x // x ∈ s }) → X
h_dyncover :
∀ (t : Fin n → { x // x... | rw [toFinset_card] | case inr.intro.inr.refine_2
X : Type u_1
T : X → X
F : Set X
F_inv : MapsTo T F F
U : Set (X × X)
U_symm : SymmetricRel U
m n : ℕ
s : Finset X
h : IsDynCoverOf T F U m ↑s
x✝ : Nonempty X
s_nemp : (↑s).Nonempty
x : X
x_F : x ∈ F
m_pos : m > 0
dyncover : (Fin n → { x // x ∈ s }) → X
h_dyncover :
∀ (t : Fin n → { x // x... | de3b71772507431f |
Field.finSepDegree_eq | Mathlib/FieldTheory/PurelyInseparable/Basic.lean | theorem finSepDegree_eq [Algebra.IsAlgebraic F E] :
finSepDegree F E = Cardinal.toNat (sepDegree F E) | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
this✝ : Algebra.IsAlgebraic (↥(separableClosure F E)) E
h : finSepDegree F E = finSepDegree F ↥(separableClosure F E) * finSepDegree (↥(separableClosure F E)) E
this : Algebra.IsSeparable F ↥(separableClosure F ... | haveI := separableClosure.isPurelyInseparable F E | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
this✝¹ : Algebra.IsAlgebraic (↥(separableClosure F E)) E
h : finSepDegree F E = finSepDegree F ↥(separableClosure F E) * finSepDegree (↥(separableClosure F E)) E
this✝ : Algebra.IsSeparable F ↥(separableClosure ... | d0a1c1c6ab7e184b |
CategoryTheory.Adjunction.commShift_of_leftAdjoint | Mathlib/CategoryTheory/Shift/Adjunction.lean | lemma commShift_of_leftAdjoint [F.CommShift A] :
letI := adj.rightAdjointCommShift A
adj.CommShift A | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
this : G.CommShift A := adj.rightAdjointCommShift A
⊢ adj.CommShift A | refine CommShift.mk' _ _ ⟨fun a ↦ ?_⟩ | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
this : G.CommShift A := adj.rightAdjointCommShift A
a : A
⊢ ((𝟭 C).commShiftIso a).hom ≫ whiskerRigh... | 363720ac1972606e |
ModuleCat.MonoidalCategory.hexagon_forward | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | theorem hexagon_forward (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).hom ≫ (braiding X _).hom ≫ (α_ Y Z X).hom =
(braiding X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (braiding X Z).hom | R : Type u
inst✝ : CommRing R
X Y Z : ModuleCat R
⊢ (α_ X Y Z).hom ≫ (X.braiding (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
(X.braiding Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (X.braiding Z).hom | ext : 1 | case hf
R : Type u
inst✝ : CommRing R
X Y Z : ModuleCat R
⊢ Hom.hom ((α_ X Y Z).hom ≫ (X.braiding (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom) =
Hom.hom ((X.braiding Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (X.braiding Z).hom) | d0fb5a4bfd6a54b4 |
Submodule.fg_unit | Mathlib/RingTheory/Finiteness/Subalgebra.lean | theorem fg_unit {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (I : (Submodule R A)ˣ) :
(I : Submodule R A).FG | case intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
I : (Submodule R A)ˣ
T T' : Finset A
hT : ↑T ⊆ ↑↑I
hT' : ↑T' ⊆ ↑↑I⁻¹
one_mem : 1 ∈ span R (↑T * ↑T')
⊢ ↑1 ≤ span R ↑T * span R ↑T' | rwa [Units.val_one, span_mul_span, one_le] | no goals | 6d75b8e62532e31b |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | case inr.inl.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc... | by_cases l.1.1 = i.1 | case pos
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsiz... | 02ca1706b397b254 |
AntilipschitzWith.properSpace | Mathlib/Topology/MetricSpace/Antilipschitz.lean | theorem properSpace {α : Type*} [MetricSpace α] {K : ℝ≥0} {f : α → β} [ProperSpace α]
(hK : AntilipschitzWith K f) (f_cont : Continuous f) (hf : Function.Surjective f) :
ProperSpace β | β : Type u_2
inst✝² : PseudoMetricSpace β
α : Type u_4
inst✝¹ : MetricSpace α
K : ℝ≥0
f : α → β
inst✝ : ProperSpace α
hK : AntilipschitzWith K f
f_cont : Continuous f
hf : Function.Surjective f
x₀ : β
r : ℝ
⊢ IsCompact (closedBall x₀ r) | let K := f ⁻¹' closedBall x₀ r | β : Type u_2
inst✝² : PseudoMetricSpace β
α : Type u_4
inst✝¹ : MetricSpace α
K✝ : ℝ≥0
f : α → β
inst✝ : ProperSpace α
hK : AntilipschitzWith K✝ f
f_cont : Continuous f
hf : Function.Surjective f
x₀ : β
r : ℝ
K : Set α := f ⁻¹' closedBall x₀ r
⊢ IsCompact (closedBall x₀ r) | a9c5d30f9c2d03d1 |
CategoryTheory.ShortComplex.Splitting.ext_s | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | lemma ext_s (s s' : S.Splitting) (h : s.s = s'.s) : s = s' | C : Type u_1
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
s s' : S.Splitting
h : s.s = s'.s
⊢ s = s' | have := s.mono_f | C : Type u_1
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
s s' : S.Splitting
h : s.s = s'.s
this : Mono S.f
⊢ s = s' | 500be6a8e246a01c |
LinearMap.trace_prodMap | Mathlib/LinearAlgebra/Trace.lean | theorem trace_prodMap :
trace R (M × N) ∘ₗ prodMapLinear R M N M N R =
(coprod id id : R × R →ₗ[R] R) ∘ₗ prodMap (trace R M) (trace R N) | case hr.a.h.h
R : Type u_1
inst✝⁸ : CommRing R
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u_3
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Free R M
inst✝² : Module.Finite R M
inst✝¹ : Free R N
inst✝ : Module.Finite R N
e : (Dual R M ⊗[R] M × Dual R N ⊗[R] N) ≃ₗ[R] (M →ₗ[R] M) × (N →ₗ[R] N... | simp only [e, dualTensorHomEquiv, LinearEquiv.coe_prod, dualTensorHomEquivOfBasis_toLinearMap,
AlgebraTensorModule.curry_apply, curry_apply, coe_restrictScalars, coe_comp, coe_inr,
Function.comp_apply, prodMap_apply, map_zero, prodMapLinear_apply, zero_prodMap_dualTensorHom,
trace_eq_contract_apply, contractLeft_... | no goals | 9f1ec8a0dddb51eb |
Vector.attachWith_map_subtype_val | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Attach.lean | theorem attachWith_map_subtype_val {p : α → Prop} (l : Vector α n) (H : ∀ a ∈ l, p a) :
(l.attachWith p H).map Subtype.val = l | α : Type u_1
n : Nat
p : α → Prop
l : Vector α n
H : ∀ (a : α), a ∈ l → p a
⊢ map Subtype.val (l.attachWith p H) = l | cases l | case mk
α : Type u_1
n : Nat
p : α → Prop
toArray✝ : Array α
size_toArray✝ : toArray✝.size = n
H : ∀ (a : α), a ∈ { toArray := toArray✝, size_toArray := size_toArray✝ } → p a
⊢ map Subtype.val ({ toArray := toArray✝, size_toArray := size_toArray✝ }.attachWith p H) =
{ toArray := toArray✝, size_toArray := size_toArr... | 1bc629b9ac251ef2 |
Multiset.le_iff_count | Mathlib/Data/Multiset/Count.lean | theorem le_iff_count {s t : Multiset α} : s ≤ t ↔ ∀ a, count a s ≤ count a t :=
Quotient.inductionOn₂ s t fun _ _ ↦ by simp [subperm_iff_count]
| α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
x✝¹ x✝ : List α
⊢ ⟦x✝¹⟧ ≤ ⟦x✝⟧ ↔ ∀ (a : α), count a ⟦x✝¹⟧ ≤ count a ⟦x✝⟧ | simp [subperm_iff_count] | no goals | c3a48cb66046c34d |
contMDiff_isOpenEmbedding | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | /-- If the `ChartedSpace` structure on a manifold `M` is given by an open embedding `e : M → H`,
then `e` is `C^n`. -/
lemma contMDiff_isOpenEmbedding [Nonempty M] :
haveI := h.singletonChartedSpace; ContMDiff I I n e | case right
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : WithTop ℕ∞
inst✝ : Nonempty M
this : IsManif... | apply mem_of_subset_of_mem _ hz.1 | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : WithTop ℕ∞
inst✝ : Nonempty M
this : IsManifold I ω M
x... | d917ee07659392c5 |
IsSemisimpleModule.finite_tfae | Mathlib/RingTheory/FiniteLength.lean | theorem IsSemisimpleModule.finite_tfae [IsSemisimpleModule R M] :
List.TFAE [Module.Finite R M, IsNoetherian R M, IsArtinian R M, IsFiniteLength R M,
∃ s : Set (Submodule R M), s.Finite ∧ sSupIndep s ∧
sSup s = ⊤ ∧ ∀ m ∈ s, IsSimpleModule R m] | R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s✝ : Set (Submodule R M)
hs : sSupIndep s✝ ∧ sSup s✝ = ⊤ ∧ ∀ m ∈ s✝, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧... | rw [← isNoetherian_top_iff, ← Submodule.topEquiv.isArtinian_iff,
← sSup_eq_top, sSup_eq_iSup, ← iSup_subtype''] | R : Type u_1
inst✝³ : Ring R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
s✝ : Set (Submodule R M)
hs : sSupIndep s✝ ∧ sSup s✝ = ⊤ ∧ ∀ m ∈ s✝, IsSimpleModule R ↥m
tfae_1_iff_2 : Module.Finite R M ↔ IsNoetherian R M
tfae_2_to_5 : IsNoetherian R M → ∃ s, s.Finite ∧ sSupIndep s ∧... | 5a414e5a28b27fdc |
LinearPMap.inverse_domain | Mathlib/LinearAlgebra/LinearPMap.lean | theorem inverse_domain : (inverse f).domain = LinearMap.range f.toFun | R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
⊢ Submodule.map (LinearMap.fst R F E) (Submodule.map (LinearEquiv.prodComm R E F) f.graph) =
Submodule.map (LinearMap.fst R F E) (Submodule.map (↑(LinearEquiv.... | rfl | no goals | 7f78cbb1bf953aab |
CategoryTheory.extensiveTopology.mem_sieves_iff_contains_colimit_cofan | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.lean | lemma extensiveTopology.mem_sieves_iff_contains_colimit_cofan {X : C} (S : Sieve X) :
S ∈ (extensiveTopology C) X ↔
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
Nonempty (IsColimit (Cofan.mk X π)) ∧ (∀ a : α, (S.arrows) (π a))) | case mp.of.intro.intro.intro.intro.intro.refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : FinitaryPreExtensive C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (Sigma.desc π)
⊢ Nonempty (IsColimit (Cofan.mk X π)) | have : IsIso (Sigma.desc (Cofan.mk X π).inj) := by simpa using h' | case mp.of.intro.intro.intro.intro.intro.refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : FinitaryPreExtensive C
X✝ : C
S✝ : Sieve X✝
X : C
S : Presieve X
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : S = Presieve.ofArrows Y π
h' : IsIso (Sigma.desc π)
this : IsIso (Sigma.desc (Cofan.mk X π).in... | 3464c4e8e1c1caf4 |
GenContFract.IntFractPair.one_le_succ_nth_stream_b | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | theorem one_le_succ_nth_stream_b {ifp_succ_n : IntFractPair K}
(succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n) : 1 ≤ ifp_succ_n.b | K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp_succ_n : IntFractPair K
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n
⊢ 1 ≤ ifp_succ_n.b | obtain ⟨ifp_n, nth_stream_eq, stream_nth_fr_ne_zero, ⟨-⟩⟩ :
∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr ≠ 0
∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n :=
succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq | case intro.intro.intro.refl
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp_n : IntFractPair K
nth_stream_eq : IntFractPair.stream v n = some ifp_n
stream_nth_fr_ne_zero : ifp_n.fr ≠ 0
succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr⁻¹)
⊢ 1 ≤ (IntFractPa... | 27c1666628e6ae6e |
List.exists_of_kerase | Mathlib/Data/List/Sigma.lean | theorem exists_of_kerase {a : α} {l : List (Sigma β)} (h : a ∈ l.keys) :
∃ (b : β a) (l₁ l₂ : List (Sigma β)),
a ∉ l₁.keys ∧ l = l₁ ++ ⟨a, b⟩ :: l₂ ∧ kerase a l = l₁ ++ l₂ | α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
hd : Sigma β
tl : List (Sigma β)
ih : a ∈ tl.keys → ∃ b l₁ l₂, a ∉ l₁.keys ∧ tl = l₁ ++ ⟨a, b⟩ :: l₂ ∧ kerase a tl = l₁ ++ l₂
e : ¬a = hd.fst
h : a ∈ tl.keys
b : β a
tl₁ tl₂ : List (Sigma β)
h₁ : a ∉ tl₁.keys
h₂ : tl = tl₁ ++ ⟨a, b⟩ :: tl₂
h₃ : kerase a tl = tl₁ ++ ... | (rw [h₂]; rfl) | no goals | 5c3119286dce7759 |
Subgroup.card_dvd_of_injective | Mathlib/GroupTheory/Coset/Card.lean | theorem card_dvd_of_injective (f : α →* H) (hf : Function.Injective f) :
Nat.card α ∣ Nat.card H | α : Type u_1
inst✝¹ : Group α
H : Type u_2
inst✝ : Group H
f : α →* H
hf : Function.Injective ⇑f
⊢ Nat.card α ∣ Nat.card H | calc
Nat.card α = Nat.card (f.range : Subgroup H) := Nat.card_congr (Equiv.ofInjective f hf)
_ ∣ Nat.card H := card_subgroup_dvd_card _ | no goals | b5f2c3eff6f37705 |
Field.Emb.cardinal_eq_two_pow_rank | Mathlib/FieldTheory/CardinalEmb.lean | theorem cardinal_eq_two_pow_rank [Algebra.IsSeparable F E]
(rank_inf : ℵ₀ ≤ Module.rank F E) : #(Field.Emb F E) = 2 ^ Module.rank F E | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
this : Fact (ℵ₀ ≤ Module.rank F E)
⊢ #(Emb F E) = 2 ^ Module.rank F E | rw [Emb.Cardinal.embEquivPi.cardinal_eq, mk_pi] | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
rank_inf : ℵ₀ ≤ Module.rank F E
this : Fact (ℵ₀ ≤ Module.rank F E)
⊢ (Cardinal.prod fun i => #(Cardinal.factor ↑i)) = 2 ^ Module.rank F E | 2b63d180bde4e1e0 |
CategoryTheory.NatTrans.exchange | Mathlib/CategoryTheory/Functor/Category.lean | theorem exchange {I J K : D ⥤ E} (α : F ⟶ G) (β : G ⟶ H) (γ : I ⟶ J) (δ : J ⟶ K) :
(α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ β ◫ δ | C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F G H : C ⥤ D
I J K : D ⥤ E
α : F ⟶ G
β : G ⟶ H
γ : I ⟶ J
δ : J ⟶ K
⊢ (α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ β ◫ δ | aesop_cat | no goals | 33b66374d103eb01 |
toIocMod_add_zsmul | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem toIocMod_add_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b + m • p) = toIocMod hp a b | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
m : ℤ
⊢ b + m • p - (toIocDiv hp a b • p + m • p) = b - toIocDiv hp a b • p | abel | no goals | 55e2429e4a674b4c |
MeasureTheory.unifIntegrable_of' | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem unifIntegrable_of' (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, StronglyMeasurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0, 0 < C ∧
∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UnifIntegrable f p μ | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0... | rw [← Set.indicator_indicator] | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0... | fc6184377b9b455c |
MvPolynomial.support_esymm'' | Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.lean | theorem support_esymm'' [DecidableEq σ] [Nontrivial R] (n : ℕ) :
(esymm σ R n).support =
(powersetCard n (univ : Finset σ)).biUnion fun t =>
(Finsupp.single (∑ i ∈ t, Finsupp.single i 1) (1 : R)).support | σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i => (Finsupp.single i 1).support) = s.biUnion fun i => (Finsupp.single i 1).support
s✝ : ... | rw [Finsupp.support_single_ne_zero x one_ne_zero] | no goals | 91d19fd65d77a5f5 |
List.getElem?_ofFn | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/OfFn.lean | theorem getElem?_ofFn (f : Fin n → α) (i) : (ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none :=
if h : i < (ofFn f).length
then by
rw [getElem?_eq_getElem h, List.getElem_ofFn]
· simp only [length_ofFn] at h; simp [h]
else by
rw [dif_neg] <;>
simpa using h
| n : Nat
α : Type u_1
f : Fin n → α
i : Nat
h✝ : i < (ofFn f).length
h : i < n
⊢ some (f ⟨i, ⋯⟩) = if h : i < n then some (f ⟨i, h⟩) else none | simp [h] | no goals | bc4f0a777be4b9a1 |
BoxIntegral.hasIntegralIndicatorConst | Mathlib/Analysis/BoxIntegral/Integrability.lean | theorem hasIntegralIndicatorConst (l : IntegrationParams) (hl : l.bRiemann = false)
{s : Set (ι → ℝ)} (hs : MeasurableSet s) (I : Box ι) (y : E) (μ : Measure (ι → ℝ))
[IsLocallyFiniteMeasure μ] :
HasIntegral.{u, v, v} I l (s.indicator fun _ => y) μ.toBoxAdditive.toSMul
((μ (s ∩ I)).toReal • y) | case intro.intro.intro.intro.intro.intro.intro.intro.h
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ... | have htU : t ⊆ U ∩ I := by
simp only [t, TaggedPrepartition.iUnion_def, iUnion_subset_iff, TaggedPrepartition.mem_filter,
and_imp]
refine fun J hJ hJs x hx => ⟨hrsU _ ⟨hJs, π.tag_mem_Icc J⟩ ?_, π.le_of_mem' J hJ hx⟩
simpa only [r, s.piecewise_eq_of_mem _ _ hJs] using hπ.1 J hJ (Box.coe_subset_Icc hx) | case intro.intro.intro.intro.intro.intro.intro.intro.h
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
hl : l.bRiemann = false
s : Set (ι → ℝ)
hs : MeasurableSet s
I : Box ι
y : E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
ε : ℝ≥0
ε0 : 0 < ... | 14c4b865d3538e42 |
zpow_two | Mathlib/Algebra/Group/Defs.lean | @[to_additive two_zsmul] lemma zpow_two (a : G) : a ^ (2 : ℤ) = a * a | G : Type u_1
inst✝ : DivInvMonoid G
a : G
⊢ a ^ 2 = a * a | rw [zpow_ofNat, pow_two] | no goals | 51e1617214e09dd3 |
List.filterMapM_toArray | Mathlib/.lake/packages/batteries/Batteries/Data/Array/Monadic.lean | theorem filterMapM_toArray [Monad m] [LawfulMonad m] (l : List α) (f : α → m (Option β)) :
l.toArray.filterMapM f = toArray <$> l.filterMapM f | m : Type u_1 → Type u_2
α β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : List α
f : α → m (Option β)
⊢ Array.filterMapM f l.toArray = toArray <$> filterMapM f l | simp [Array.filterMapM, filterMapM] | m : Type u_1 → Type u_2
α β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : List α
f : α → m (Option β)
⊢ List.foldlM
(fun x y => do
let __do_lift ← f y
match __do_lift with
| some b => pure (x.push b)
| none => pure x)
#[] l =
toArray <$> filterMapM.loop f l [] | ab3e95d5effd2cfd |
Field.Emb.Cardinal.strictMono_leastExt | Mathlib/FieldTheory/CardinalEmb.lean | theorem strictMono_leastExt : StrictMono φ := fun i j h ↦ by
have least := isLeast_leastExt (F := F) (E := E)
by_contra!
obtain eq | lt := this.eq_or_lt
· exact (least j).1 (subset_adjoin _ _ ⟨i, h, congr_arg b eq.symm⟩)
· refine ((least i).2 <| mt (adjoin.mono _ _ _ (image_mono ?_) ·) (least j).1).not_lt lt
... | case inl
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
i j : (Module.rank F E).ord.toType
h : i < j
least : ∀ (i : (Module.rank F E).ord.toType), IsLeast {k | b k ∉ adjoin F (⇑b ∘ φ '' Iio i)} (φ i)
this : φ j ≤ φ i
eq... | exact (least j).1 (subset_adjoin _ _ ⟨i, h, congr_arg b eq.symm⟩) | no goals | 8c3fabb094089744 |
NNReal.inner_le_Lp_mul_Lq_tsum | Mathlib/Analysis/MeanInequalities.lean | theorem inner_le_Lp_mul_Lq_tsum {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsConjExponent q)
(hf : Summable fun i => f i ^ p) (hg : Summable fun i => g i ^ q) :
(Summable fun i => f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) | case refine_2
ι : Type u
f g : ι → ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : Summable fun i => f i ^ p
hg : Summable fun i => g i ^ q
s : Finset ι
⊢ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≤ (∑' (i : ι), g i ^ q) ^ (1 / q) | rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)] | case refine_2
ι : Type u
f g : ι → ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : Summable fun i => f i ^ p
hg : Summable fun i => g i ^ q
s : Finset ι
⊢ ∑ i ∈ s, g i ^ q ≤ ∑' (i : ι), g i ^ q | a6de81c9fde72ed0 |
Finset.disjSum_eq_iff | Mathlib/Data/Finset/Sum.lean | lemma disjSum_eq_iff : s.disjSum t = u ↔ s = u.toLeft ∧ t = u.toRight :=
⟨fun h => by simp [← h], fun h => by simp [h, toLeft_disjSum_toRight]⟩
| α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
u : Finset (α ⊕ β)
h : s.disjSum t = u
⊢ s = u.toLeft ∧ t = u.toRight | simp [← h] | no goals | 41e6af2ef95a364f |
isOpen_singleton_of_finite_mem_nhds | Mathlib/Topology/Separation/Basic.lean | theorem isOpen_singleton_of_finite_mem_nhds [T1Space X] (x : X)
{s : Set X} (hs : s ∈ 𝓝 x) (hsf : s.Finite) : IsOpen ({x} : Set X) | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
s : Set X
hs : s ∈ 𝓝 x
hsf : s.Finite
A : {x} ⊆ s
B : IsClosed (s \ {x})
C : (s \ {x})ᶜ ∈ 𝓝 x
⊢ IsOpen {x} | have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
s : Set X
hs : s ∈ 𝓝 x
hsf : s.Finite
A : {x} ⊆ s
B : IsClosed (s \ {x})
C : (s \ {x})ᶜ ∈ 𝓝 x
D : {x} ∈ 𝓝 x
⊢ IsOpen {x} | f081030dc6b388bb |
LTSeries.apply_add_index_le_apply_add_index_int | Mathlib/Order/RelSeries.lean | /--
In ℤ, two entries in an `LTSeries` differ by at least the difference of their indices.
(Expressed in a way that avoids subtraction).
-/
lemma apply_add_index_le_apply_add_index_int (p : LTSeries ℤ) (i j : Fin (p.length + 1))
(hij : i ≤ j) : p i + j ≤ p j + i | case succ
p : LTSeries ℤ
i✝ j✝¹ : Fin (p.length + 1)
i : ℕ
hi : i < p.length + 1
j✝ j : ℕ
_hij : i ≤ j
hj : j + 1 < p.length + 1
ih : p.toFun ⟨i, hi⟩ + ↑j ≤ p.toFun ⟨j, ⋯⟩ + ↑i
step : p.toFun ⟨j, ⋯⟩ < p.toFun ⟨j + 1, ⋯⟩
⊢ p.toFun ⟨i, hi⟩ + ↑(j + 1) ≤ p.toFun ⟨j + 1, hj⟩ + ↑i | omega | no goals | 12eb8f68589a7c04 |
List.mem_set | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem mem_set (l : List α) (n : Nat) (h : n < l.length) (a : α) :
a ∈ l.set n a | α : Type u_1
l : List α
n : Nat
h : n < l.length
a : α
⊢ (l.set n a)[n] = a | simp | no goals | 5d562f2aba1de18d |
isPrimePow_iff_unique_prime_dvd | Mathlib/Data/Nat/Factorization/PrimePow.lean | theorem isPrimePow_iff_unique_prime_dvd {n : ℕ} : IsPrimePow n ↔ ∃! p : ℕ, p.Prime ∧ p ∣ n | case mp
n : ℕ
⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) → ∃! p, Nat.Prime p ∧ p ∣ n | rintro ⟨p, k, hp, hk, rfl⟩ | case mp.intro.intro.intro.intro
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
⊢ ∃! p_1, Nat.Prime p_1 ∧ p_1 ∣ p ^ k | 7ee0910c616e837c |
TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso | Mathlib/Topology/Sheaves/Stalks.lean | theorem app_isIso_of_stalkFunctor_map_iso {F G : Sheaf C X} (f : F ⟶ G) (U : Opens X)
[∀ x : U, IsIso ((stalkFunctor C x.val).map f.1)] : IsIso (f.1.app (op U)) | case h
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝⁴ : PreservesFilteredColimits (forget C)
inst✝³ : HasLimits C
inst✝² : PreservesLimits (forget C)
inst✝¹ : (forget ... | apply (isIso_iff_bijective _).mp | case h
C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝⁴ : PreservesFilteredColimits (forget C)
inst✝³ : HasLimits C
inst✝² : PreservesLimits (forget C)
inst✝¹ : (forget ... | ee0919d7877d7d44 |
Primrec.nat_omega_rec' | Mathlib/Computability/Primrec.lean | theorem nat_omega_rec' (f : β → σ) {m : β → ℕ} {l : β → List β} {g : β → List σ → Option σ}
(hm : Primrec m) (hl : Primrec l) (hg : Primrec₂ g)
(Ord : ∀ b, ∀ b' ∈ l b, m b' < m b)
(H : ∀ b, g b ((l b).map f) = some (f b)) : Primrec f | β : Type u_2
σ : Type u_4
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
f : β → σ
m : β → ℕ
l : β → List β
g : β → List σ → Option σ
hm : Primrec m
hl : Primrec l
hg : Primrec₂ g
Ord : ∀ (b b' : β), b' ∈ l b → m b' < m b
H : ∀ (b : β), g b (List.map f (l b)) = some (f b)
this✝ : DecidableEq β
mapGraph : List (β × σ) → L... | simp [graph_eq_map_bindList (m b + 1) (Nat.le_refl _), bindList] | no goals | 4ce92daa13848025 |
Mathlib.Meta.NormNum.isNat_intCast | Mathlib/Tactic/NormNum/Basic.lean | theorem isNat_intCast {R} [Ring R] (n : ℤ) (m : ℕ) :
IsNat n m → IsNat (n : R) m | R : Type u_1
inst✝ : Ring R
m : ℕ
⊢ ↑↑m = ↑m | simp | no goals | 8e008b6c70dc18e6 |
FiniteDimensional.of_isCompact_closedBall₀ | Mathlib/Analysis/Normed/Module/FiniteDimension.lean | theorem FiniteDimensional.of_isCompact_closedBall₀ {r : ℝ} (rpos : 0 < r)
(h : IsCompact (Metric.closedBall (0 : E) r)) : FiniteDimensional 𝕜 E | case intro.intro.intro.intro
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
r : ℝ
rpos : 0 < r
h : IsCompact (closedBall 0 r)
hfin : ¬FiniteDimensional 𝕜 E
R : ℝ
f : ℕ → E
Rgt : 1 < R
fle : ∀ (n : ℕ), ‖f n‖ ≤ R
lef : Pairwise ... | obtain ⟨c, hc⟩ : ∃ c : 𝕜, 0 < ‖c‖ ∧ ‖c‖ < r / R := NormedField.exists_norm_lt _ rRpos | case intro.intro.intro.intro.intro
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
r : ℝ
rpos : 0 < r
h : IsCompact (closedBall 0 r)
hfin : ¬FiniteDimensional 𝕜 E
R : ℝ
f : ℕ → E
Rgt : 1 < R
fle : ∀ (n : ℕ), ‖f n‖ ≤ R
lef : Pai... | c91708e1e8291456 |
Stirling.stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq | Mathlib/Analysis/SpecialFunctions/Stirling.lean | theorem stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq (n : ℕ) (hn : n ≠ 0) :
stirlingSeq n ^ 4 / stirlingSeq (2 * n) ^ 2 * (n / (2 * n + 1)) = Wallis.W n | n : ℕ
hn : n ≠ 0
this : 4 = 2 * 2
⊢ (↑n ! ^ 2) ^ 2 * rexp (2 * (2 * ↑n)) * (2 * (2 * ↑n) * ((2 * ↑n) ^ (2 * n)) ^ 2) * ↑n *
(↑(2 * n)! ^ 2 * (2 * ↑n + 1)) =
2 ^ (4 * n) * ↑n ! ^ 4 * ((2 * ↑n) ^ 2 * ((↑n ^ n) ^ 2) ^ 2 * (↑(2 * n)! ^ 2 * rexp (2 * (2 * ↑n))) * (2 * ↑n + 1)) | ring_nf | no goals | 3bce3cf189c245f4 |
InnerProductSpace.Core.inner_mul_inner_self_le | Mathlib/Analysis/InnerProductSpace/Defs.lean | theorem inner_mul_inner_self_le (x y : F) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ | 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫_𝕜‖ * t + normSq y | by_cases hzero : ⟪x, y⟫ = 0 | case pos
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
hzero : ⟪x, y⟫_𝕜 = 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫_𝕜‖ * t + normSq y
case neg
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Modu... | 3490ee6e9e583d18 |
Polynomial.isMaximal_comap_C_of_isMaximal | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isMaximal_comap_C_of_isMaximal [IsJacobsonRing R] [Nontrivial R]
(hP' : ∀ x : R, C x ∈ P → x = 0) :
IsMaximal (comap (C : R →+* R[X]) P : Ideal R) | R : Type u_1
inst✝² : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝¹ : IsJacobsonRing R
inst✝ : Nontrivial R
hP' : ∀ (x : R), C x ∈ P → x = 0
P' : Ideal R := comap C P
hP'_prime : P'.IsPrime
m : R[X]
hmem_P : m ∈ P
hm : ⟨m, hmem_P⟩ ≠ 0
hm' : m ≠ 0
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
a : R ⧸ P' := (map (Idea... | rwa [Ideal.map_bot] | no goals | 0ec58dabe1e295b5 |
MeasureTheory.Measure.restrict_restrict_of_subset | Mathlib/MeasureTheory/Measure/Restrict.lean | theorem restrict_restrict_of_subset (h : s ⊆ t) : (μ.restrict t).restrict s = μ.restrict s | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
⊢ (μ.restrict t).restrict s = μ.restrict s | ext1 u hu | case h
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
h : s ⊆ t
u : Set α
hu : MeasurableSet u
⊢ ((μ.restrict t).restrict s) u = (μ.restrict s) u | 32a6b3c33c034485 |
CategoryTheory.Limits.preservesFiniteLimits_of_preservesFiniteLimitsOfSize | Mathlib/CategoryTheory/Limits/Preserves/Finite.lean | /-- We can always derive `PreservesFiniteLimits C` by showing that we are preserving limits at an
arbitrary universe. -/
lemma preservesFiniteLimits_of_preservesFiniteLimitsOfSize (F : C ⥤ D)
(h :
∀ (J : Type w) {𝒥 : SmallCategory J} (_ : @FinCategory J 𝒥), PreservesLimitsOfShape J F) :
PreservesFiniteL... | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
h : ∀ (J : Type w) {𝒥 : SmallCategory J}, FinCategory J → PreservesLimitsOfShape J F
J : Type
x✝¹ : SmallCategory J
x✝ : FinCategory J
⊢ PreservesLimitsOfShape J F | haveI := h (ULiftHom (ULift J)) CategoryTheory.finCategoryUlift | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
h : ∀ (J : Type w) {𝒥 : SmallCategory J}, FinCategory J → PreservesLimitsOfShape J F
J : Type
x✝¹ : SmallCategory J
x✝ : FinCategory J
this : PreservesLimitsOfShape (ULiftHom (ULift.{w, 0} J)) F
⊢ PreservesLimitsOfShape J F | 3e985a47bd7d04ff |
ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_nonneg | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma IsMeasurableRatCDF.stieltjesFunctionAux_nonneg (a : α) (r : ℝ) :
0 ≤ IsMeasurableRatCDF.stieltjesFunctionAux f a r | α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
r : ℝ
⊢ Nonempty { r' // r < ↑r' } | obtain ⟨r, hrx⟩ := exists_rat_gt r | case intro
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
r✝ : ℝ
r : ℚ
hrx : r✝ < ↑r
⊢ Nonempty { r' // r✝ < ↑r' } | 31ed3d79177ecb0c |
MeasureTheory.Measure.exists_isOpen_everywherePosSubset_eq_diff | Mathlib/MeasureTheory/Measure/EverywherePos.lean | /-- The everywhere positive subset of a set is obtained by removing an open set. -/
lemma exists_isOpen_everywherePosSubset_eq_diff (μ : Measure α) (s : Set α) :
∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u | α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ IsOpen {x | ∃ n ∈ 𝓝[s] x, μ n = 0} | rw [isOpen_iff_mem_nhds] | α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ ∀ x ∈ {x | ∃ n ∈ 𝓝[s] x, μ n = 0}, {x | ∃ n ∈ 𝓝[s] x, μ n = 0} ∈ 𝓝 x | 7270b311930e87cb |
CategoryTheory.isVanKampenColimit_of_evaluation | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J D] (F : J ⥤ C ⥤ D)
(c : Cocone F) (hc : ∀ x : C, IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)) :
IsVanKampenColimit c | J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : HasPullbacks D
inst✝ : HasColimitsOfShape J D
F : J ⥤ C ⥤ D
c : Cocone F
hc : ∀ (x : C), IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)
⊢ IsVanKampenColimit c | intro F' c' α f e hα | J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : HasPullbacks D
inst✝ : HasColimitsOfShape J D
F : J ⥤ C ⥤ D
c : Cocone F
hc : ∀ (x : C), IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)
F' : J ⥤ C ⥤ D
c' : Cocone F'
α : F' ⟶ F
... | feaac997c4176d8a |
Fin.one_succAbove_succ | Mathlib/Data/Fin/Basic.lean | @[simp] lemma one_succAbove_succ {n : ℕ} (j : Fin n) :
(1 : Fin (n + 2)).succAbove j.succ = j.succ.succ | n : ℕ
j : Fin n
this : (succ 0).succAbove j.succ = (succAbove 0 j).succ
⊢ succAbove 1 j.succ = j.succ.succ | rwa [succ_zero_eq_one, zero_succAbove] at this | no goals | 511a15190140a694 |
AlgebraicGeometry.Scheme.Hom.range_subset_ker_support | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma Hom.range_subset_ker_support (f : X.Hom Y) :
Set.range f.base ⊆ f.ker.support | case intro.intro.intro.intro.intro.intro
X Y : Scheme
f : X.Hom Y
x : ↑↑X.toPresheafedSpace
U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace
hU : U ∈ Y.affineOpens
hxU : (ConcreteCategory.hom f.base) x ∈ ↑U
⊢ (ConcreteCategory.hom f.base) x ∈ Y.zeroLocus ↑(f.ker.ideal ⟨U, hU⟩) | simp only [Scheme.mem_zeroLocus_iff, SetLike.mem_coe] | case intro.intro.intro.intro.intro.intro
X Y : Scheme
f : X.Hom Y
x : ↑↑X.toPresheafedSpace
U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace
hU : U ∈ Y.affineOpens
hxU : (ConcreteCategory.hom f.base) x ∈ ↑U
⊢ ∀ f_1 ∈ f.ker.ideal ⟨U, hU⟩, (ConcreteCategory.hom f.base) x ∉ Y.basicOpen f_1 | 9d460a55329e679b |
Pell.IsFundamental.eq_pow_of_nonneg | Mathlib/NumberTheory/Pell.lean | theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
(hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n | case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 = a.y
⊢ ∃ n, a = a₁ ^ n | refine ⟨0, ?_⟩ | case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 = a.y
⊢ a = a₁ ^ 0 | e2cabc4b76df1e43 |
exists_dual_vector | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | theorem exists_dual_vector (x : E) (h : x ≠ 0) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖ | case intro.refine_2
𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : x ≠ 0
p : Submodule 𝕜 E := span 𝕜 {x}
f : ↥(span 𝕜 {x}) →L[𝕜] 𝕜 := ↑‖x‖ • coord 𝕜 x h
g : E →L[𝕜] 𝕜
hg : (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖
⊢ g x = ↑‖x‖ | calc
g x = g (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) := by rw [coe_mk]
_ = ((‖x‖ : 𝕜) • coord 𝕜 x h) (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) := by rw [← hg.1]
_ = ‖x‖ := by simp [-algebraMap_smul] | no goals | 0b5b2636d12164f0 |
Array.getElem_eq_getElem?_get | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_eq_getElem?_get (a : Array α) (i : Nat) (h : i < a.size) :
a[i] = a[i]?.get (by simp [getElem?_eq_getElem, h]) | α : Type u_1
a : Array α
i : Nat
h : i < a.size
⊢ a[i] = a[i]?.get ⋯ | simp [getElem_eq_iff] | no goals | 25b4d33b28d20b25 |
SimpleGraph.ComponentCompl.hom_eq_iff_not_disjoint | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | theorem hom_eq_iff_not_disjoint (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K) :
C.hom h = D ↔ ¬Disjoint (C : Set V) (D : Set V) | V : Type u
G : SimpleGraph V
K L : Set V
C : G.ComponentCompl L
h : K ⊆ L
D : G.ComponentCompl K
⊢ hom h C = D ↔ ¬Disjoint ↑C ↑D | rw [Set.not_disjoint_iff] | V : Type u
G : SimpleGraph V
K L : Set V
C : G.ComponentCompl L
h : K ⊆ L
D : G.ComponentCompl K
⊢ hom h C = D ↔ ∃ x ∈ ↑C, x ∈ ↑D | 506fd1049e0c4f34 |
le_limsup_add | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | lemma le_limsup_add (h₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u | ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : CovariantClass α α (fun a b => a + b) fun x1 x2 => x1 ≤ x2
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u) _auto✝
h₂ : autoParam (IsCobound... | refine ((frequently_lt_of_lt_limsup h₂ a_u).and_eventually
(eventually_lt_of_lt_liminf b_v h₄)).mono fun _ ab_x ↦ ?_ | ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommGroup α
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : DenselyOrdered α
inst✝¹ : CovariantClass α α (fun a b => a + b) fun x1 x2 => x1 ≤ x2
f : Filter ι
inst✝ : f.NeBot
u v : ι → α
h₁ : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u) _auto✝
h₂ : autoParam (IsCobound... | 4ac72417e1f191a1 |
List.sublists_append | Mathlib/Data/List/Sublists.lean | theorem sublists_append (l₁ l₂ : List α) :
sublists (l₁ ++ l₂) = (sublists l₂) >>= (fun x => (sublists l₁).map (· ++ x)) | case cons
α : Type u
l₂ : List α
a : α
l₁ : List α
ih :
foldr (fun a acc => flatMap (fun x => [x, a :: x]) acc)
(foldr (fun a acc => flatMap (fun x => [x, a :: x]) acc) [[]] l₂) l₁ =
do
let x ← foldr (fun a acc => flatMap (fun x => [x, a :: x]) acc) [[]] l₂
map (fun x_1 => x_1 ++ x) (foldr (fun a ac... | simp [List.flatMap, flatten_flatten, Function.comp_def] | no goals | 35a7e9ec7d618eee |
Fin.append_assoc | Mathlib/Data/Fin/Tuple/Basic.lean | theorem append_assoc {p : ℕ} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) :
append (append a b) c = append a (append b c) ∘ Fin.cast (Nat.add_assoc ..) | m n : ℕ
α : Sort u_1
p : ℕ
a : Fin m → α
b : Fin n → α
c : Fin p → α
⊢ append (append a b) c = append a (append b c) ∘ Fin.cast ⋯ | ext i | case h
m n : ℕ
α : Sort u_1
p : ℕ
a : Fin m → α
b : Fin n → α
c : Fin p → α
i : Fin (m + n + p)
⊢ append (append a b) c i = (append a (append b c) ∘ Fin.cast ⋯) i | f0cf68d1bdb47d70 |
Convex.convex_remove_iff_not_mem_convexHull_remove | Mathlib/Analysis/Convex/Hull.lean | theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex 𝕜 s) (x : E) :
Convex 𝕜 (s \ {x}) ↔ x ∉ convexHull 𝕜 (s \ {x}) | case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
hx : x ∉ (convexHull 𝕜) (s \ {x})
⊢ s \ {x} = (convexHull 𝕜) (s \ {x}) | exact
Subset.antisymm (subset_convexHull 𝕜 _) fun y hy =>
⟨convexHull_min diff_subset hs hy, by
rintro (rfl : y = x)
exact hx hy⟩ | no goals | 2f0adfe15b540bae |
AlgebraicGeometry.Spec.preimage_id | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | @[simp]
lemma Spec.preimage_id {R : CommRingCat} : Spec.preimage (𝟙 (Spec R)) = 𝟙 R :=
Spec.map_injective (by simp)
| R : CommRingCat
⊢ map (preimage (𝟙 (Spec R))) = map (𝟙 R) | simp | no goals | 146d724cfe901cfc |
Padic.exi_rat_seq_conv | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem exi_rat_seq_conv {ε : ℚ} (hε : 0 < ε) :
∃ N, ∀ i ≥ N, padicNormE (f i - (limSeq f i : ℚ_[p]) : ℚ_[p]) < ε | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ padicNormE (↑f i - ↑(limSeq f i)) < ε | refine lt_of_lt_of_le h ((div_le_iff₀' <| mod_cast succ_pos _).mpr ?_) | p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
⊢ 1 ≤ (↑i + 1) * ε | a4713483bdf9656c |
isAdic_iff | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | theorem isAdic_iff [top : TopologicalSpace R] [IsTopologicalRing R] {J : Ideal R} :
IsAdic J ↔
(∀ n : ℕ, IsOpen ((J ^ n : Ideal R) : Set R)) ∧
∀ s ∈ 𝓝 (0 : R), ∃ n : ℕ, ((J ^ n : Ideal R) : Set R) ⊆ s | case mp
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H : IsAdic J
⊢ (∀ (n : ℕ), IsOpen ↑(J ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s | change _ = _ at H | case mp
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H : top = J.adicTopology
⊢ (∀ (n : ℕ), IsOpen ↑(J ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s | 5361f5c3aa3b31e4 |
CategoryTheory.Pretriangulated.contractible_distinguished₂ | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | /-- Obvious triangles `X ⟶ 0 ⟶ X⟦1⟧ ⟶ X⟦1⟧` are distinguished -/
lemma contractible_distinguished₂ (X : C) :
Triangle.mk (0 : X ⟶ 0) 0 (𝟙 (X⟦1⟧)) ∈ distTriang C | 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
X : C
⊢ Triangle.mk 0 0 (𝟙 ((shiftFunctor C 1).obj X)) ≅ (Triangle.mk 0 (𝟙 ((shiftFunctor C 1).obj X)) 0).invRotate | exact Triangle.isoMk _ _ ((shiftEquiv C (1 : ℤ)).unitIso.app X) (Iso.refl _) (Iso.refl _)
(by simp) (by simp)
(by dsimp; simp only [shift_shiftFunctorCompIsoId_inv_app, id_comp]) | no goals | 493ef9790a6907d4 |
Polynomial.isIntegral_isLocalization_polynomial_quotient | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Idea... | case intro.mk.mk.refine_1.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R... | simp only [Set.mem_insert_iff] at hp | case intro.mk.mk.refine_1.intro.intro
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R... | 4f255f198e9b3f92 |
Commute.smul_left_iff | Mathlib/Algebra/Group/Action/Defs.lean | @[simp] lemma Commute.smul_left_iff : Commute (g • a) b ↔ Commute a b | G : Type u_3
H : Type u_4
inst✝⁴ : Group G
g : G
inst✝³ : Mul H
inst✝² : MulAction G H
inst✝¹ : SMulCommClass G H H
inst✝ : IsScalarTower G H H
a b : H
⊢ Commute (g • a) b ↔ Commute a b | rw [Commute.symm_iff, Commute.smul_right_iff, Commute.symm_iff] | no goals | 207eae4a90e87c9b |
Cardinal.ofENat_eq_nat | Mathlib/SetTheory/Cardinal/ENat.lean | @[simp] lemma ofENat_eq_nat {m : ℕ∞} {n : ℕ} : (m : Cardinal) = n ↔ m = n | m : ℕ∞
n : ℕ
⊢ ↑m = ↑n ↔ m = ↑n | norm_cast | no goals | bb0a79dfbdb40694 |
Complex.tan_neg | Mathlib/Data/Complex/Trigonometric.lean | theorem tan_neg : tan (-x) = -tan x | x : ℂ
⊢ tan (-x) = -tan x | simp [tan, neg_div] | no goals | cc85babdcbd82ef9 |
CategoryTheory.Subobject.inf_pullback | Mathlib/CategoryTheory/Subobject/Lattice.lean | theorem inf_pullback {X Y : C} (g : X ⟶ Y) (f₁ f₂) :
(pullback g).obj (f₁ ⊓ f₂) = (pullback g).obj f₁ ⊓ (pullback g).obj f₂ | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
X Y : C
g : X ⟶ Y
f₂ : Subobject Y
⊢ ∀ (f₁ : Subobject Y), (pullback g).obj (f₁ ⊓ f₂) = (pullback g).obj f₁ ⊓ (pullback g).obj f₂ | apply Quotient.ind' | case h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
X Y : C
g : X ⟶ Y
f₂ : Subobject Y
⊢ ∀ (a : MonoOver Y), (pullback g).obj (Quotient.mk'' a ⊓ f₂) = (pullback g).obj (Quotient.mk'' a) ⊓ (pullback g).obj f₂ | dd4e8497ffc97474 |
Real.arctan_neg | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | theorem arctan_neg (x : ℝ) : arctan (-x) = -arctan x | x : ℝ
⊢ arctan (-x) = -arctan x | simp [arctan_eq_arcsin, neg_div] | no goals | cda6acc33d5938ab |
WithTop.isLUB_sSup' | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | theorem isLUB_sSup' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)}
(hs : s.Nonempty) : IsLUB s (sSup s) | case pos
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ∈ lowerBounds (upperBounds s) | rintro (⟨⟩ | b) hb | case pos.none
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ : ⊤ ∉ s
h₂ : BddAbove ((fun a => ↑a) ⁻¹' s)
hb : none ∈ upperBounds s
⊢ ↑(sSup ((fun a => ↑a) ⁻¹' s)) ≤ none
case pos.some
β : Type u_5
inst✝ : ConditionallyCompleteLattice β
s : Set (WithTop β)
hs : s.Nonempty
h₁ ... | 8c5594b3d704c24c |
List.get_insertIdx_self | Mathlib/Data/List/InsertIdx.lean | theorem get_insertIdx_self (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length)
(hn' : n < (insertIdx n x l).length :=
(by rwa [length_insertIdx_of_le_length hn, Nat.lt_succ_iff])) :
(insertIdx n x l).get ⟨n, hn'⟩ = x | α : Type u
β : Type v
a : α
l : List α
x : α
n : ℕ
hn : n ≤ l.length
⊢ n < (insertIdx n x l).length | rwa [length_insertIdx_of_le_length hn, Nat.lt_succ_iff] | no goals | ac7fe648ca6a7050 |
HomologicalComplex.mono_homologyMap_of_mono_of_not_rel | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | lemma mono_homologyMap_of_mono_of_not_rel (φ : K ⟶ L) (j : ι)
[K.HasHomology j] [L.HasHomology j] [Mono (φ.f j)] (hj : ∀ i, ¬ c.Rel i j) :
Mono (homologyMap φ j) :=
((MorphismProperty.monomorphisms C).arrow_mk_iso_iff
(Arrow.isoMk (K.isoHomologyπ _ j rfl (shape _ _ _ (by tauto)))
(L.isoHomologyπ _ j... | C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L : HomologicalComplex C c
φ : K ⟶ L
j : ι
inst✝² : K.HasHomology j
inst✝¹ : L.HasHomology j
inst✝ : Mono (φ.f j)
hj : ∀ (i : ι), ¬c.Rel i j
⊢ ¬c.Rel (c.prev j) j | tauto | no goals | 884a796921f4b669 |
IsFractionRing.lift_mk' | Mathlib/RingTheory/Localization/FractionRing.lean | theorem lift_mk' (hg : Injective g) (x) (y : nonZeroDivisors A) :
lift hg (mk' K x y) = g x / g y | A : Type u_4
inst✝⁴ : CommRing A
K : Type u_5
inst✝³ : Field K
L : Type u_7
inst✝² : Field L
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
g : A →+* L
hg : Injective ⇑g
x : A
y : ↥(nonZeroDivisors A)
⊢ (lift hg) (mk' K x y) = g x / g ↑y | simp only [mk'_eq_div, map_div₀, lift_algebraMap] | no goals | d9b3f1360532103d |
BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_isClosedEmbedding | Mathlib/Topology/TietzeExtension.lean | theorem exists_extension_forall_exists_le_ge_of_isClosedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
{e : X → Y} (he : IsClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f | case intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
b : ℝ
hb : IsLUB (range ⇑f)... | simp [dg0 (Or.inr hc), (hg_mem y).2] | no goals | c0f84dcf09f7662d |
GenContFract.fib_le_of_contsAux_b | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | theorem fib_le_of_contsAux_b :
n ≤ 1 ∨ ¬(of v).TerminatedAt (n - 2) → (fib n : K) ≤ ((of v).contsAux n).b :=
Nat.strong_induction_on n
(by
intro n IH hyp
rcases n with (_ | _ | n)
· simp [fib_add_two, contsAux] -- case n = 0
· simp [fib_add_two, contsAux] -- case n = 1
· let g :=... | case zero
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
IH : ∀ m < 0, m ≤ 1 ∨ ¬(of v).TerminatedAt (m - 2) → ↑(fib m) ≤ ((of v).contsAux m).b
hyp : 0 ≤ 1 ∨ ¬(of v).TerminatedAt (0 - 2)
⊢ ↑(fib 0) ≤ ((of v).contsAux 0).b | simp [fib_add_two, contsAux] | no goals | 3d07880c022731b8 |
Multiset.revzip_powersetAux_perm_aux' | Mathlib/Data/Multiset/Powerset.lean | theorem revzip_powersetAux_perm_aux' {l : List α} :
revzip (powersetAux l) ~ revzip (powersetAux' l) | α : Type u_1
l : List α
this : DecidableEq α
⊢ (powersetAux l).revzip ~ (powersetAux' l).revzip | rw [revzip_powersetAux_lemma l revzip_powersetAux, revzip_powersetAux_lemma l revzip_powersetAux'] | α : Type u_1
l : List α
this : DecidableEq α
⊢ List.map (fun x => (x, ↑l - x)) (powersetAux l) ~ List.map (fun x => (x, ↑l - x)) (powersetAux' l) | a56dcf86bbe1b699 |
TensorProduct.congr_zpow | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | theorem congr_zpow (f : M ≃ₗ[R] M) (g : N ≃ₗ[R] N) (n : ℤ) :
congr f g ^ n = congr (f ^ n) (g ^ n) | case ofNat
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_5
N : Type u_6
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
f : M ≃ₗ[R] M
g : N ≃ₗ[R] N
n : ℕ
⊢ congr f g ^ Int.ofNat n = congr (f ^ Int.ofNat n) (g ^ Int.ofNat n) | exact congr_pow _ _ _ | no goals | 8e2a13d8ee0a8ff1 |
Finmap.induction_on | Mathlib/Data/Finmap.lean | theorem induction_on {C : Finmap β → Prop} (s : Finmap β) (H : ∀ a : AList β, C ⟦a⟧) : C s | α : Type u
β : α → Type v
C : Finmap β → Prop
s : Finmap β
H : ∀ (a : AList β), C ⟦a⟧
⊢ C s | rcases s with ⟨⟨a⟩, h⟩ | case mk.mk
α : Type u
β : α → Type v
C : Finmap β → Prop
H : ∀ (a : AList β), C ⟦a⟧
entries✝ : Multiset (Sigma β)
a : List (Sigma β)
h : Multiset.NodupKeys (Quot.mk (⇑(isSetoid (Sigma β))) a)
⊢ C { entries := Quot.mk (⇑(isSetoid (Sigma β))) a, nodupKeys := h } | 7a847f2698068eb7 |
CauSeq.neg_limZero | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) | α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : f.LimZero
⊢ (-1 * f).LimZero | exact mul_limZero_right _ hf | no goals | f53493ac67187faf |
SemiNormedGrp₁.inv_hom_apply | Mathlib/Analysis/Normed/Group/SemiNormedGrp.lean | @[simp]
lemma inv_hom_apply {M N : SemiNormedGrp₁} (e : M ≅ N) (r : M) : e.inv (e.hom r) = r | M N : SemiNormedGrp₁
e : M ≅ N
r : M.carrier
⊢ ↑(Hom.hom (e.hom ≫ e.inv)) r = r | simp | no goals | a8b0ec5c7466d80f |
CFC.log_exp | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog.lean | lemma log_exp (a : A) (ha : IsSelfAdjoint a | A : Type u_1
inst✝⁴ : NormedRing A
inst✝³ : StarRing A
inst✝² : NormedAlgebra ℝ A
inst✝¹ : ContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝ : CompleteSpace A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
hcont : ContinuousOn Real.log (Real.exp '' spectrum ℝ a)
⊢ cfc (fun x => Real.log (Real.exp x)) a = a | simp [cfc_id' (R := ℝ) a] | no goals | 78bf966fd06131b2 |
Polynomial.ringHom_ext | Mathlib/Algebra/Polynomial/Monomial.lean | theorem ringHom_ext {S} [Semiring S] {f g : R[X] →+* S} (h₁ : ∀ a, f (C a) = g (C a))
(h₂ : f X = g X) : f = g | R : Type u
inst✝¹ : Semiring R
S : Type u_1
inst✝ : Semiring S
f g : R[X] →+* S
h₁ : ∀ (a : R), f (C a) = g (C a)
h₂ : f X = g X
f' : AddMonoidAlgebra R ℕ →+* S := f.comp (toFinsuppIso R).symm.toRingHom
hf' : f' = f.comp (toFinsuppIso R).symm.toRingHom
⊢ f = g | set g' := g.comp (toFinsuppIso R).symm.toRingHom with hg' | R : Type u
inst✝¹ : Semiring R
S : Type u_1
inst✝ : Semiring S
f g : R[X] →+* S
h₁ : ∀ (a : R), f (C a) = g (C a)
h₂ : f X = g X
f' : AddMonoidAlgebra R ℕ →+* S := f.comp (toFinsuppIso R).symm.toRingHom
hf' : f' = f.comp (toFinsuppIso R).symm.toRingHom
g' : AddMonoidAlgebra R ℕ →+* S := g.comp (toFinsuppIso R).symm.toR... | 0a7f72cbc18ffb89 |
LieAlgebra.ExtendScalars.bracket_lie_self | Mathlib/Algebra/Lie/BaseChange.lean | theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 | case refine_2
R : Type u_1
A : Type u_2
L : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x : A ⊗[R] L
a : A
l : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L L) (a ⊗ₜ[R] l)) (a ⊗ₜ[R] l) = 0 | simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self] | no goals | ce1e0c51b9871d3e |
Finset.memberSubfamily_union_nonMemberSubfamily | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | theorem memberSubfamily_union_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) :
𝒜.memberSubfamily a ∪ 𝒜.nonMemberSubfamily a = 𝒜.image fun s => s.erase a | α : Type u_1
inst✝ : DecidableEq α
a : α
𝒜 : Finset (Finset α)
s : Finset α
hs : s ∈ 𝒜
ha : a ∈ s
⊢ insert a (s.erase a) ∈ 𝒜 | rwa [insert_erase ha] | no goals | e84640d1d865ba15 |
Sylow.card_eq_multiplicity | Mathlib/GroupTheory/Sylow.lean | theorem card_eq_multiplicity [Finite G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
Nat.card P = p ^ Nat.factorization (Nat.card G) p | case intro
G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Nat.card ↥↑P = p ^ n
⊢ Nat.card ↥↑P ∣ Nat.card G | exact P.1.card_subgroup_dvd_card | no goals | 30073896caf384fa |
Std.Tactic.BVDecide.BVExpr.bitblast.blastConst.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Const.lean | theorem go_denote_eq (aig : AIG α) (c : BitVec w) (assign : α → Bool)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
⟦
(go aig c curr s hcurr).aig,
(go aig c curr s hcurr).vec.get idx hidx1,
assign... | case isTrue.inl
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec 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✝ : curr < w
hgo :
go (aig.mkConstCached (c.getLsbD curr)).aig c (curr + 1) ((s.cast... | rw [go_denote_mem_prefix] | case isTrue.inl
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
c : BitVec 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✝ : curr < w
hgo :
go (aig.mkConstCached (c.getLsbD curr)).aig c (curr + 1) ((s.cast... | bf1ceda9c97ad4d3 |
MvPowerSeries.le_lexOrder_mul | Mathlib/RingTheory/MvPowerSeries/LexOrder.lean | theorem le_lexOrder_mul (φ ψ : MvPowerSeries σ R) :
lexOrder φ + lexOrder ψ ≤ lexOrder (φ * ψ) | case inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hv : ↑(toLex v) < ψ.lexOrder
⊢ (coeff R u) φ * (coeff R v) ψ = 0 | rw [coeff_eq_zero_of_lt_lexOrder hv, mul_zero] | no goals | a292b68486513728 |
ModP.v_p_lt_preVal | Mathlib/RingTheory/Perfection.lean | theorem v_p_lt_preVal {x : ModP O p} : v p < preVal K v O p x ↔ x ≠ 0 | K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
x : ModP O p
h : x ≠ 0
hp : preVal K v O p x ≤ v ↑p
⊢ x = 0 | obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x | case intro
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
r : O
h : (Ideal.Quotient.mk (Ideal.span {↑p})) r ≠ 0
hp : preVal K v O p ((Ideal.Quotient.mk (Ideal.span {↑p})) r) ≤ v ↑p
⊢ (Ideal.Quotient.mk (Ideal.span {↑p})) r = 0 | a79ee61ac0b09228 |
AlgebraicGeometry.RingedSpace.isUnit_of_isUnit_germ | Mathlib/Geometry/RingedSpace/Basic.lean | theorem isUnit_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U))
(h : ∀ (x) (hx : x ∈ U), IsUnit (X.presheaf.germ U x hx f)) : IsUnit f | X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
h : ∀ (x : ↑↑X.toPresheafedSpace) (hx : x ∈ U), IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)
V : ↥U → Opens ↑↑X.toPresheafedSpace
iVU : (x : ↥U) → V x ⟶ U
m : ∀ (x : ↥U), ↑x ∈ V x
h_unit : ∀ (x : ↥U), IsUnit ((ConcreteCategory.ho... | obtain ⟨gl, gl_spec, -⟩ :
∃ gl : X.presheaf.obj (op U), (∀ i, ((sheaf X).val.map (iVU i).op) gl = g i) ∧ _ :=
X.sheaf.existsUnique_gluing' V U iVU hcover g ic | case intro.intro
X : RingedSpace
U : Opens ↑↑X.toPresheafedSpace
f : ↑(X.presheaf.obj (op U))
h : ∀ (x : ↑↑X.toPresheafedSpace) (hx : x ∈ U), IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)
V : ↥U → Opens ↑↑X.toPresheafedSpace
iVU : (x : ↥U) → V x ⟶ U
m : ∀ (x : ↥U), ↑x ∈ V x
h_unit : ∀ (x : ↥U), IsUnit ((Co... | 06a9663935eb6e01 |
Std.DHashMap.Internal.Raw₀.toListModel_insert | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem toListModel_insert [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α] {m : Raw₀ α β}
(h : Raw.WFImp m.1) {a : α} {b : β a} :
Perm (toListModel (m.insert a b).1.2) (insertEntry a b (toListModel m.1.2)) | α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
h : Raw.WFImp m.val
a : α
b : β a
⊢ toListModel (m.insertₘ a b).val.buckets ~ insertEntry a b (toListModel m.val.buckets) | exact toListModel_insertₘ h | no goals | aa6f496b074b4568 |
Dynamics.coverEntropy_eq_iSup_netEntropyEntourage | Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean | theorem coverEntropy_eq_iSup_netEntropyEntourage :
coverEntropy T F = ⨆ U ∈ 𝓤 X, netEntropyEntourage T F U | case a
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
⊢ netEntropyEntourage T F U ≤ coverEntropy T F | apply (netEntropyEntourage_antitone T F (symmetrizeRel_subset_self U)).trans | case a
X : Type u_1
inst✝ : UniformSpace X
T : X → X
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
⊢ (fun U => netEntropyEntourage T F U) (symmetrizeRel U) ≤ coverEntropy T F | 0de0c2d5957d7ac9 |
Besicovitch.TauPackage.color_lt | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N | case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A✝ : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall... | rw [← color_G a (Nat.lt_succ_iff.1 a.2), ← color_G b (Nat.lt_succ_iff.1 b.2), H] at A | case inr
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A✝ : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall... | 70d4d5d8aa8a7de2 |
Matrix.one_submatrix_mul | Mathlib/Data/Matrix/Mul.lean | theorem one_submatrix_mul [Fintype m] [Finite o] [NonAssocSemiring α] [DecidableEq o] (e₁ : l → o)
(e₂ : m ≃ o) (M : Matrix m n α) :
((1 : Matrix o o α).submatrix e₁ e₂) * M = submatrix M (e₂.symm ∘ e₁) id | case intro
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type v
inst✝³ : Fintype m
inst✝² : Finite o
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq o
e₁ : l → o
e₂ : m ≃ o
M : Matrix m n α
val✝ : Fintype o
A : Matrix o n α := M.submatrix (⇑e₂.symm) id
⊢ submatrix 1 e₁ ⇑e₂ * M = M.submatrix (⇑e₂.symm ∘ e₁) id | have : M = A.submatrix e₂ id := by
simp only [A, submatrix_submatrix, Function.comp_id, submatrix_id_id, Equiv.symm_comp_self] | case intro
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type v
inst✝³ : Fintype m
inst✝² : Finite o
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq o
e₁ : l → o
e₂ : m ≃ o
M : Matrix m n α
val✝ : Fintype o
A : Matrix o n α := M.submatrix (⇑e₂.symm) id
this : M = A.submatrix (⇑e₂) id
⊢ submatrix 1 e₁ ⇑e₂ * M ... | aca44bff6cd3c6ef |
Polynomial.reflect_mul_induction | Mathlib/Algebra/Polynomial/Reverse.lean | theorem reflect_mul_induction (cf cg : ℕ) :
∀ N O : ℕ,
∀ f g : R[X],
#f.support ≤ cf.succ →
#g.support ≤ cg.succ →
f.natDegree ≤ N →
g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g | case neg.a
R : Type u_1
inst✝ : Semiring R
cg : ℕ
hcg :
∀ (N O : ℕ) (f g : R[X]),
#f.support ≤ Nat.succ 0 →
#g.support ≤ cg.succ → f.natDegree ≤ N → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g
N O : ℕ
f g : R[X]
Cf : #f.support ≤ Nat.succ 0
Cg : #g.support ≤ (cg + 1).succ
Nf : f.na... | exact le_add_left card_support_C_mul_X_pow_le_one | no goals | 0a2af005637b9810 |
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