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 |
|---|---|---|---|---|---|---|
map_invOf | Mathlib/Algebra/Group/Invertible/Basic.lean | theorem map_invOf {R : Type*} {S : Type*} {F : Type*} [MulOneClass R] [Monoid S]
[FunLike F R S] [MonoidHomClass F R S] (f : F) (r : R)
[Invertible r] [ifr : Invertible (f r)] :
f (⅟ r) = ⅟ (f r) :=
have h : ifr = Invertible.map f r := Subsingleton.elim _ _
by subst h; rfl
| R : Type u_1
S : Type u_2
F : Type u_3
inst✝⁴ : MulOneClass R
inst✝³ : Monoid S
inst✝² : FunLike F R S
inst✝¹ : MonoidHomClass F R S
f : F
r : R
inst✝ : Invertible r
⊢ f ⅟r = ⅟(f r) | rfl | no goals | b56e565dc22160c7 |
Finset.sum_smul_vsub_eq_affineCombination_vsub | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem sum_smul_vsub_eq_affineCombination_vsub (w : ι → k) (p₁ p₂ : ι → P) :
(∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i)) =
s.affineCombination k p₁ w -ᵥ s.affineCombination k p₂ w | k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
w : ι → k
p₁ p₂ : ι → P
⊢ ∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i) =
(s.weightedVSubOfPoint p₁ (Classical.choice ⋯)) w - (s.weightedVSubOfPoint p₂ (Classical.choice ⋯)) w | exact s.sum_smul_vsub_eq_weightedVSubOfPoint_sub _ _ _ _ | no goals | b90ff7b7d863fa7c |
Std.DHashMap.Raw.insertMany_nil | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem insertMany_nil [EquivBEq α] [LawfulHashable α] (h : m.WF) :
m.insertMany [] = m | α : Type u
β : α → Type v
m : Raw α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
⊢ m.insertMany [] = m | simp_to_raw | α : Type u
β : α → Type v
m : Raw α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
⊢ (Raw₀.insertMany ⟨m, ⋯⟩ []).val.val = m | 56ee60b486ec2b7a |
CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.prod_uniq | Mathlib/CategoryTheory/Localization/Prod.lean | lemma prod_uniq (F₁ F₂ : (W₁.Localization × W₂.Localization ⥤ E))
(h : (W₁.Q.prod W₂.Q) ⋙ F₁ = (W₁.Q.prod W₂.Q) ⋙ F₂) :
F₁ = F₂ | case h.h.h
C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F₁ F₂ : W₁.Localization × W₂.Localization ⥤ E
h : W₁.Q.prod W₂.Q ⋙ F₁ = W₁.Q.prod W₂.Q ⋙ F₂
⊢ (W₁.Q ⋙ curry.obj F₁).flip = (W₁.Q ⋙ cu... | apply Construction.uniq | case h.h.h.h
C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
E : Type u₅
inst✝ : Category.{v₅, u₅} E
F₁ F₂ : W₁.Localization × W₂.Localization ⥤ E
h : W₁.Q.prod W₂.Q ⋙ F₁ = W₁.Q.prod W₂.Q ⋙ F₂
⊢ W₂.Q ⋙ (W₁.Q ⋙ curry.obj F₁).flip = W... | 77a9b99162fee2d1 |
Matroid.Indep.exists_insert_of_not_maximal | Mathlib/Data/Matroid/Basic.lean | theorem Indep.exists_insert_of_not_maximal (M : Matroid α) ⦃I B : Set α⦄ (hI : M.Indep I)
(hInotmax : ¬ Maximal M.Indep I) (hB : Maximal M.Indep B) :
∃ x ∈ B \ I, M.Indep (insert x I) | case refine_2
α : Type u_1
M : Matroid α
I B : Set α
hI : M.Indep I
hB : M.Indep B ∧ ∀ ⦃t : Set α⦄, M.Indep t → B ⊆ t → B = t
hInotmax : ∃ x, M.Indep x ∧ I ⊆ x ∧ ¬I = x
⊢ M.IsBase B | exact hB.1.isBase_of_maximal fun J hJ hBJ ↦ hB.2 hJ hBJ | no goals | 1004af68e323d321 |
Ordinal.unbounded_of_unbounded_sUnion | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem unbounded_of_unbounded_sUnion (r : α → α → Prop) [wo : IsWellOrder α r] {s : Set (Set α)}
(h₁ : Unbounded r <| ⋃₀ s) (h₂ : #s < Order.cof (swap rᶜ)) : ∃ x ∈ s, Unbounded r x | α : Type u
r : α → α → Prop
wo : IsWellOrder α r
s : Set (Set α)
h₁ : Unbounded r (⋃₀ s)
h₂ : #↑s < Order.cof (swap rᶜ)
h : ∀ x ∈ s, ¬Unbounded r x
⊢ False | simp_rw [not_unbounded_iff] at h | α : Type u
r : α → α → Prop
wo : IsWellOrder α r
s : Set (Set α)
h₁ : Unbounded r (⋃₀ s)
h₂ : #↑s < Order.cof (swap rᶜ)
h : ∀ x ∈ s, Bounded r x
⊢ False | 0b92ba7b3f6226a0 |
genericPoint_spec | Mathlib/Topology/Sober.lean | theorem genericPoint_spec [QuasiSober α] [IrreducibleSpace α] :
IsGenericPoint (genericPoint α) univ | α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : QuasiSober α
inst✝ : IrreducibleSpace α
⊢ IsGenericPoint (genericPoint α) univ | simpa using (IrreducibleSpace.isIrreducible_univ α).isGenericPoint_genericPoint_closure | no goals | 639e0a10edbf1834 |
MeasureTheory.Submartingale.sum_mul_sub | Mathlib/Probability/Martingale/Basic.lean | theorem Submartingale.sum_mul_sub [IsFiniteMeasure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ}
(hf : Submartingale f 𝒢 μ) (hξ : Adapted 𝒢 ξ) (hbdd : ∀ n ω, ξ n ω ≤ R)
(hnonneg : ∀ n ω, 0 ≤ ξ n ω) :
Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ | Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
R : ℝ
ξ f : ℕ → Ω → ℝ
hf : Submartingale f 𝒢 μ
hξ : Adapted 𝒢 ξ
hbdd : ∀ (n : ℕ) (ω : Ω), ξ n ω ≤ R
hnonneg : ∀ (n : ℕ) (ω : Ω), 0 ≤ ξ n ω
hξbdd : ∀ (i : ℕ), ∃ C, ∀ (ω : Ω), |ξ i ω| ≤ C
hint : ∀ (m : ℕ), Integrable (∑ k ∈... | exact EventuallyLE.trans (EventuallyLE.mul_nonneg (Eventually.of_forall (hnonneg _))
(hf.condExp_sub_nonneg (Nat.le_succ _))) (condExp_mul_of_stronglyMeasurable_left (hξ _)
(((hf.integrable _).sub (hf.integrable _)).bdd_mul
hξ.stronglyMeasurable.aestronglyMeasurable (hξbdd _))
((hf.integrable _).sub (hf.integ... | no goals | 3f6a4e1e736663ea |
TopCat.range_prod_map | Mathlib/Topology/Category/TopCat/Limits/Products.lean | theorem range_prod_map {W X Y Z : TopCat.{u}} (f : W ⟶ Y) (g : X ⟶ Z) :
Set.range (Limits.prod.map f g) =
(Limits.prod.fst : Y ⨯ Z ⟶ _) ⁻¹' Set.range f ∩
(Limits.prod.snd : Y ⨯ Z ⟶ _) ⁻¹' Set.range g | case h.a.mk.right
W X Y Z : TopCat
f : W ⟶ Y
g : X ⟶ Z
x : ↑(Y ⨯ Z)
x₁ : ↑W
hx₁ : (ConcreteCategory.hom f) x₁ = (ConcreteCategory.hom prod.fst) x
x₂ : ↑X
hx₂ : (ConcreteCategory.hom g) x₂ = (ConcreteCategory.hom prod.snd) x
⊢ (ConcreteCategory.hom (limit.π (pair Y Z) { as := WalkingPair.right }))
((ConcreteCatego... | rw [← ConcreteCategory.comp_apply] | case h.a.mk.right
W X Y Z : TopCat
f : W ⟶ Y
g : X ⟶ Z
x : ↑(Y ⨯ Z)
x₁ : ↑W
hx₁ : (ConcreteCategory.hom f) x₁ = (ConcreteCategory.hom prod.fst) x
x₂ : ↑X
hx₂ : (ConcreteCategory.hom g) x₂ = (ConcreteCategory.hom prod.snd) x
⊢ (ConcreteCategory.hom (prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.right }))
(... | 453c44c742ff64ce |
Equiv.Perm.support_ofSubtype | Mathlib/GroupTheory/Perm/Support.lean | theorem support_ofSubtype {p : α → Prop} [DecidablePred p] (u : Perm (Subtype p)) :
(ofSubtype u).support = u.support.map (Function.Embedding.subtype p) | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
p : α → Prop
inst✝ : DecidablePred p
u : Perm (Subtype p)
⊢ (ofSubtype u).support = map (Embedding.subtype p) u.support | ext x | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Fintype α
p : α → Prop
inst✝ : DecidablePred p
u : Perm (Subtype p)
x : α
⊢ x ∈ (ofSubtype u).support ↔ x ∈ map (Embedding.subtype p) u.support | dd98ff51cd7a9116 |
CoxeterMatrix.reindex_relationsSet | Mathlib/GroupTheory/Coxeter/Basic.lean | theorem reindex_relationsSet :
(M.reindex e).relationsSet =
FreeGroup.freeGroupCongr e '' M.relationsSet := let M' := M.reindex e; calc
Set.range (uncurry M'.relation)
_ = Set.range (uncurry M'.relation ∘ Prod.map e e) | B : Type u_1
B' : Type u_2
M : CoxeterMatrix B
e : B ≃ B'
M' : CoxeterMatrix B' := CoxeterMatrix.reindex e M
⊢ Set.range (⇑(FreeGroup.freeGroupCongr e) ∘ uncurry M.relation) = ⇑(FreeGroup.freeGroupCongr e) '' M.relationsSet | simp [Set.range_comp, relationsSet] | no goals | 806e1030005fed22 |
Nat.le_minFac | Mathlib/Data/Nat/Prime/Defs.lean | theorem le_minFac {m n : ℕ} : n = 1 ∨ m ≤ minFac n ↔ ∀ p, Prime p → p ∣ n → m ≤ p :=
⟨fun h p pp d =>
h.elim (by rintro rfl; cases pp.not_dvd_one d) fun h =>
le_trans h <| minFac_le_of_dvd pp.two_le d,
fun H => or_iff_not_imp_left.2 fun n1 => H _ (minFac_prime n1) (minFac_dvd _)⟩
| m p : ℕ
pp : Prime p
h : 1 = 1 ∨ m ≤ minFac 1
d : p ∣ 1
⊢ m ≤ p | cases pp.not_dvd_one d | no goals | f767657106202d49 |
Ring.toNonUnitalRing_injective | Mathlib/Algebra/Ring/Ext.lean | theorem toNonUnitalRing_injective :
Function.Injective (@toNonUnitalRing R) | R : Type u
a₁✝ a₂✝ : Ring R
h : toNonUnitalRing = toNonUnitalRing
⊢ a₁✝ = a₂✝ | ext x y | case h_add.h.h
R : Type u
a₁✝ a₂✝ : Ring R
h : toNonUnitalRing = toNonUnitalRing
x y : R
⊢ x + y = x + y
case h_mul.h.h
R : Type u
a₁✝ a₂✝ : Ring R
h : toNonUnitalRing = toNonUnitalRing
x y : R
⊢ x * y = x * y | ae697a04b351b7ce |
ProjectiveSpectrum.basicOpen_eq_union_of_projection | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | theorem basicOpen_eq_union_of_projection (f : A) :
basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=
TopologicalSpace.Opens.ext <|
Set.ext fun z => by
rw [mem_coe_basicOpen, mem_coe, iSup, TopologicalSpace.Opens.mem_sSup]
constructor <;> intro hz
· rcases show ∃ i, Graded... | R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
z : ProjectiveSpectrum 𝒜
H : ∀ (i : ℕ), (GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal
⊢ f ∈ z.asHomogeneousIdeal | rw [← DirectSum.sum_support_decompose 𝒜 f] | R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
z : ProjectiveSpectrum 𝒜
H : ∀ (i : ℕ), (GradedAlgebra.proj 𝒜 i) f ∈ z.asHomogeneousIdeal
⊢ ∑ i ∈ DFinsupp.support ((decompose 𝒜) f), ↑(((decompose 𝒜) f) i) ∈ z.asHomogene... | 6fd1b9f7e9f59840 |
Basis.ofEquivFun_equivFun | Mathlib/LinearAlgebra/Basis/Defs.lean | theorem Basis.ofEquivFun_equivFun [Finite ι] (v : Basis ι R M) :
Basis.ofEquivFun v.equivFun = v :=
Basis.repr_injective <| by ext; rfl
| case h.h
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Finite ι
v : Basis ι R M
x✝ : M
a✝ : ι
⊢ ((ofEquivFun v.equivFun).repr x✝) a✝ = (v.repr x✝) a✝ | rfl | no goals | 33b6ae518dfc84ad |
MvPolynomial.map_bind₂ | Mathlib/Algebra/MvPolynomial/Monad.lean | theorem map_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* T) (φ : MvPolynomial σ R) :
map g (bind₂ f φ) = bind₂ ((map g).comp f) φ | σ : Type u_1
R : Type u_3
S : Type u_4
T : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : CommSemiring T
f : R →+* MvPolynomial σ S
g : S →+* T
φ : MvPolynomial σ R
⊢ (map g) ((bind₂ f) φ) = (bind₂ ((map g).comp f)) φ | simp only [bind₂, eval₂_comp_right, coe_eval₂Hom, eval₂_map] | σ : Type u_1
R : Type u_3
S : Type u_4
T : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : CommSemiring T
f : R →+* MvPolynomial σ S
g : S →+* T
φ : MvPolynomial σ R
⊢ eval₂ ((map g).comp f) (⇑(map g) ∘ X) φ = eval₂ ((map g).comp f) X φ | 5dd4e347498a42df |
Real.tan_nonpos_of_nonpos_of_neg_pi_div_two_le | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem tan_nonpos_of_nonpos_of_neg_pi_div_two_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -(π / 2) ≤ x) :
tan x ≤ 0 :=
neg_nonneg.1 (tan_neg x ▸ tan_nonneg_of_nonneg_of_le_pi_div_two (by linarith) (by linarith))
| x : ℝ
hx0 : x ≤ 0
hpx : -(π / 2) ≤ x
⊢ -x ≤ π / 2 | linarith | no goals | 6f0b3450e88de690 |
CategoryTheory.Limits.Types.isLimit_iff | Mathlib/CategoryTheory/Limits/Types.lean | theorem isLimit_iff (c : Cone F) :
Nonempty (IsLimit c) ↔ ∀ s ∈ F.sections, ∃! x : c.pt, ∀ j, c.π.app j x = s j | case refine_1
J : Type v
inst✝ : Category.{w, v} J
F : J ⥤ Type u
c : Cone F
x✝ : Nonempty (IsLimit c)
s : (j : J) → F.obj j
hs : s ∈ F.sections
t : IsLimit c
⊢ ∃! x, ∀ (j : J), c.π.app j x = s j | let cs := coneOfSection hs | case refine_1
J : Type v
inst✝ : Category.{w, v} J
F : J ⥤ Type u
c : Cone F
x✝ : Nonempty (IsLimit c)
s : (j : J) → F.obj j
hs : s ∈ F.sections
t : IsLimit c
cs : Cone F := coneOfSection ⋯
⊢ ∃! x, ∀ (j : J), c.π.app j x = s j | c5a98652924d8bd7 |
Profinite.NobelingProof.GoodProducts.smaller_mono | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem smaller_mono {o₁ o₂ : Ordinal} (h : o₁ ≤ o₂) : smaller C o₁ ⊆ smaller C o₂ | case h.intro.mk
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
l : Products I
gl : Products.isGood (π C fun x => ord I x < o₁) l
⊢ (πs' C h) (eval (π C fun x => ord I x < o₁) ⟨l, gl⟩) ∈ range (π C fun x => ord I x < o₂) ∧
(πs C o₂) ((πs' C h) (eval (π C ... | refine ⟨?_, ?_⟩ | case h.intro.mk.refine_1
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
l : Products I
gl : Products.isGood (π C fun x => ord I x < o₁) l
⊢ (πs' C h) (eval (π C fun x => ord I x < o₁) ⟨l, gl⟩) ∈ range (π C fun x => ord I x < o₂)
case h.intro.mk.refine_2
I :... | bc2bd3c700f8cd86 |
MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_eLpNorm | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem completeSpace_lp_of_cauchy_complete_eLpNorm [hp : Fact (1 ≤ p)]
(H :
∀ (f : ℕ → α → E) (_ : ∀ n, MemLp (f n) p μ) (B : ℕ → ℝ≥0∞) (_ : ∑' i, B i < ∞)
(_ : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N),
∃ (f_lim : α → E), MemLp f_lim p μ ∧
atTop.Tendsto (fun n => ... | case intro
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
∀ (f : ℕ → α → E),
(∀ (n : ℕ), MemLp (f n) p μ) →
∀ (B : ℕ → ℝ≥0∞),
∑' (i : ℕ), B i < ⊤ →
(∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
... | refine H f1 (fun n => Lp.memLp (f n)) B1 hB1 fun N n m hn hm => ?_ | case intro
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
∀ (f : ℕ → α → E),
(∀ (n : ℕ), MemLp (f n) p μ) →
∀ (B : ℕ → ℝ≥0∞),
∑' (i : ℕ), B i < ⊤ →
(∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
... | 3439de49b0148c06 |
AddDissociated.randomisation | Mathlib/Combinatorics/Additive/Randomisation.lean | /-- One can randomise by a function of dissociated support. -/
lemma AddDissociated.randomisation (c : AddChar G ℂ → ℝ) (d : AddChar G ℂ → ℂ)
(hcd : AddDissociated {ψ | d ψ ≠ 0}) : 𝔼 a, ∏ ψ, (c ψ + (d ψ * ψ a).re) = ∏ ψ, c ψ | G : Type u_1
inst✝¹ : Fintype G
inst✝ : AddCommGroup G
c : AddChar G ℂ → ℝ
d : AddChar G ℂ → ℂ
hcd : AddDissociated {ψ | d ψ ≠ 0}
t : Finset (AddChar G ℂ)
ht : t ≠ ∅
⊢ ∑ j ∈ t.powerset,
𝔼 i : G, (∏ i_1 ∈ j, d i_1 * i_1 i) * ∏ i_1 ∈ t \ j, (starRingEnd ℂ) (d i_1) * (starRingEnd ℂ) (i_1 i) =
0 | refine sum_eq_zero fun u _ ↦ ?_ | G : Type u_1
inst✝¹ : Fintype G
inst✝ : AddCommGroup G
c : AddChar G ℂ → ℝ
d : AddChar G ℂ → ℂ
hcd : AddDissociated {ψ | d ψ ≠ 0}
t : Finset (AddChar G ℂ)
ht : t ≠ ∅
u : Finset (AddChar G ℂ)
x✝ : u ∈ t.powerset
⊢ 𝔼 i : G, (∏ i_1 ∈ u, d i_1 * i_1 i) * ∏ i_1 ∈ t \ u, (starRingEnd ℂ) (d i_1) * (starRingEnd ℂ) (i_1 i) = 0 | fbcadeff10f3d7bb |
ProjectiveSpectrum.basicOpen_mul_le_right | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | theorem basicOpen_mul_le_right (f g : A) : basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g | R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f g : A
⊢ basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g | rw [basicOpen_mul 𝒜 f g] | R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f g : A
⊢ basicOpen 𝒜 f ⊓ basicOpen 𝒜 g ≤ basicOpen 𝒜 g | 8f952c2146bdb648 |
spinGroup.star_mem | Mathlib/LinearAlgebra/CliffordAlgebra/SpinGroup.lean | theorem star_mem {x : CliffordAlgebra Q} (hx : x ∈ spinGroup Q) : star x ∈ spinGroup Q | case intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
hx₁ : x ∈ pinGroup Q
hx₂ : x ∈ even Q
⊢ star x ∈ even Q | dsimp only [CliffordAlgebra.even] at hx₂ ⊢ | case intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
hx₁ : x ∈ pinGroup Q
hx₂ : x ∈ (evenOdd Q 0).toSubalgebra ⋯ ⋯
⊢ star x ∈ (evenOdd Q 0).toSubalgebra ⋯ ⋯ | c11cc64e31318b0a |
AlgebraicGeometry.HasRingHomProperty.of_source_openCover | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | theorem of_source_openCover [IsAffine Y]
(𝒰 : X.OpenCover) [∀ i, IsAffine (𝒰.obj i)] (H : ∀ i, Q ((𝒰.map i ≫ f).appTop.hom)) :
P f | case openCover
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝² : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsAffine Y
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
H✝ : ∀ (i : 𝒰.J), Q (CommRingCat.Hom.hom (Scheme.Hom.appTo... | rintro r | case openCover
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝² : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsAffine Y
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
H✝ : ∀ (i : 𝒰.J), Q (CommRingCat.Hom.hom (Scheme.Hom.appTo... | 14bf5a1ed5917276 |
Matroid.dual_isBase_iff' | Mathlib/Data/Matroid/Dual.lean | theorem dual_isBase_iff' : M✶.IsBase B ↔ M.IsBase (M.E \ B) ∧ B ⊆ M.E :=
(em (B ⊆ M.E)).elim (fun h ↦ by rw [dual_isBase_iff, and_iff_left h])
(fun h ↦ iff_of_false (h ∘ (fun h' ↦ h'.subset_ground)) (h ∘ And.right))
| α : Type u_1
M : Matroid α
B : Set α
h : B ⊆ M.E
⊢ M✶.IsBase B ↔ M.IsBase (M.E \ B) ∧ B ⊆ M.E | rw [dual_isBase_iff, and_iff_left h] | no goals | 8aaf1aa15f91ec12 |
Filter.HasBasis.inf' | Mathlib/Order/Filter/Bases.lean | theorem HasBasis.inf' (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') :
(l ⊓ l').HasBasis (fun i : PProd ι ι' => p i.1 ∧ p' i.2) fun i => s i.1 ∩ s' i.2 :=
⟨by
intro t
constructor
· simp only [mem_inf_iff, hl.mem_iff, hl'.mem_iff]
rintro ⟨t, ⟨i, hi, ht⟩, t', ⟨i', hi', ht'⟩, rfl⟩
exact ⟨⟨i,... | α : Type u_1
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s : ι → Set α
p' : ι' → Prop
s' : ι' → Set α
hl : l.HasBasis p s
hl' : l'.HasBasis p' s'
⊢ ∀ (t : Set α), t ∈ l ⊓ l' ↔ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t | intro t | α : Type u_1
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s : ι → Set α
p' : ι' → Prop
s' : ι' → Set α
hl : l.HasBasis p s
hl' : l'.HasBasis p' s'
t : Set α
⊢ t ∈ l ⊓ l' ↔ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t | 51e3b9dfa5727c3a |
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) | 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 ... | let B₀ := volume A | 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 ... | e1b4f22b1d3b4e6d |
PFunctor.M.mk_dest | Mathlib/Data/PFunctor/Univariate/M.lean | theorem mk_dest (x : M F) : M.mk (dest x) = x | case H.succ
F : PFunctor.{u}
x : F.M
n : ℕ
a✝ : Approx.sMk x.dest n = x.approx n
⊢ (CofixA.intro (head' (x.approx 1)) fun i => (x.children i).approx n) = x.approx (n + 1) | rcases h : x.approx (succ n) with - | ⟨hd, ch⟩ | case H.succ.intro
F : PFunctor.{u}
x : F.M
n : ℕ
a✝ : Approx.sMk x.dest n = x.approx n
hd : F.A
ch : F.B hd → CofixA F n
h : x.approx n.succ = CofixA.intro hd ch
⊢ (CofixA.intro (head' (x.approx 1)) fun i => (x.children i).approx n) = CofixA.intro hd ch | 475d78b79340343f |
isLocallyInjective_iff_isOpen_diagonal | Mathlib/Topology/SeparatedMap.lean | theorem isLocallyInjective_iff_isOpen_diagonal {f : X → Y} :
IsLocallyInjective f ↔ IsOpen f.pullbackDiagonal | X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ (∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U) ↔
∀ (a b : X) (b_1 : f a = f b),
⟨(a, b), b_1⟩ ∈ Function.pullbackDiagonal f → ∃ t ∈ 𝓝 a ×ˢ 𝓝 b, Subtype.val ⁻¹' t ⊆ Function.pullbackDiagonal f | refine ⟨?_, fun h x ↦ ?_⟩ | case refine_1
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
⊢ (∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U) →
∀ (a b : X) (b_1 : f a = f b),
⟨(a, b), b_1⟩ ∈ Function.pullbackDiagonal f → ∃ t ∈ 𝓝 a ×ˢ 𝓝 b, Subtype.val ⁻¹' t ⊆ Function.pullbackDiagonal f
case refine_2
X : Type u_1
Y : Type u_2
inst... | 81b1cbf2345d2091 |
MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 | case refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ ... | exact hf_zero _ (hs.inter hf.measurableSet) hμs | no goals | 0f1e714a9354b744 |
lowerClosure_image | Mathlib/Order/UpperLower/Basic.lean | theorem lowerClosure_image (f : α ≃o β) :
lowerClosure (f '' s) = LowerSet.map f (lowerClosure s) | α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
f : α ≃o β
⊢ lowerClosure (⇑f '' s) = (LowerSet.map f.symm).symm (lowerClosure s) | ext | case a.h
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
s : Set α
f : α ≃o β
x✝ : β
⊢ x✝ ∈ ↑(lowerClosure (⇑f '' s)) ↔ x✝ ∈ ↑((LowerSet.map f.symm).symm (lowerClosure s)) | c48ad1d3f84d1cab |
weightedVSub_mem_vectorSpan_pair | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem weightedVSub_mem_vectorSpan_pair {p : ι → P} (h : AffineIndependent k p) {w w₁ w₂ : ι → k}
{s : Finset ι} (hw : ∑ i ∈ s, w i = 0) (hw₁ : ∑ i ∈ s, w₁ i = 1)
(hw₂ : ∑ i ∈ s, w₂ i = 1) :
s.weightedVSub p w ∈
vectorSpan k ({s.affineCombination k p w₁, s.affineCombination k p w₂} : Set P) ↔
... | case refine_1.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 0
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
r : k
hr : r • ((affineCombin... | rw [s.affineCombination_vsub, ← s.weightedVSub_const_smul, ← sub_eq_zero, ← map_sub] at hr | case refine_1.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 0
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
r : k
hr : (s.weightedVSub p)... | 6e1484c0d97194ef |
exists_maximalIdeal_pow_eq_of_principal | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [IsLocalRing R] [IsDomain R]
(h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
∃ n : ℕ, I = maximalIdeal R ^ n | R : Type u_1
inst✝³ : CommRing R
inst✝² : IsNoetherianRing R
inst✝¹ : IsLocalRing R
inst✝ : IsDomain R
I : Ideal R
hI : I ≠ ⊥
h : ¬IsField R
x : R
hx : maximalIdeal R = Ideal.span {x}
hI' : ¬I = ⊤
H : ∀ (r : R), ¬IsUnit r ↔ x ∣ r
this : x ≠ 0
hx' : Irreducible x
f : Multiset R
hf₁ : ∀ b ∈ f, Irreducible b
hf₂ : f ≠ ∅
h... | exact Irreducible.associated_of_dvd hx' (hf₁ b hb) ((H b).mp (hf₁ b hb).1) | no goals | 70b9173581ff941a |
Function.IsPeriodicPt.comp | Mathlib/Dynamics/PeriodicPts/Defs.lean | theorem comp {g : α → α} (hco : Commute f g) (hf : IsPeriodicPt f n x) (hg : IsPeriodicPt g n x) :
IsPeriodicPt (f ∘ g) n x | α : Type u_1
f : α → α
x : α
n : ℕ
g : α → α
hco : Function.Commute f g
hf : IsPeriodicPt f n x
hg : IsPeriodicPt g n x
⊢ IsFixedPt (f^[n] ∘ g^[n]) x | exact IsFixedPt.comp hf hg | no goals | 4c76f2f6b5c2adcd |
AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono_naturality | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | theorem mapMono_naturality (i : Δ ⟶ Δ') [Mono i] :
mapMono K i ≫ f.f Δ.len = f.f Δ'.len ≫ mapMono K' i | case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' : SimplexCategory
i : Δ ⟶ Δ'
inst✝ : Mono i
h : ¬Δ' = Δ
h✝ : ¬Isδ₀ i
⊢ 0 ≫ f.f Δ.len = f.f Δ'.len ≫ 0 | rw [zero_comp, comp_zero] | no goals | 06c5a28184e2d168 |
segment_inter_eq_endpoint_of_linearIndependent_of_ne | Mathlib/Analysis/Convex/Segment.lean | lemma segment_inter_eq_endpoint_of_linearIndependent_of_ne [OrderedCommRing 𝕜] [NoZeroDivisors 𝕜]
[AddCommGroup E] [Module 𝕜 E]
{x y : E} (h : LinearIndependent 𝕜 ![x, y]) {s t : 𝕜} (hs : s ≠ t) (c : E) :
[c + x -[𝕜] c + t • y] ∩ [c + x -[𝕜] c + s • y] = {c + x} | case h
𝕜 : Type u_1
E : Type u_2
inst✝³ : OrderedCommRing 𝕜
inst✝² : NoZeroDivisors 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y : E
h : LinearIndependent 𝕜 ![x, y]
s t : 𝕜
c : E
hs : -1 * s - t * -1 = 0
⊢ s = t | apply Eq.symm | case h.h
𝕜 : Type u_1
E : Type u_2
inst✝³ : OrderedCommRing 𝕜
inst✝² : NoZeroDivisors 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y : E
h : LinearIndependent 𝕜 ![x, y]
s t : 𝕜
c : E
hs : -1 * s - t * -1 = 0
⊢ t = s | c33bd0ba7e6d724a |
AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean | @[reassoc]
lemma evaluation_naturality {V : Opens Y} (x : (Opens.map f.base).obj V) :
Y.evaluation ⟨f.base x, x.property⟩ ≫ residueFieldMap f x.val =
f.c.app (op V) ≫ X.evaluation x | case hf.a
X Y : LocallyRingedSpace
f : X ⟶ Y
V : Opens ↑Y.toTopCat
x : ↥((Opens.map f.base).obj V)
a : ↑(Y.presheaf.obj (op V))
⊢ (CommRingCat.Hom.hom
(Y.presheaf.germ V ((ConcreteCategory.hom f.base) ↑x) ⋯ ≫
CommRingCat.ofHom (IsLocalRing.residue ↑(Y.presheaf.stalk ((ConcreteCategory.hom f.base) ↑x))... | simp only [CommRingCat.comp_apply] | case hf.a
X Y : LocallyRingedSpace
f : X ⟶ Y
V : Opens ↑Y.toTopCat
x : ↥((Opens.map f.base).obj V)
a : ↑(Y.presheaf.obj (op V))
⊢ (ConcreteCategory.hom (CommRingCat.ofHom (IsLocalRing.ResidueField.map (CommRingCat.Hom.hom (Hom.stalkMap f ↑x)))))
((ConcreteCategory.hom
(CommRingCat.ofHom (IsLocalRing.res... | d8b83a9f05602449 |
Grp_.isPullback | Mathlib/CategoryTheory/Monoidal/Grp_.lean | theorem isPullback (A : Grp_ C) :
IsPullback (A.mul ▷ A.X) ((α_ A.X A.X A.X).hom ≫ (A.X ◁ A.mul)) A.mul A.mul where
w | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : ChosenFiniteProducts C
A : Grp_ C
s : PullbackCone A.mul A.mul
⊢ ((fun s =>
lift (lift (s.snd ≫ fst A.X A.X) (lift (s.snd ≫ fst A.X A.X ≫ A.inv) (s.fst ≫ fst A.X A.X) ≫ A.mul))
(s.fst ≫ snd A.X A.X))
s ≫
(α_ A.X A.X A.X).hom ≫ ... | simp | no goals | d4cb31ef2ed58598 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_go_eq_divRec_q | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_go_eq_divRec_q (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case zero
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
aig : AIG α
rbv qbv : BitVec w
falseRef trueRef : aig.Ref
n d q r : aig.RefVec w
wn wr : Nat
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx... | simp [go, hq] | no goals | edb5936f3962e4a9 |
integrableOn_Icc_iff_integrableOn_Ioc' | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ | case neg
α : Type u_1
E : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : μ {a} ≠ ⊤
hab : ¬a ≤ b
⊢ ¬a < b | contrapose! hab | case neg
α : Type u_1
E : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : μ {a} ≠ ⊤
hab : a < b
⊢ a ≤ b | ed88dff1321302d6 |
TopCat.GlueData.ι_eq_iff_rel | Mathlib/Topology/Gluing.lean | theorem ι_eq_iff_rel (i j : D.J) (x : D.U i) (y : D.U j) :
𝖣.ι i x = 𝖣.ι j y ↔ D.Rel ⟨i, x⟩ ⟨j, y⟩ | case mpr.intro.intro
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
z : ↑(D.V (⟨i, x⟩.fst, ⟨j, y⟩.fst))
e₁ : (ConcreteCategory.hom (D.f i j)) z = x
e₂ : (ConcreteCategory.hom (D.f j i)) ((ConcreteCategory.hom (D.t i j)) z) = y
⊢ (ConcreteCategory.hom (D.ι i)) x = (ConcreteCategory.hom (D.ι j)) y | rw [← e₁, ← e₂] at * | case mpr.intro.intro
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
z : ↑(D.V (⟨i, x⟩.fst, ⟨j, y⟩.fst))
e₁ : (ConcreteCategory.hom (D.f i j)) z = (ConcreteCategory.hom (D.f i j)) z
e₂ :
(ConcreteCategory.hom (D.f j i)) ((ConcreteCategory.hom (D.t i j)) z) =
(ConcreteCategory.hom (D.f j i)) ((ConcreteCategory.ho... | e0a110dda282edcf |
Std.Sat.AIG.RefVec.ite.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/If.lean | theorem go_get_aux {w : Nat} (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (discr : Ref aig)
(lhs rhs : RefVec aig w) (s : RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr) (hfoo),
(go aig curr hcurr discr lhs rhs s).vec.get idx (by omega)
=
(s.get idx hidx).cast hfoo | case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α w
h✝ : curr < w
hgo :
go (aig.mkIfCached { discr := discr, lhs := lhs.get curr h✝, rhs := rhs.get cu... | rw [go_get_aux] | case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α w
h✝ : curr < w
hgo :
go (aig.mkIfCached { discr := discr, lhs := lhs.get curr h✝, rhs := rhs.get cu... | c08e573c9b942d4a |
CondensedSet.isDiscrete_tfae | Mathlib/Condensed/Discrete/Characterization.lean | theorem isDiscrete_tfae (X : CondensedSet.{u}) :
TFAE
[ X.IsDiscrete
, IsIso ((Condensed.discreteUnderlyingAdj _).counit.app X)
, (Condensed.discrete _).essImage X
, CondensedSet.LocallyConstant.functor.essImage X
, IsIso (CondensedSet.LocallyConstant.adjunction.counit.app X)
, Sheaf.IsCons... | X : CondensedSet
tfae_1_iff_2 : Condensed.IsDiscrete X ↔ IsIso ((discreteUnderlyingAdj (Type (u + 1))).counit.app X)
tfae_1_iff_3 : Condensed.IsDiscrete X ↔ (discrete (Type (u + 1))).essImage X
tfae_1_iff_4 : Condensed.IsDiscrete X ↔ LocallyConstant.functor.essImage X
tfae_1_iff_5 : Condensed.IsDiscrete X ↔ IsIso (Loca... | tfae_finish | no goals | b6f90d9e4c32554e |
Order.IsSuccPrelimit.isMin_of_noMax | Mathlib/Order/SuccPred/Limit.lean | theorem IsSuccPrelimit.isMin_of_noMax (h : IsSuccPrelimit a) : IsMin a | case intro.zero
α : Type u_1
inst✝³ : Preorder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : NoMaxOrder α
b : α
h : IsSuccPrelimit (succ^[0] b)
hb : b ≤ succ^[0] b
⊢ succ^[0] b ≤ b | exact le_rfl | no goals | c030a4a9f9ed950a |
PresheafOfModules.freeYoneda.isSeparating | Mathlib/Algebra/Category/ModuleCat/Presheaf/Generator.lean | lemma isSeparating : IsSeparating (freeYoneda R) | case h.op.hf.h
C : Type u
inst✝ : Category.{v, u} C
R : Cᵒᵖ ⥤ RingCat
M N : PresheafOfModules R
f₁ f₂ : M ⟶ N
h : ∀ G ∈ freeYoneda R, ∀ (h : G ⟶ M), h ≫ f₁ = h ≫ f₂
X : C
m : ↑(M.obj (Opposite.op X))
⊢ (ModuleCat.Hom.hom (f₁.app (Opposite.op X))) m = (ModuleCat.Hom.hom (f₂.app (Opposite.op X))) m | obtain ⟨g, rfl⟩ := freeYonedaEquiv.surjective m | case h.op.hf.h.intro
C : Type u
inst✝ : Category.{v, u} C
R : Cᵒᵖ ⥤ RingCat
M N : PresheafOfModules R
f₁ f₂ : M ⟶ N
h : ∀ G ∈ freeYoneda R, ∀ (h : G ⟶ M), h ≫ f₁ = h ≫ f₂
X : C
g : (free R).obj (yoneda.obj X) ⟶ M
⊢ (ModuleCat.Hom.hom (f₁.app (Opposite.op X))) (freeYonedaEquiv g) =
(ModuleCat.Hom.hom (f₂.app (Opposi... | 39cfcb40038c27d5 |
Filter.eventually_lt_of_lt_liminf | Mathlib/Order/LiminfLimsup.lean | theorem eventually_lt_of_lt_liminf {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
{b : β} (h : b < liminf u f)
(hu : f.IsBoundedUnder (· ≥ ·) u | α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) _auto✝
⊢ ∀ᶠ (a : α) in f, b < u a | obtain ⟨c, hc, hbc⟩ : ∃ (c : β) (_ : c ∈ { c : β | ∀ᶠ n : α in f, c ≤ u n }), b < c := by
simp_rw [exists_prop]
exact exists_lt_of_lt_csSup hu h | case intro.intro
α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) _auto✝
c : β
hc : c ∈ {c | ∀ᶠ (n : α) in f, c ≤ u n}
hbc : b < c
⊢ ∀ᶠ (a : α) in f, b < u a | 82ef5ffe328c509b |
Nat.lt_of_testBit | Mathlib/Data/Nat/Bitwise.lean | theorem lt_of_testBit {n m : ℕ} (i : ℕ) (hn : testBit n i = false) (hm : testBit m i = true)
(hnm : ∀ j, i < j → testBit n j = testBit m j) : n < m | case neg
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), n.testBit i = false → m.testBit i = true → (∀ (j : ℕ), i < j → n.testBit j = m.testBit j) → n < m
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
(bit b n).testBit i = false →
m.testBit i = true → (∀ (j : ℕ), i < j → (bit b n).testBit j = m.testBit j) → bit b n < m
i : ℕ
... | obtain ⟨i', rfl⟩ := exists_eq_succ_of_ne_zero hi | case neg.intro
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), n.testBit i = false → m.testBit i = true → (∀ (j : ℕ), i < j → n.testBit j = m.testBit j) → n < m
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
(bit b n).testBit i = false →
m.testBit i = true → (∀ (j : ℕ), i < j → (bit b n).testBit j = m.testBit j) → bit b n < m
... | 6ffde8c2e514a295 |
Array.getElem_insertIdx_loop_gt | Mathlib/.lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | theorem getElem_insertIdx_loop_gt {as : Array α} {i : Nat} {j : Nat} {hj : j < as.size}
{k : Nat} {h} (w : i < k) :
(insertIdx.loop i as ⟨j, hj⟩)[k] =
if k ≤ j then as[k-1]'(by simp at h; omega) else as[k]'(by simpa using h) | case isTrue.isFalse.isTrue
α : Type u_1
as : Array α
i j : Nat
hj : j < as.size
k : Nat
h : k < (insertIdx.loop i as ⟨j, hj⟩).size
w : i < k
h₁ : i < j
this : j - 1 < j
h₂ : ¬k ≤ j - 1
h₃ : k = j
⊢ as[j - 1] = if k ≤ j then as[k - 1] else as[k] | simp [h₃] | no goals | ab494e4c94494354 |
List.Vector.reverse_snoc | Mathlib/Data/Vector/Snoc.lean | theorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) | case mk
α : Type u_1
n : ℕ
x : α
val✝ : List α
property✝ : val✝.length = n
⊢ (snoc ⟨val✝, property✝⟩ x).reverse = x ::ᵥ reverse ⟨val✝, property✝⟩ | simp only [reverse, snoc, cons, toList_mk] | case mk
α : Type u_1
n : ℕ
x : α
val✝ : List α
property✝ : val✝.length = n
⊢ ⟨(append ⟨val✝, property✝⟩ ⟨[x], ⋯⟩).toList.reverse, ⋯⟩ = ⟨x :: val✝.reverse, ⋯⟩ | a4d200fb11dfb707 |
Real.cos_le_one_div_sqrt_sq_add_one | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | theorem cos_le_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2) :
cos x ≤ (1 : ℝ) / √(x ^ 2 + 1) | x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
⊢ cos x ≤ 1 / √(x ^ 2 + 1) | rcases eq_or_ne x 0 with (rfl | hx3) | case inl
hx1 : -(3 * π / 2) ≤ 0
hx2 : 0 ≤ 3 * π / 2
⊢ cos 0 ≤ 1 / √(0 ^ 2 + 1)
case inr
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
⊢ cos x ≤ 1 / √(x ^ 2 + 1) | df34c19df6902c91 |
IsPreirreducible.subset_irreducible | Mathlib/Topology/Irreducible.lean | theorem IsPreirreducible.subset_irreducible {S U : Set X} (ht : IsPreirreducible t)
(hU : U.Nonempty) (hU' : IsOpen U) (h₁ : U ⊆ S) (h₂ : S ⊆ t) : IsIrreducible S | case intro
X : Type u_1
inst✝ : TopologicalSpace X
t S U : Set X
hU' : IsOpen U
h₁ : U ⊆ S
h₂ : S ⊆ t
z : X
hz : z ∈ U
ht : IsIrreducible t
⊢ IsPreirreducible S | rintro u v hu hv ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩ | case intro.intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
t S U : Set X
hU' : IsOpen U
h₁ : U ⊆ S
h₂ : S ⊆ t
z : X
hz : z ∈ U
ht : IsIrreducible t
u v : Set X
hu : IsOpen u
hv : IsOpen v
x : X
hx : x ∈ S
hx' : x ∈ u
y : X
hy : y ∈ S
hy' : y ∈ v
⊢ (S ∩ (u ∩ v)).Nonempty | 1b6e5fec17eba115 |
seminormFromBounded_le | Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean | theorem seminormFromBounded_le (f_nonneg : 0 ≤ f)
(f_mul : ∀ x y : R, f (x * y) ≤ c * f x * f y) (x : R) :
seminormFromBounded' f x ≤ c * f x | case inr
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x y : R
hy : 0 y < f y
⊢ f (x * y) / f y ≤ c * f x | rw [div_le_iff₀ hy] | case inr
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x y : R
hy : 0 y < f y
⊢ f (x * y) ≤ c * f x * f y | d145aa743e0a78d0 |
MeasurableEmbedding.rnDeriv_map_aux | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | lemma _root_.MeasurableEmbedding.rnDeriv_map_aux (hf : MeasurableEmbedding f)
(hμν : μ ≪ ν) [SigmaFinite μ] [SigmaFinite ν] :
(fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν | case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ⊤
⊢ ∫⁻ (a : β), (map f μ).rnDeriv (map f ν) a ∂map f (ν.restrict s) = μ s | have hs_eq : s = f ⁻¹' (f '' s) := by rw [hf.injective.preimage_image] | case refine_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
hμν : μ ≪ ν
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
s : Set α
x✝¹ : MeasurableSet s
x✝ : ν s < ⊤
hs_eq : s = f ⁻¹' (f '' s)
⊢ ∫⁻ (a : β), (map f μ).rnDeriv (map f ν) a ∂map f (ν... | 57b2d54254d50ee8 |
HasFiniteFPowerSeriesOnBall.add | Mathlib/Analysis/Analytic/CPolynomial.lean | theorem HasFiniteFPowerSeriesOnBall.add (hf : HasFiniteFPowerSeriesOnBall f pf x n r)
(hg : HasFiniteFPowerSeriesOnBall g pg x m r) :
HasFiniteFPowerSeriesOnBall (f + g) (pf + pg) x (max n m) r :=
⟨hf.1.add hg.1, fun N hN ↦ by
rw [Pi.add_apply, hf.finite _ ((le_max_left n m).trans hN),
hg.finite _... | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
n m : ℕ
hf : HasFiniteFPowerSeriesOnBall f pf x n r
hg : HasFiniteF... | rw [Pi.add_apply, hf.finite _ ((le_max_left n m).trans hN),
hg.finite _ ((le_max_right n m).trans hN), zero_add] | no goals | 4efa65f57ea6e764 |
PartialHomeomorph.extend_preimage_mem_nhds | Mathlib/Geometry/Manifold/IsManifold/ExtChartAt.lean | theorem extend_preimage_mem_nhds {x : M} (h : x ∈ f.source) (ht : t ∈ 𝓝 x) :
(f.extend I).symm ⁻¹' t ∈ 𝓝 (f.extend I x) | 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
t : Set M
x : M
h : x ∈ f.source
ht : t ∈ 𝓝 x
⊢ ↑(f.extend I).sym... | apply (continuousAt_extend_symm f h).preimage_mem_nhds | 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
t : Set M
x : M
h : x ∈ f.source
ht : t ∈ 𝓝 x
⊢ t ∈ 𝓝 (↑(f.exten... | 63da83ff986fd329 |
AkraBazziRecurrence.growsPolynomially_log | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma growsPolynomially_log : GrowsPolynomially Real.log | b : ℝ
hb : b ∈ Set.Ioo 0 1
hb₀ : 0 < b
h_tendsto : Tendsto (fun x => 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ x ≠ 0 | positivity | no goals | 441cf9a25600ec9c |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRupUnits_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem assignmentsInvariant_insertRupUnits_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRupUnits f units).1 | case neg
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize... | apply Or.inl | case neg.h
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsi... | 7537b9cac0fe9952 |
List.intercalate_splitOn | Mathlib/Data/List/SplitOn.lean | theorem intercalate_splitOn (x : α) [DecidableEq α] : [x].intercalate (xs.splitOn x) = xs | case pos
α : Type u_1
xs : List α
inst✝ : DecidableEq α
hd : α
tl hd' : List α
tl' : List (List α)
ih : (intersperse [hd] (hd' :: tl')).flatten = tl
h' : splitOnP (fun x => x == hd) tl = hd' :: tl'
⊢ (intersperse [hd] ([] :: splitOnP (fun x => x == hd) tl)).flatten = hd :: tl | simp [ih, flatten, h'] | no goals | c6930e97e5414b52 |
List.foldr_max_of_ne_nil | Mathlib/Data/List/MinMax.lean | theorem foldr_max_of_ne_nil (h : l ≠ []) : ↑(l.foldr max ⊥) = l.maximum | case neg
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : OrderBot α
l : List α
hd : α
tl : List α
IH : tl ≠ [] → ↑(foldr max ⊥ tl) = tl.maximum
h✝ : hd :: tl ≠ []
h : ¬tl = []
⊢ ↑hd ⊔ ↑(foldr max ⊥ tl) = ↑hd ⊔ tl.maximum | simp [IH h] | no goals | ad38d95326b26928 |
PFun.fix_fwd_eq | Mathlib/Data/PFun.lean | theorem fix_fwd_eq {f : α →. β ⊕ α} {a a' : α} (ha' : Sum.inr a' ∈ f a) : f.fix a = f.fix a' | case H.mpr
α : Type u_1
β : Type u_2
f : α →. β ⊕ α
a a' : α
ha' : Sum.inr a' ∈ f a
b : β
h : b ∈ f.fix a'
⊢ Sum.inl b ∈ f a ∨ ∃ a', Sum.inr a' ∈ f a ∧ b ∈ f.fix a' | exact Or.inr ⟨a', ha', h⟩ | no goals | daf82381c1d645b7 |
Ordinal.nfpFamily_le_apply | Mathlib/SetTheory/Ordinal/FixedPoint.lean | theorem nfpFamily_le_apply [Nonempty ι] [Small.{u} ι] (H : ∀ i, IsNormal (f i)) {a b} :
(∃ i, nfpFamily f a ≤ f i b) ↔ nfpFamily f a ≤ b | ι : Type u_1
f : ι → Ordinal.{u} → Ordinal.{u}
inst✝¹ : Nonempty ι
inst✝ : Small.{u, u_1} ι
H : ∀ (i : ι), IsNormal (f i)
a b : Ordinal.{u}
⊢ (¬∃ i, nfpFamily f a ≤ f i b) ↔ ¬nfpFamily f a ≤ b | push_neg | ι : Type u_1
f : ι → Ordinal.{u} → Ordinal.{u}
inst✝¹ : Nonempty ι
inst✝ : Small.{u, u_1} ι
H : ∀ (i : ι), IsNormal (f i)
a b : Ordinal.{u}
⊢ (∀ (i : ι), f i b < nfpFamily f a) ↔ b < nfpFamily f a | 195555708ee22d6f |
LocallyConstant.exists_eq_const | Mathlib/Topology/LocallyConstant/Basic.lean | theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] (f : LocallyConstant X Y) :
∃ y, f = const X y | case inl.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : PreconnectedSpace X
inst✝ : Nonempty Y
f : LocallyConstant X Y
x : X
⊢ ∃ y, f = const X y | exact ⟨f x, f.eq_const x⟩ | no goals | 90c6654a6065eac3 |
PseudoMetricSpace.le_two_mul_dist_ofPreNNDist | Mathlib/Topology/Metrizable/Uniformity.lean | theorem le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x)
(hd : ∀ x₁ x₂ x₃ x₄, d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
↑(d x y) ≤ 2 * @dist X
(@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self d... | case intro.intro.refine_3
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * (d x₁ x₂ ⊔ (d x₂ x₃ ⊔ d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length... | rcases hMl.eq_or_lt with (rfl | hMl) | case intro.intro.refine_3.inl
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * (d x₁ x₂ ⊔ (d x₂ x₃ ⊔ d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < l.le... | a217eb6776c388b0 |
Matrix.cons_mulVec | Mathlib/Data/Matrix/Notation.lean | theorem cons_mulVec [Fintype n'] (v : n' → α) (A : Fin m → n' → α) (w : n' → α) :
(of <| vecCons v A) *ᵥ w = vecCons (dotProduct v w) (of A *ᵥ w) | α : Type u
m : ℕ
n' : Type uₙ
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : Fintype n'
v : n' → α
A : Fin m → n' → α
w : n' → α
⊢ of (vecCons v A) *ᵥ w = vecCons (v ⬝ᵥ w) (of A *ᵥ w) | ext i | case h
α : Type u
m : ℕ
n' : Type uₙ
inst✝¹ : NonUnitalNonAssocSemiring α
inst✝ : Fintype n'
v : n' → α
A : Fin m → n' → α
w : n' → α
i : Fin m.succ
⊢ (of (vecCons v A) *ᵥ w) i = vecCons (v ⬝ᵥ w) (of A *ᵥ w) i | f9db0601dbd48e70 |
List.Vector.mapAccumr_bisim | Mathlib/Data/Vector/MapLemmas.lean | theorem mapAccumr_bisim {f₁ : α → σ₁ → σ₁ × β} {f₂ : α → σ₂ → σ₂ × β} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {s q} a, R s q → R (f₁ a s).1 (f₂ a q).1 ∧ (f₁ a s).2 = (f₂ a q).2) :
R (mapAccumr f₁ xs s₁).fst (mapAccumr f₂ xs s₂).fst
∧ (mapAccumr f₁ xs s₁).snd = (mapAccumr f₂ xs s₂... | case snoc
α : Type u_1
β : Type u_2
σ₁ : Type u_6
σ₂ : Type u_7
n : ℕ
xs : Vector α n
f₁ : α → σ₁ → σ₁ × β
f₂ : α → σ₂ → σ₂ × β
R : σ₁ → σ₂ → Prop
hR : ∀ {s : σ₁} {q : σ₂} (a : α), R s q → R (f₁ a s).1 (f₂ a q).1 ∧ (f₁ a s).2 = (f₂ a q).2
n✝ : ℕ
xs✝ : Vector α n✝
x✝ : α
a✝ :
∀ {s₁ : σ₁} {s₂ : σ₂},
R s₁ s₂ → R (ma... | next xs x ih =>
rcases (hR x h₀) with ⟨hR, _⟩
simp only [mapAccumr_snoc, ih hR, true_and]
congr 1 | no goals | 011fe96bad5078a7 |
DiscreteQuotient.exists_of_compat | Mathlib/Topology/DiscreteQuotient.lean | theorem exists_of_compat [CompactSpace X] (Qs : (Q : DiscreteQuotient X) → Q)
(compat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs _) = Qs _) :
∃ x : X, ∀ Q : DiscreteQuotient X, Q.proj x = Qs _ | X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
Qs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid
compat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B
⊢ ∃ x, ∀ (Q : DiscreteQuotient X), Q.proj x = Qs Q | have H₁ : ∀ Q₁ Q₂, Q₁ ≤ Q₂ → proj Q₁ ⁻¹' {Qs Q₁} ⊆ proj Q₂ ⁻¹' {Qs Q₂} := fun _ _ h => by
rw [← compat _ _ h]
exact fiber_subset_ofLE _ _ | X : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
Qs : (Q : DiscreteQuotient X) → Quotient Q.toSetoid
compat : ∀ (A B : DiscreteQuotient X) (h : A ≤ B), ofLE h (Qs A) = Qs B
H₁ : ∀ (Q₁ Q₂ : DiscreteQuotient X), Q₁ ≤ Q₂ → Q₁.proj ⁻¹' {Qs Q₁} ⊆ Q₂.proj ⁻¹' {Qs Q₂}
⊢ ∃ x, ∀ (Q : DiscreteQuotient X), Q.proj x... | ebe53f3c1d232b5c |
Batteries.HashMap.Imp.WF.filterMap | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean | theorem WF.filterMap {α β γ} {f : α → β → Option γ} [BEq α] [Hashable α]
{m : Imp α β} (H : WF m) : WF (filterMap f m) | case refl.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → Option γ
inst✝¹ : BEq α
inst✝ : Hashable α
m : Imp α β
H : m.WF
g₁ : AssocList α β → List (α × γ) :=
fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l.toList
H1 :
∀ (l : AssocList α β) (n : ULift Nat) (acc ... | simp only [List.forall_mem_map, List.toList_toAssocList] | case refl.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → Option γ
inst✝¹ : BEq α
inst✝ : Hashable α
m : Imp α β
H : m.WF
g₁ : AssocList α β → List (α × γ) :=
fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l.toList
H1 :
∀ (l : AssocList α β) (n : ULift Nat) (acc ... | e5de3aeb49b27505 |
MeasureTheory.IsSetSemiring.disjointOfUnion_props | Mathlib/MeasureTheory/SetSemiring.lean | theorem disjointOfUnion_props (hC : IsSetSemiring C) (h1 : ↑J ⊆ C) :
∃ K : Set α → Finset (Set α),
PairwiseDisjoint J K
∧ (∀ i ∈ J, ↑(K i) ⊆ C)
∧ PairwiseDisjoint (⋃ x ∈ J, (K x : Set (Set α))) id
∧ (∀ j ∈ J, ⋃₀ K j ⊆ j)
∧ (∀ j ∈ J, ∅ ∉ K j)
∧ ⋃₀ J = ⋃₀ (⋃ x ∈ J, (K x : Set (Set ... | case h
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ K j) ∧ ⋃₀... | refine ⟨?_, ⟨hC.disjointOfDiffUnion_subset h1.1 h1.2, ?_⟩, ?_,
⟨hC.subset_of_mem_disjointOfDiffUnion h1.1 h1.2, ?_⟩, ?_, ?_⟩ | case h.refine_1
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ ... | defbf789029f4cb3 |
Finset.gcd_eq_zero_iff | Mathlib/Algebra/GCDMonoid/Finset.lean | theorem gcd_eq_zero_iff : s.gcd f = 0 ↔ ∀ x : β, x ∈ s → f x = 0 | case mp
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Finset β
f : β → α
h : ∀ x ∈ Multiset.map f s.val, x = 0
⊢ ∀ x ∈ s, f x = 0 | intro b bs | case mp
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Finset β
f : β → α
h : ∀ x ∈ Multiset.map f s.val, x = 0
b : β
bs : b ∈ s
⊢ f b = 0 | c2b830459b9b3360 |
Matroid.comap_closure_eq | Mathlib/Data/Matroid/Closure.lean | @[simp] lemma comap_closure_eq {β : Type*} (M : Matroid β) (f : α → β) (X : Set α) :
(M.comap f).closure X = f ⁻¹' M.closure (f '' X) | case intro.intro.intro
α : Type u_2
β : Type u_3
M : Matroid β
f : α → β
X I : Set α
hI : (M.comap f).IsBasis' I X
hI' : M.IsBasis' (f '' I) (f '' X)
hIinj : InjOn f I
x : α
hxE : f x ∈ M.E
⊢ M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I ↔ M.Indep (f '' insert x I) → ∃ x_1 ∈ I, f x_1 = f x | by_cases hxI : x ∈ I | case pos
α : Type u_2
β : Type u_3
M : Matroid β
f : α → β
X I : Set α
hI : (M.comap f).IsBasis' I X
hI' : M.IsBasis' (f '' I) (f '' X)
hIinj : InjOn f I
x : α
hxE : f x ∈ M.E
hxI : x ∈ I
⊢ M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I ↔ M.Indep (f '' insert x I) → ∃ x_1 ∈ I, f x_1 = f x
case neg
α : Type u... | c2748b7c903f9450 |
Projectivization.orthogonal_cross_right | Mathlib/LinearAlgebra/Projectivization/Constructions.lean | lemma orthogonal_cross_right {v w : ℙ F (Fin 3 → F)} (h : v ≠ w) :
w.orthogonal (cross v w) | F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v w : ℙ F (Fin 3 → F)
h : v ≠ w
⊢ w.orthogonal (v.cross w) | rw [orthogonal_comm] | F : Type u_1
inst✝¹ : Field F
inst✝ : DecidableEq F
v w : ℙ F (Fin 3 → F)
h : v ≠ w
⊢ (v.cross w).orthogonal w | ccf88516d768ed1c |
Ordnode.balancedSz_up | Mathlib/Data/Ordmap/Ordset.lean | theorem balancedSz_up {l r₁ r₂ : ℕ} (h₁ : r₁ ≤ r₂) (h₂ : l + r₂ ≤ 1 ∨ r₂ ≤ delta * l)
(H : BalancedSz l r₁) : BalancedSz l r₂ | case inr
l r₁ r₂ : ℕ
h₁ : r₁ ≤ r₂
h₂ : l + r₂ ≤ 1 ∨ r₂ ≤ delta * l
h : ¬l + r₂ ≤ 1
H : l ≤ delta * r₁ ∧ r₁ ≤ delta * l
⊢ l ≤ delta * r₂ | exact le_trans H.1 (Nat.mul_le_mul_left _ h₁) | no goals | e077dcdaf0ad010d |
Multiset.le_bind | Mathlib/Data/Multiset/Bind.lean | theorem le_bind {α β : Type*} {f : α → Multiset β} (S : Multiset α) {x : α} (hx : x ∈ S) :
f x ≤ S.bind f | case intro
α : Type u_4
β : Type u_5
f : α → Multiset β
S : Multiset α
x : α
hx : x ∈ S
a : β
m' : Multiset ℕ
hm' : map (fun b => count a (f b)) S = (fun b => count a (f b)) x ::ₘ m'
⊢ count a (f x) ≤ count a (S.bind f) | rw [count_bind, hm', sum_cons] | case intro
α : Type u_4
β : Type u_5
f : α → Multiset β
S : Multiset α
x : α
hx : x ∈ S
a : β
m' : Multiset ℕ
hm' : map (fun b => count a (f b)) S = (fun b => count a (f b)) x ::ₘ m'
⊢ count a (f x) ≤ (fun b => count a (f b)) x + m'.sum | 2e61a8e5a411e4b9 |
CategoryTheory.toSheafify_plusPlusIsoSheafify_hom | Mathlib/CategoryTheory/Sites/LeftExact.lean | @[reassoc (attr := simp)]
lemma toSheafify_plusPlusIsoSheafify_hom (P : Cᵒᵖ ⥤ D) :
J.toSheafify P ≫ (plusPlusIsoSheafify J D P).hom = toSheafify J P | case h.e'_2.h.h.e'_6.h.h.e
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁷ : Category.{max v u, w} D
inst✝⁶ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type (max v u)
ins... | ext1 P | case h.e'_2.h.h.e'_6.h.h.e.h
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁷ : Category.{max v u, w} D
inst✝⁶ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type (max v u)
i... | 62e88ebee2d49ea9 |
WithSeminorms.hasBasis_zero_ball | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem WithSeminorms.hasBasis_zero_ball (hp : WithSeminorms p) :
(𝓝 (0 : E)).HasBasis
(fun sr : Finset ι × ℝ => 0 < sr.2) fun sr => (sr.1.sup p).ball 0 sr.2 | case mpr.intro.intro.intro
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
V : Set E
s : Finset ι
r : ℝ
hr : 0 < r
hV : (s.sup p).ball 0 r ⊆ V
⊢ ∃ i, (∃ i_1 r, 0 < r... | exact ⟨_, ⟨s, r, hr, rfl⟩, hV⟩ | no goals | b36e4cf97c255670 |
IsPerfectClosure.equiv_symm_apply | Mathlib/FieldTheory/IsPerfectClosure.lean | theorem equiv_symm_apply (x : M) (n : ℕ) (y : K) (h : j y = x ^ p ^ n) :
(equiv i j p).symm x = (iterateFrobeniusEquiv L p n).symm (i y) | K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁹ : CommRing K
inst✝⁸ : CommRing L
inst✝⁷ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝⁶ : ExpChar M p
inst✝⁵ : ExpChar K p
inst✝⁴ : ExpChar L p
inst✝³ : PerfectRing L p
inst✝² : IsPerfectClosure i p
inst✝¹ : PerfectRing M p
inst✝ : IsPerfectClosure j p
x : M
n : ℕ
y : K
h... | rw [equiv_symm, equiv_apply j i p _ _ _ h] | no goals | 213f7a54125d0be8 |
IsConformalMap.injective | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | theorem injective {f : M' →L[R] N} (h : IsConformalMap f) : Function.Injective f | case intro.intro.intro
R : Type u_1
N : Type u_3
M' : Type u_5
inst✝⁴ : NormedField R
inst✝³ : SeminormedAddCommGroup N
inst✝² : NormedSpace R N
inst✝¹ : NormedAddCommGroup M'
inst✝ : NormedSpace R M'
c : R
hc : c ≠ 0
li : M' →ₗᵢ[R] N
⊢ Injective ⇑(c • li.toContinuousLinearMap) | exact (smul_right_injective _ hc).comp li.injective | no goals | 21c17507d1b14276 |
ONote.nf_repr_split | Mathlib/SetTheory/Ordinal/Notation.lean | theorem nf_repr_split {o o' m} [NF o] (h : split o = (o', m)) : NF o' ∧ repr o = repr o' + m | case mk.intro
o : ONote
m : ℕ
inst✝ : o.NF
a : ONote
s₁ : a.NF
e : o.split' = (a, m)
s₂ : o.repr = ω * a.repr + ↑m
⊢ (scale 1 a).NF ∧ o.repr = (scale 1 a).repr + ↑m | simp only [repr_scale, repr_one, Nat.cast_one, opow_one, ← s₂, and_true] | case mk.intro
o : ONote
m : ℕ
inst✝ : o.NF
a : ONote
s₁ : a.NF
e : o.split' = (a, m)
s₂ : o.repr = ω * a.repr + ↑m
⊢ (scale 1 a).NF | fbed572193a7606e |
TendstoUniformlyOnFilter.uniformCauchySeqOnFilter | Mathlib/Topology/UniformSpace/UniformConvergence.lean | theorem TendstoUniformlyOnFilter.uniformCauchySeqOnFilter (hF : TendstoUniformlyOnFilter F f p p') :
UniformCauchySeqOnFilter F p p' | α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
hF : TendstoUniformlyOnFilter F f p p'
⊢ UniformCauchySeqOnFilter F p p' | intro u hu | α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
hF : TendstoUniformlyOnFilter F f p p'
u : Set (β × β)
hu : u ∈ 𝓤 β
⊢ ∀ᶠ (m : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', (F m.1.1 m.2, F m.1.2 m.2) ∈ u | 3ecc2f337644b7e7 |
ProbabilityTheory.Kernel.meas_countablePartitionSet_le_of_fst_le | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma meas_countablePartitionSet_le_of_fst_le (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ)
(s : Set β) :
κ a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (κ.fst a) (countablePartitionSet n x) | rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
⊢ (κ a) (countablePartitionSet n x ×ˢ s) ≤ (κ a) {p | p.1 ∈ countablePartitionSet n x} | 32c0a56db3e1e141 |
Polynomial.coeff_expand | Mathlib/Algebra/Polynomial/Expand.lean | theorem coeff_expand {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff n = if p ∣ n then f.coeff (n / p) else 0 | case pos.h₁
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : 0 < p
f : R[X]
n : ℕ
h : p ∣ n
⊢ n / p ∉ f.support → (if p * (n / p) = n then f.coeff (n / p) else 0) = 0 | intro hn | case pos.h₁
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : 0 < p
f : R[X]
n : ℕ
h : p ∣ n
hn : n / p ∉ f.support
⊢ (if p * (n / p) = n then f.coeff (n / p) else 0) = 0 | 992a628995b03431 |
LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete | Mathlib/GroupTheory/ArchimedeanDensely.lean | lemma LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete
{G : Type*} [LinearOrderedAddCommGroup G] [Nontrivial G] {g : G} :
Set.WellFoundedOn {x : G | g ≤ x} (· < ·) ↔ Nonempty (G ≃+o ℤ) | case mpr
G : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Nontrivial G
g : G
⊢ Nonempty (G ≃+o ℤ) → {x | 0 ≤ x}.WellFoundedOn fun x1 x2 => x1 < x2 | rintro ⟨f⟩ | case mpr.intro
G : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Nontrivial G
g : G
f : G ≃+o ℤ
⊢ {x | 0 ≤ x}.WellFoundedOn fun x1 x2 => x1 < x2 | 59c60e31ffce1071 |
AlgebraicTopology.DoldKan.HigherFacesVanish.on_Γ₀_summand_id | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | theorem HigherFacesVanish.on_Γ₀_summand_id (K : ChainComplex C ℕ) (n : ℕ) :
@HigherFacesVanish C _ _ (Γ₀.obj K) _ n (n + 1)
(((Γ₀.splitting K).cofan _).inj (Splitting.IndexSet.id (op ⦋n + 1⦌))) | C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ HigherFacesVanish (n + 1) (((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌))) | intro j _ | C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
j : Fin (n + 1)
a✝ : n + 1 ≤ ↑j + (n + 1)
⊢ ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫ (Γ₀.obj K).δ j.succ = 0 | a18d137f64af8b48 |
Submodule.LinearDisjoint.linearIndependent_mul_of_flat_right | Mathlib/LinearAlgebra/LinearDisjoint.lean | theorem linearIndependent_mul_of_flat_right (H : M.LinearDisjoint N) [Module.Flat R N]
{κ ι : Type*} {m : κ → M} {n : ι → N} (hm : LinearIndependent R m)
(hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1 | R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
inst✝¹ : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝ : Module.Flat R ↥N
κ : Type u_1
ι : Type u_2
m : κ → ↥M
n : ι → ↥N
hm : Function.Injective ⇑(Finsupp.linearCombination R m)
hn : Function.Injective ⇑(Finsupp.linearCombination R n)
⊢ Function.... | let i0 := (finsuppTensorFinsupp' R κ ι).symm | R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : Ring S
inst✝¹ : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝ : Module.Flat R ↥N
κ : Type u_1
ι : Type u_2
m : κ → ↥M
n : ι → ↥N
hm : Function.Injective ⇑(Finsupp.linearCombination R m)
hn : Function.Injective ⇑(Finsupp.linearCombination R n)
i0 : (κ × ι... | 4743650bb34b43cd |
Path.Homotopy.trans_assoc_reparam | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
(p.trans q).trans r =
(p.trans (q.trans r)).reparam
(fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
(Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAss... | case pos
X : Type u
inst✝ : TopologicalSpace X
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : 2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ False | linarith | no goals | fdaccc7c71c4d168 |
String.revFindAux_of_valid | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem revFindAux_of_valid (p) : ∀ l r,
revFindAux ⟨l.reverse ++ r⟩ p ⟨utf8Len l⟩ = (l.dropWhile (!p ·)).tail?.map (⟨utf8Len ·⟩)
| [], r => by unfold revFindAux List.dropWhile; simp
| c::l, r => by
unfold revFindAux List.dropWhile
rw [dif_neg (by exact Pos.ne_of_gt add_utf8Size_pos)]
have h1 := get... | p : Char → Bool
c : Char
l r : List Char
h1 : { data := l.reverse ++ c :: r }.get { byteIdx := utf8Len l.reverse } = (c :: r).headD default
h2 :
{ data := l.reverse ++ c :: r }.prev { byteIdx := utf8Len l.reverse + c.utf8Size } = { byteIdx := utf8Len l.reverse }
⊢ (let_fun this := ⋯;
let pos := { data := (c :: l)... | simp only [utf8Len_reverse, Char.reduceDefault, List.headD_cons] at h1 h2 | p : Char → Bool
c : Char
l r : List Char
h1 : { data := l.reverse ++ c :: r }.get { byteIdx := utf8Len l } = c
h2 : { data := l.reverse ++ c :: r }.prev { byteIdx := utf8Len l + c.utf8Size } = { byteIdx := utf8Len l }
⊢ (let_fun this := ⋯;
let pos := { data := (c :: l).reverse ++ r }.prev { byteIdx := utf8Len (c ::... | f68483ff30c5dbc7 |
Array.isPrefixOf_toList | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem isPrefixOf_toList [BEq α] {as bs : Array α} :
as.toList.isPrefixOf bs.toList = as.isPrefixOf bs | case mk.mk
α : Type u_1
inst✝ : BEq α
toList✝¹ toList✝ : List α
⊢ { toList := toList✝¹ }.toList.isPrefixOf { toList := toList✝ }.toList =
{ toList := toList✝¹ }.isPrefixOf { toList := toList✝ } | simp | no goals | 9b0d4b88d4767055 |
Set.range_inter_ssubset_iff_preimage_ssubset | Mathlib/Data/Set/Image.lean | theorem range_inter_ssubset_iff_preimage_ssubset {f : α → β} {S S' : Set β} :
range f ∩ S ⊂ range f ∩ S' ↔ f ⁻¹' S ⊂ f ⁻¹' S' | α : Type u_1
β : Type u_2
f : α → β
S S' : Set β
⊢ (range f ∩ S ⊆ range f ∩ S' ∧ ∃ x ∈ range f ∩ S', x ∉ range f ∩ S) ↔ f ⁻¹' S ⊆ f ⁻¹' S' ∧ ∃ x ∈ f ⁻¹' S', x ∉ f ⁻¹' S | apply and_congr ?_ (by aesop) | α : Type u_1
β : Type u_2
f : α → β
S S' : Set β
⊢ range f ∩ S ⊆ range f ∩ S' ↔ f ⁻¹' S ⊆ f ⁻¹' S' | b7b500a3517f2a64 |
SeparationQuotient.map_mk_nhds | Mathlib/Topology/Inseparable.lean | theorem map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) | X : Type u_1
inst✝ : TopologicalSpace X
x : X
⊢ Filter.map mk (𝓝 x) = 𝓝 (mk x) | rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk] | no goals | 598c6ab5f4f37658 |
Std.Sat.CNF.unsat_relabel_iff | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Relabel.lean | theorem unsat_relabel_iff {f : CNF α} {r : α → β}
(hw : ∀ {v1 v2}, Mem v1 f → Mem v2 f → r v1 = r v2 → v1 = v2) :
Unsat (relabel r f) ↔ Unsat f | α : Type u_1
β : Type u_2
f : CNF α
r : α → β
hw : ∀ {v1 v2 : α}, Mem v1 f → Mem v2 f → r v1 = r v2 → v1 = v2
a₀ : α
h✝ : (relabel r f).Unsat
em : (a : Prop) → Decidable a
g : β → α := fun b => if h : ∃ a, Mem a f ∧ r a = b then h.choose else a₀
h' : (relabel g (relabel r f)).Unsat
v : α
h : Mem v f
⊢ (if h : ∃ a, Mem ... | rw [dif_pos ⟨v, h, rfl⟩] | α : Type u_1
β : Type u_2
f : CNF α
r : α → β
hw : ∀ {v1 v2 : α}, Mem v1 f → Mem v2 f → r v1 = r v2 → v1 = v2
a₀ : α
h✝ : (relabel r f).Unsat
em : (a : Prop) → Decidable a
g : β → α := fun b => if h : ∃ a, Mem a f ∧ r a = b then h.choose else a₀
h' : (relabel g (relabel r f)).Unsat
v : α
h : Mem v f
⊢ ⋯.choose = v | 0a4721def34674f8 |
Std.DHashMap.Raw.Const.getKey_modify | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getKey_modify [EquivBEq α] [LawfulHashable α] [Inhabited α] {k k' : α} {f : β → β}
(h : m.WF) {hc : k' ∈ Const.modify m k f} :
(Const.modify m k f).getKey k' hc =
if k == k' then
k
else
haveI h' : k' ∈ m := mem_modify h |>.mp hc
m.getKey k' h' | α : Type u
inst✝⁴ : BEq α
inst✝³ : Hashable α
β : Type v
m : Raw α fun x => β
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited α
k k' : α
f : β → β
h : m.WF
hc : k' ∈ modify m k f
⊢ (modify m k f).contains k' = true → (modify m k f).getKey k' hc = if (k == k') = true then k else m.getKey k' ⋯ | simp_to_raw using Raw₀.Const.getKey_modify | no goals | 95612e3b201e00e6 |
Finset.mulIndicator_biUnion | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | @[to_additive]
lemma mulIndicator_biUnion (s : Finset ι) (t : ι → Set κ) {f : κ → β}
(hs : (s : Set ι).PairwiseDisjoint t) :
mulIndicator (⋃ i ∈ s, t i) f = fun a ↦ ∏ i ∈ s, mulIndicator (t i) f a | case cons.h
ι : Type u_1
β : Type u_4
inst✝ : CommMonoid β
κ : Type u_7
t : ι → Set κ
f : κ → β
i : ι
s : Finset ι
hi : i ∉ s
ih : (↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).mulIndicator f = fun a => ∏ i ∈ s, (t i).mulIndicator f a
hs : (↑(cons i s hi)).PairwiseDisjoint t
j : κ
⊢ (⋃ i_1 ∈ cons i s hi, t i_1).mulIndicator... | rw [coe_cons, Set.pairwiseDisjoint_insert_of_not_mem (Finset.mem_coe.not.2 hi)] at hs | case cons.h
ι : Type u_1
β : Type u_4
inst✝ : CommMonoid β
κ : Type u_7
t : ι → Set κ
f : κ → β
i : ι
s : Finset ι
hi : i ∉ s
ih : (↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).mulIndicator f = fun a => ∏ i ∈ s, (t i).mulIndicator f a
hs : (↑s).PairwiseDisjoint t ∧ ∀ j ∈ ↑s, Disjoint (t i) (t j)
j : κ
⊢ (⋃ i_1 ∈ cons i s hi... | 372ee78aaced8e83 |
Int.mul_tdiv_assoc | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem mul_tdiv_assoc (a : Int) : ∀ {b c : Int}, c ∣ b → (a * b).tdiv c = a * (b.tdiv c)
| _, c, ⟨d, rfl⟩ =>
if cz : c = 0 then by simp [cz, Int.mul_zero] else by
rw [Int.mul_left_comm, Int.mul_tdiv_cancel_left _ cz, Int.mul_tdiv_cancel_left _ cz]
| a c d : Int
cz : ¬c = 0
⊢ (a * (c * d)).tdiv c = a * (c * d).tdiv c | rw [Int.mul_left_comm, Int.mul_tdiv_cancel_left _ cz, Int.mul_tdiv_cancel_left _ cz] | no goals | c8106281c2be24e5 |
UniformSpace.ball_mem_nhdsWithin | Mathlib/Topology/UniformSpace/Defs.lean | theorem UniformSpace.ball_mem_nhdsWithin {x : α} {S : Set α} ⦃V : Set (α × α)⦄ (x_in : x ∈ S)
(V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S)) : ball x V ∈ 𝓝[S] x | α : Type ua
inst✝ : UniformSpace α
x : α
S : Set α
V : Set (α × α)
x_in : x ∈ S
V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S)
⊢ ∃ t ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S), Prod.mk x ⁻¹' t ⊆ ball x V | exact ⟨V, V_in, Subset.rfl⟩ | no goals | 479ab6e26d41b347 |
Nat.bitwise_lt_two_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem bitwise_lt_two_pow (left : x < 2^n) (right : y < 2^n) : (Nat.bitwise f x y) < 2^n | f : Bool → Bool → Bool
n : Nat
hyp : ∀ {x y : Nat}, x < 2 ^ n → y < 2 ^ n → bitwise f x y < 2 ^ n
x y : Nat
left : x < 2 ^ (n + 1)
right : y < 2 ^ (n + 1)
x_zero : ¬x = 0
y_zero : ¬y = 0
⊢ (if x = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then x else 0
else
... | simp only [x_zero, y_zero, if_neg] | f : Bool → Bool → Bool
n : Nat
hyp : ∀ {x y : Nat}, x < 2 ^ n → y < 2 ^ n → bitwise f x y < 2 ^ n
x y : Nat
left : x < 2 ^ (n + 1)
right : y < 2 ^ (n + 1)
x_zero : ¬x = 0
y_zero : ¬y = 0
⊢ (if False then if f false true = true then y else 0
else
if False then if f true false = true then x else 0
else
... | b3846a598e8ae70f |
AddCircle.coe_eq_zero_iff_of_mem_Ico | Mathlib/Topology/Instances/AddCircle.lean | lemma coe_eq_zero_iff_of_mem_Ico (ha : a ∈ Ico 0 p) :
(a : AddCircle p) = 0 ↔ a = 0 | 𝕜 : Type u_1
inst✝¹ : LinearOrderedAddCommGroup 𝕜
p : 𝕜
hp : Fact (0 < p)
a : 𝕜
inst✝ : Archimedean 𝕜
ha : a ∈ Ico 0 p
⊢ 0 ∈ Ico 0 (0 + p) | simpa [zero_add, left_mem_Ico] using hp.out | no goals | 4faacd3d488ea321 |
GenContFract.abs_sub_convs_le | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | theorem abs_sub_convs_le (not_terminatedAt_n : ¬(of v).TerminatedAt n) :
|v - (of v).convs n| ≤ 1 / ((of v).dens n * ((of v).dens <| n + 1)) | case intro.intro.intro.intro.intro.intro.right.a0
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
not_terminatedAt_n : ¬(of v).TerminatedAt n
g : GenContFract K := of v
nextConts : Pair K := g.contsAux (n + 2)
conts : Pair K := g.contsAux (n + 1)
conts_eq : conts = g.contsAux (n + 1)
pred_con... | exact this | no goals | a2279d23d2560b5e |
Real.pi_lt_sqrtTwoAddSeries | Mathlib/Data/Real/Pi/Bounds.lean | theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n | n : ℕ
⊢ π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n | have : π < (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * (2 : ℝ) ^ (n + 2) := by
rw [← div_lt_iff₀ (by norm_num), ← sin_pi_over_two_pow_succ, ← sub_lt_iff_lt_add']
calc
π / 2 ^ (n + 2) - sin (π / 2 ^ (n + 2)) < (π / 2 ^ (n + 2)) ^ 3 / 4 :=
sub_lt_comm.1 <| sin_gt_sub_cube (by positivity) <| di... | n : ℕ
this : π < (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)
⊢ π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n | 02d82b689d50a592 |
measurableSet_integrable | Mathlib/MeasureTheory/Integral/Prod.lean | theorem measurableSet_integrable [SFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} | α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | HasFiniteIntegral (f x) ν} | exact measurableSet_lt (Measurable.lintegral_prod_right hf.enorm) measurable_const | no goals | b4b9244543445929 |
ComplexShape.boundaryLE_embeddingUpIntLE_iff | Mathlib/Algebra/Homology/Embedding/Boundary.lean | lemma boundaryLE_embeddingUpIntLE_iff (p : ℤ) (n : ℕ) :
(embeddingUpIntGE p).BoundaryGE n ↔ n = 0 | case mp.succ
p : ℤ
n : ℕ
h : (embeddingUpIntGE p).BoundaryGE (n + 1)
⊢ n + 1 = 0 | have := h.2 n | case mp.succ
p : ℤ
n : ℕ
h : (embeddingUpIntGE p).BoundaryGE (n + 1)
this : ¬(up ℤ).Rel ((embeddingUpIntGE p).f n) ((embeddingUpIntGE p).f (n + 1))
⊢ n + 1 = 0 | 14489894ffc1fd9c |
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