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 |
|---|---|---|---|---|---|---|
SimpleGraph.Walk.IsTrail.even_countP_edges_iff | Mathlib/Combinatorics/SimpleGraph/Trails.lean | theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v | case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v x u✝ v✝ w✝ : V
huv : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ht : p.IsTrail ∧ s(u✝, v✝) ∉ p.edges
ih : Even (List.countP (fun e => decide (x ∈ e)) p.edges) ↔ v✝ ≠ w✝ → x ≠ v✝ ∧ x ≠ w✝
h : ¬x = u✝ ∧ ¬x = v✝
⊢ Even (List.countP (fun e => decide (x ∈ e)) p.edges + 0) ... | simp only [h.1, h.2, not_false_iff, true_and, add_zero, Ne] at ih ⊢ | case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v x u✝ v✝ w✝ : V
huv : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ht : p.IsTrail ∧ s(u✝, v✝) ∉ p.edges
h : ¬x = u✝ ∧ ¬x = v✝
ih : Even (List.countP (fun e => decide (x ∈ e)) p.edges) ↔ ¬v✝ = w✝ → ¬x = w✝
⊢ Even (List.countP (fun e => decide (x ∈ e)) p.edges) ↔ ¬u✝ = w✝ ... | a61699336dc9d9f7 |
Complex.norm_eqOn_closedBall_of_isMaxOn | Mathlib/Analysis/Complex/AbsMax.lean | theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
(hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) | case inr
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
⊢ (norm ∘ f) w = Function.const E ‖f z‖ w | set e := (lineMap z w : ℂ → E) | case inr
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
z : E
r : ℝ
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (ball z r) z
w : E
hw : dist z w ≤ r
hne : z ≠ w
e : ℂ → E := ⇑(lineMap z w)
⊢ (norm ∘ f) w = Functio... | 16e60b5e932312c6 |
IsLocalRing.basisQuotient_repr | Mathlib/RingTheory/LocalRing/Quotient.lean | lemma basisQuotient_repr {ι} [Fintype ι] (b : Basis ι R S) (x) (i) :
(basisQuotient b).repr (Ideal.Quotient.mk pS x) i =
Ideal.Quotient.mk p (b.repr x i) | case a.a
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_4
inst✝ : Fintype ι
b : Basis ι R S
x : S
i : ι
⊢ (basisQuotient b).repr.symm
((Finsupp.linearEquivFunOnFinite (R ⧸ p) (R ⧸ p) ι).symm ⇑((b... | simp only [Finsupp.linearEquivFunOnFinite_symm_coe, LinearEquiv.symm_apply_apply,
Basis.repr_symm_apply] | case a.a
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_4
inst✝ : Fintype ι
b : Basis ι R S
x : S
i : ι
⊢ (Ideal.Quotient.mk pS) x =
(Finsupp.linearCombination (R ⧸ p) ⇑(basisQuotient b))
((F... | fc5762d27995bd54 |
Submodule.prod_sup_prod | Mathlib/LinearAlgebra/Span/Basic.lean | theorem prod_sup_prod : prod p q₁ ⊔ prod p' q₁' = prod (p ⊔ p') (q₁ ⊔ q₁') | R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
p p' : Submodule R M
M' : Type u_9
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
q₁ q₁' : Submodule R M'
⊢ p.prod q₁ ⊔ p'.prod q₁' = (p ⊔ p').prod (q₁ ⊔ q₁') | refine le_antisymm
(sup_le (prod_mono le_sup_left le_sup_left) (prod_mono le_sup_right le_sup_right)) ?_ | R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
p p' : Submodule R M
M' : Type u_9
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
q₁ q₁' : Submodule R M'
⊢ (p ⊔ p').prod (q₁ ⊔ q₁') ≤ p.prod q₁ ⊔ p'.prod q₁' | 6bc60ca9bdd6f33f |
eVariationOn.add_point | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
(us : ∀ i, u i ∈ s) (n : ℕ) :
∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f ... | case e_a
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i => if i < N then u i else if i = ... | congr 1 | case e_a.e_a
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i => if i < N then u i else if ... | 760e94573f7a043d |
CliffordAlgebra.evenOdd_mul_le | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | theorem evenOdd_mul_le (i j : ZMod 2) : evenOdd Q i * evenOdd Q j ≤ evenOdd Q (i + j) | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
i j : ZMod 2
⊢ evenOdd Q i * evenOdd Q j ≤ evenOdd Q (i + j) | simp_rw [evenOdd, Submodule.iSup_eq_span, Submodule.span_mul_span] | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
i j : ZMod 2
⊢ Submodule.span R ((⋃ i_1, ↑(LinearMap.range (ι Q) ^ ↑i_1)) * ⋃ i, ↑(LinearMap.range (ι Q) ^ ↑i)) ≤
Submodule.span R (⋃ i_1, ↑(LinearMap.range (ι Q) ^ ↑i_1)) | ee4ee497f619176d |
ite_and | Mathlib/Logic/Basic.lean | theorem ite_and : ite (P ∧ Q) a b = ite P (ite Q a b) b | α : Sort u_1
P Q : Prop
inst✝¹ : Decidable P
a b : α
inst✝ : Decidable Q
⊢ (if P ∧ Q then a else b) = if P then if Q then a else b else b | by_cases hp : P <;> by_cases hq : Q <;> simp [hp, hq] | no goals | c458c06f9ae57769 |
Batteries.RBNode.zoom_toList | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem _root_.Batteries.RBNode.zoom_toList {t : RBNode α} (eq : t.zoom cut = (t', p')) :
p'.withList t'.toList = t.toList | α : Type u_1
cut : α → Ordering
t' : RBNode α
p' : Path α
t : RBNode α
eq : zoom cut t = (t', p')
⊢ (root.fill t).toList = t.toList | rfl | no goals | 54920de27d8142db |
Complex.differentiableAt_Gamma | Mathlib/Analysis/SpecialFunctions/Gamma/Deriv.lean | theorem differentiableAt_Gamma (s : ℂ) (hs : ∀ m : ℕ, s ≠ -m) : DifferentiableAt ℂ Gamma s | s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ Gamma s | let n := ⌊1 - s.re⌋₊ + 1 | s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
n : ℕ := ⌊1 - s.re⌋₊ + 1
⊢ DifferentiableAt ℂ Gamma s | d883d9d56c34495c |
CategoryTheory.Presieve.isSheaf_coverage | Mathlib/CategoryTheory/Sites/Coverage.lean | theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type*) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) | case mp
C : Type u_2
inst✝ : Category.{u_3, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ K.covering X
⊢ IsSheafFor P (Sieve.generate R).arrows | apply H _ <| Saturate.of _ _ hR | no goals | b0ee5f44dbe2a8c4 |
MeasureTheory.Lp.dist_def | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem dist_def (f g : Lp E p μ) : dist f g = (eLpNorm (⇑f - ⇑g) p μ).toReal | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : ↥(Lp E p μ)
⊢ dist f g = (eLpNorm (↑↑f - ↑↑g) p μ).toReal | simp_rw [dist, norm_def] | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : ↥(Lp E p μ)
⊢ (eLpNorm (↑↑(f - g)) p μ).toReal = (eLpNorm (↑↑f - ↑↑g) p μ).toReal | 95e354d102df0f62 |
PFunctor.supp_eq | Mathlib/Data/PFunctor/Univariate/Basic.lean | theorem supp_eq {α : Type u} (a : P.A) (f : P.B a → α) :
@supp.{u} P.Obj _ α (⟨a, f⟩ : P α) = f '' univ | case h.mpr.intro
P : PFunctor.{u}
α : Type u
a : P.A
f : P.B a → α
x : α
w✝ : P.B a
h✝ : f w✝ = x
⊢ ∀ ⦃p : α → Prop⦄, (∀ (i : P.B a), p (f i)) → p x | subst x | case h.mpr.intro
P : PFunctor.{u}
α : Type u
a : P.A
f : P.B a → α
w✝ : P.B a
⊢ ∀ ⦃p : α → Prop⦄, (∀ (i : P.B a), p (f i)) → p (f w✝) | 1e4b6dfd73ae68eb |
Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable | Mathlib/Analysis/Complex/RemovableSingularity.lean | theorem two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable {U : Set ℂ}
(hU : IsOpen U) {c w₀ : ℂ} {R : ℝ} {f : ℂ → E} (hc : closedBall c R ⊆ U)
(hf : DifferentiableOn ℂ f U) (hw₀ : w₀ ∈ ball c R) :
((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • f z) = deriv f w₀ | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U : Set ℂ
hU : IsOpen U
c w₀ : ℂ
R : ℝ
f : ℂ → E
hc : closedBall c R ⊆ U
hf : DifferentiableOn ℂ f U
hw₀ : w₀ ∈ ball c R
hf' : DifferentiableOn ℂ (dslope f w₀) U
h0 : ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w₀)⁻¹ • dslope f w... | simpa using circleIntegral.integral_sub_zpow_of_ne (by decide : (-2 : ℤ) ≠ -1) c w₀ R | no goals | 61aad2c5fca47fad |
CategoryTheory.GrothendieckTopology.pullback_close | Mathlib/CategoryTheory/Sites/Closed.lean | theorem pullback_close {X Y : C} (f : Y ⟶ X) (S : Sieve X) :
J₁.close (S.pullback f) = (J₁.close S).pullback f | case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Sieve X
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f (J₁.close S)).arrows g
⊢ (J₁.close (Sieve.pullback f S)).arrows g | change _ ∈ J₁ _ | case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Sieve X
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f (J₁.close S)).arrows g
⊢ Sieve.pullback g (Sieve.pullback f S) ∈ J₁ Z | 9c330639664df589 |
CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁' | Mathlib/CategoryTheory/Sites/OneHypercover.lean | lemma sieve₁_eq_pullback_sieve₁' {W : C} (p₁ : W ⟶ E.X i₁) (p₂ : W ⟶ E.X i₂)
(w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂) :
E.sieve₁ p₁ p₂ = (E.sieve₁' i₁ i₂).pullback (pullback.lift _ _ w) | C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
i₁ i₂ : E.I₀
inst✝ : HasPullback (E.f i₁) (E.f i₂)
W : C
p₁ : W ⟶ E.X i₁
p₂ : W ⟶ E.X i₂
w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂
Z : C
g : Z ⟶ W
Y : C
j : E.I₁ i₁ i₂
h : Z ⟶ E.Y j
fac : h ≫ E.toPullback j = g ≫ pullback.lift p₁ p₂ w
⊢ g ≫ p₁ = h ≫ E.p₁ j | simpa using fac.symm =≫ pullback.fst _ _ | no goals | 790b2b0bc3710f19 |
hasFTaylorSeriesUpToOn_top_iff_right | Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean | theorem hasFTaylorSeriesUpToOn_top_iff_right (hN : ∞ ≤ N) :
HasFTaylorSeriesUpToOn N f p s ↔
(∀ x ∈ s, (p x 0).curry0 = f x) ∧
(∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
HasFTaylorSeriesUpToOn N (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
(... | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
N : WithTop ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
h :
(∀ x ∈ s, (p x 0).curry0 = f x) ∧
(∀ x ∈ s... | rw [hasFTaylorSeriesUpToOn_succ_nat_iff_right] | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
N : WithTop ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
h :
(∀ x ∈ s, (p x 0).curry0 = f x) ∧
(∀ x ∈ s... | 4e208dd9d3c072db |
RootPairing.eq_zero_of_mem_rootSpan_of_rootForm_self_eq_zero | Mathlib/LinearAlgebra/RootSystem/Finite/Nondegenerate.lean | lemma eq_zero_of_mem_rootSpan_of_rootForm_self_eq_zero {x : M}
(hx : x ∈ P.rootSpan) (hx' : P.RootForm x x = 0) :
x = 0 | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : Fintype ι
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : LinearOrderedCommRing R
inst✝¹ : Module R M
inst✝ : Module R N
P : RootPairing ι R M N
x : M
hx : x ∈ P.rootSpan
hx' : (P.RootForm x) x = 0
this : x ∈ P.rootSpan ⊓ LinearMap.ker P.RootForm
⊢ x... | simpa [P.disjoint_rootSpan_ker_rootForm.eq_bot] using this | no goals | a7d97ae79ee318b5 |
CategoryTheory.Sheaf.isSheafFor_bind | Mathlib/CategoryTheory/Sites/Canonical.lean | theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄, U f → Sieve Y)
(hU : Presieve.IsSheafFor P (U : Presieve X))
(hB : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (hf : U f), Presieve.IsSheafFor P (B hf : Presieve Y))
(hB' : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (h : U f) ⦃Z⦄ (g : Z ⟶ Y),
Presieve.IsSeparatedFor P (((... | C : Type u
inst✝ : Category.{v, u} C
X : C
P : Cᵒᵖ ⥤ Type v
U : Sieve X
B : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y
hU : Presieve.IsSheafFor P U.arrows
hB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows
hB' :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), Presieve.IsSe... | conv_lhs at h => congr; rw [assoc, assoc] | C : Type u
inst✝ : Category.{v, u} C
X : C
P : Cᵒᵖ ⥤ Type v
U : Sieve X
B : ⦃Y : C⦄ → ⦃f : Y ⟶ X⦄ → U.arrows f → Sieve Y
hU : Presieve.IsSheafFor P U.arrows
hB : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (hf : U.arrows f), Presieve.IsSheafFor P (B hf).arrows
hB' :
∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄ (h : U.arrows f) ⦃Z : C⦄ (g : Z ⟶ Y), Presieve.IsSe... | a03f863f17e76c08 |
Std.DHashMap.Internal.List.getEntry?_insertEntryIfNew | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getEntry?_insertEntryIfNew [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)} {k a : α}
{v : β k} : getEntry? a (insertEntryIfNew k v l) =
if k == a && !containsKey k l then some ⟨k, v⟩ else getEntry? a l | case false
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
l : List ((a : α) × β a)
k a : α
v : β k
h : containsKey k l = false
⊢ getEntry? a (insertEntryIfNew k v l) = if (k == a && !false) = true then some ⟨k, v⟩ else getEntry? a l | simp [insertEntryIfNew_of_containsKey_eq_false h, getEntry?_cons] | no goals | 99eee3066f595b86 |
Transcendental.aeval_of_transcendental | Mathlib/RingTheory/Algebraic/Basic.lean | theorem Transcendental.aeval_of_transcendental {r : A} (H : Transcendental R r)
{f : R[X]} (hf : Transcendental R f) : Transcendental R (Polynomial.aeval r f) | R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : A
H : ∀ (p : R[X]), (Polynomial.aeval r) p = 0 → p = 0
f : R[X]
hf : ∀ (p : R[X]), (Polynomial.aeval f) p = 0 → p = 0
p : R[X]
hp : (Polynomial.aeval ((Polynomial.aeval r) f)) p = 0
⊢ p = 0 | exact hf _ (H _ (by rwa [← aeval_comp, comp_eq_aeval] at hp)) | no goals | c1e75fcc09584408 |
AlgebraicGeometry.QuasiSeparated.of_comp | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | theorem QuasiSeparated.of_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [QuasiSeparated (f ≫ g)] :
QuasiSeparated f | X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝ : QuasiSeparated (f ≫ g)
𝒰 : Scheme.Cover (@IsOpenImmersion) Y :=
(Scheme.Cover.pullbackCover Z.affineCover g).bind fun x =>
((Scheme.Cover.pullbackCover Z.affineCover g).obj x).affineCover
i : 𝒰.J
⊢ IsAffine ((pullback g (Z.affineCover.map i.fst)).affineCover.obj i.snd) | infer_instance | no goals | 96ecf9408a067b28 |
WeierstrassCurve.Affine.CoordinateRing.XYIdeal_eq₂ | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | lemma XYIdeal_eq₂ {x₁ x₂ y₁ y₂ : F} (h₁ : W.Equation x₁ y₁)
(h₂ : W.Equation x₂ y₂) (hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)) :
XYIdeal W x₂ (C y₂) = XYIdeal W x₂ (linePolynomial x₁ y₁ <| W.slope x₁ x₂ y₁ y₂) | F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hy₂ : y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))
⊢ span
{(mk W) (C (X - C x₂)),
(mk W)
(Y - C (C (eval x₂ (C (W.slope x₁ x₂ y₁ y₂) * (X - C x₁) ... | apply congr_arg (_ ∘ _ ∘ _ ∘ _) | F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hy₂ : y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))
⊢ Y - C (C (eval x₂ (C (W.slope x₁ x₂ y₁ y₂) * (X - C x₁) + C y₁))) + C (X - C x₂) * C (C (-W.slope x₁ x₂ y₁ y₂)) =
... | 673ede9f8602b947 |
Array.getElem?_zipWith' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Zip.lean | theorem getElem?_zipWith' {f : α → β → γ} {i : Nat} :
(zipWith f l₁ l₂)[i]? = (l₁[i]?.map f).bind fun g => l₂[i]?.map g | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
l₂ : Array β
f : α → β → γ
i : Nat
toList✝ : List α
⊢ (zipWith f { toList := toList✝ } l₂)[i]? = (Option.map f { toList := toList✝ }[i]?).bind fun g => Option.map g l₂[i]? | cases l₂ | case mk.mk
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
i : Nat
toList✝¹ : List α
toList✝ : List β
⊢ (zipWith f { toList := toList✝¹ } { toList := toList✝ })[i]? =
(Option.map f { toList := toList✝¹ }[i]?).bind fun g => Option.map g { toList := toList✝ }[i]? | 2bb9a06929cc592a |
LieModuleHom.coe_linear_mk | Mathlib/Algebra/Lie/Basic.lean | theorem coe_linear_mk (f : M →ₗ[R] N) (h) : ((⟨f, h⟩ : M →ₗ⁅R,L⁆ N) : M →ₗ[R] N) = f | R : Type u
L : Type v
M : Type w
N : Type w₁
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : LieRingModule L M
inst✝ : LieRingModule L N
f : M →ₗ[R] N
h : ∀ {x : L} {m : M}, f.toFun ⁅x, m⁆ = ⁅x, f.toFun m⁆
⊢ ↑{ toLinearMap := f, map... | rfl | no goals | 3945ab9904f091ce |
DomMulAct.stabilizer_card' | Mathlib/GroupTheory/Perm/DomMulAct.lean | theorem stabilizer_card':
Fintype.card {g : Perm α // f ∘ g = f} =
∏ i ∈ Finset.univ.image f, (Fintype.card ({a // f a = i}))! | α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
a : α
⊢ f a ∈ ↑(Finset.image f Finset.univ) | simp | no goals | eb29591c1a59346a |
Subgroup.mulSingle_mem_pi | Mathlib/Algebra/Group/Subgroup/Basic.lean | theorem mulSingle_mem_pi [DecidableEq η] {I : Set η} {H : ∀ i, Subgroup (f i)} (i : η) (x : f i) :
Pi.mulSingle i x ∈ pi I H ↔ i ∈ I → x ∈ H i | case pos
η : Type u_7
f : η → Type u_8
inst✝¹ : (i : η) → Group (f i)
inst✝ : DecidableEq η
I : Set η
H : (i : η) → Subgroup (f i)
i : η
x : f i
h : i ∈ I → x ∈ H i
j : η
hj : j ∈ I
heq : j = i
⊢ Pi.mulSingle i x j ∈ (fun i => ((fun i => (H i).toSubmonoid) i).carrier) j | subst heq | case pos
η : Type u_7
f : η → Type u_8
inst✝¹ : (i : η) → Group (f i)
inst✝ : DecidableEq η
I : Set η
H : (i : η) → Subgroup (f i)
j : η
hj : j ∈ I
x : f j
h : j ∈ I → x ∈ H j
⊢ Pi.mulSingle j x j ∈ (fun i => ((fun i => (H i).toSubmonoid) i).carrier) j | 1c8d5b24fe240cda |
ProbabilityTheory.Kernel.setIntegral_densityProcess | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma setIntegral_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
(hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, densityProcess κ ν n a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
this : IsFiniteKernel κ
S : Finset (Set γ)
hS_subset : ↑S ⊆ co... | intro u v huv | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
this : IsFiniteKernel κ
S : Finset (Set γ)
hS_subset : ↑S ⊆ co... | fa7b3883c5aa09d1 |
MeasureTheory.generateFrom_generateSetAlgebra_eq | Mathlib/MeasureTheory/SetAlgebra.lean | theorem generateFrom_generateSetAlgebra_eq :
generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 | case empty
α : Type u_1
𝒜 : Set (Set α)
s : Set α
⊢ MeasurableSet ∅ | exact MeasurableSpace.measurableSet_empty _ | no goals | 48402cc40fc3a030 |
mem_analyticGroupoid_of_boundaryless | Mathlib/Geometry/Manifold/AnalyticManifold.lean | theorem mem_analyticGroupoid_of_boundaryless [I.Boundaryless] (e : PartialHomeomorph H H) :
e ∈ analyticGroupoid I ↔ AnalyticOnNhd 𝕜 (I ∘ e ∘ I.symm) (I '' e.source) ∧
AnalyticOnNhd 𝕜 (I ∘ e.symm ∘ I.symm) (I '' e.target) | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ e ∈ analyticGroupoid I ↔
AnalyticOnNhd 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I '' e.source... | simp only [mem_analyticGroupoid, I.range_eq_univ, inter_univ, I.image_eq] | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
inst✝ : I.Boundaryless
e : PartialHomeomorph H H
⊢ AnalyticOn 𝕜 (↑I ∘ ↑e ∘ ↑I.symm) (↑I.symm ⁻¹' e.source) ∧
AnalyticOn 𝕜 (↑... | 01f781ea9f478848 |
FirstOrder.Language.card_functions_sum_skolem₁ | Mathlib/ModelTheory/Skolem.lean | theorem card_functions_sum_skolem₁ :
#(Σ n, (L.sum L.skolem₁).Functions n) = #(Σ n, L.BoundedFormula Empty (n + 1)) | L : Language
⊢ ((sum fun i => lift.{max u v, u} #(L.Functions i)) +
sum fun i => lift.{u, max u v} #(L.BoundedFormula Empty (i + 1))) =
sum fun i => #(L.BoundedFormula Empty (i + 1)) | conv_lhs => enter [2, 1, i]; rw [lift_id'.{u, v}] | L : Language
⊢ ((sum fun i => lift.{max u v, u} #(L.Functions i)) + sum fun i => #(L.BoundedFormula Empty (i + 1))) =
sum fun i => #(L.BoundedFormula Empty (i + 1)) | 3452a51d6291fd03 |
PhragmenLindelof.right_half_plane_of_bounded_on_real | Mathlib/Analysis/Complex/PhragmenLindelof.lean | theorem right_half_plane_of_bounded_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
(hexp : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c))
(hre : IsBoundedUnder (· ≤ ·) atTop fun x : ℝ => ‖f x‖) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
(hz : 0 ≤ z.re) : ‖f z‖ ≤ C | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
z✝ : ℂ
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * ‖z‖ ^ c)
hre : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) atTop fun x => ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z✝.re
ε : ℝ
ε₀ : ε < 0
g : ℂ → E :... | rw [norm_smul, norm_exp, re_ofReal_mul] | no goals | bf76d759dd8394c1 |
Finsupp.supportedEquivFinsupp_symm_apply_coe | Mathlib/LinearAlgebra/Finsupp/Supported.lean | theorem supportedEquivFinsupp_symm_apply_coe (s : Set α) [DecidablePred (· ∈ s)]
(f : s →₀ M) : (supportedEquivFinsupp (R := R) s).symm f = f.extendDomain | α : Type u_1
M : Type u_2
R : Type u_5
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
f : ↑s →₀ M
⊢ ↑((supportedEquivFinsupp s).symm f) = f.extendDomain | convert restrictSupportEquiv_symm_apply_coe .. | no goals | 297030a93a136f53 |
IsLocalization.isAlgebraic | Mathlib/RingTheory/Algebraic/Basic.lean | theorem IsLocalization.isAlgebraic [Nontrivial R] (M : Submonoid R) [IsLocalization M S] :
Algebra.IsAlgebraic R S where
isAlgebraic x | R : Type u
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial R
M : Submonoid R
inst✝ : IsLocalization M S
x : S
⊢ IsAlgebraic R x | obtain rfl | hx := eq_or_ne x 0 | case inl
R : Type u
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial R
M : Submonoid R
inst✝ : IsLocalization M S
⊢ IsAlgebraic R 0
case inr
R : Type u
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial R
M : Submonoid R
inst✝ : Is... | 4a59a48759350df5 |
Nat.mul_dvd_of_dvd_div | Mathlib/Data/Nat/Init.lean | lemma mul_dvd_of_dvd_div (hcb : c ∣ b) (h : a ∣ b / c) : c * a ∣ b :=
have ⟨d, hd⟩ := h
⟨d, by simpa [Nat.mul_comm, Nat.mul_left_comm] using Nat.eq_mul_of_div_eq_left hcb hd⟩
| a b c : ℕ
hcb : c ∣ b
h : a ∣ b / c
d : ℕ
hd : b / c = a * d
⊢ b = c * a * d | simpa [Nat.mul_comm, Nat.mul_left_comm] using Nat.eq_mul_of_div_eq_left hcb hd | no goals | 93b26754f3259100 |
Substring.Valid.nextn_stop | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem nextn_stop : ∀ {s}, Valid s → ∀ n, s.nextn n ⟨s.bsize⟩ = ⟨s.bsize⟩
| _, h, n => let ⟨l, m, r, h⟩ := h.validFor; by simp [h.bsize, h.nextn_stop]
| x✝ : Substring
h✝ : x✝.Valid
n : Nat
l m r : List Char
h : ValidFor l m r x✝
⊢ x✝.nextn n { byteIdx := x✝.bsize } = { byteIdx := x✝.bsize } | simp [h.bsize, h.nextn_stop] | no goals | 11cc58e5d13d8851 |
WittVector.map_teichmullerFun | Mathlib/RingTheory/WittVector/Teichmuller.lean | theorem map_teichmullerFun (f : R →+* S) (r : R) :
map f (teichmullerFun p r) = teichmullerFun p (f r) | case h
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
n : ℕ
⊢ ((map f) (teichmullerFun p r)).coeff n = (teichmullerFun p (f r)).coeff n | cases n | case h.zero
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
⊢ ((map f) (teichmullerFun p r)).coeff 0 = (teichmullerFun p (f r)).coeff 0
case h.succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S... | 48b7544a3f2c048c |
WithBot.unbot_le_iff | Mathlib/Order/WithBot.lean | lemma unbot_le_iff (hx : x ≠ ⊥) : unbot x hx ≤ b ↔ x ≤ b | case intro
α : Type u_1
b : α
inst✝ : LE α
x : α
hx : ↑x ≠ ⊥
⊢ (↑x).unbot hx ≤ b ↔ ↑x ≤ ↑b | simp | no goals | 76001aedab5074c3 |
LowerSemicontinuousWithinAt.add' | Mathlib/Topology/Semicontinuous.lean | theorem LowerSemicontinuousWithinAt.add' {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
(hg : LowerSemicontinuousWithinAt g s x)
(hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
LowerSemicontinuousWithinAt (fun z => f z + g z) s x | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
x : α
s : Set α
γ : Type u_4
inst✝² : LinearOrderedAddCommMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p =... | by_cases hx₁ : ∃ l, l < f x | case pos
α : Type u_1
inst✝³ : TopologicalSpace α
x : α
s : Set α
γ : Type u_4
inst✝² : LinearOrderedAddCommMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x)
y : γ
hy... | 4df49a8a802d230c |
Std.DHashMap.Internal.Raw₀.wfImp_insertₘ | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem wfImp_insertₘ [BEq α] [Hashable α] [EquivBEq α] [LawfulHashable α] {m : Raw₀ α β}
(h : Raw.WFImp m.1) {a : α} {b : β a} : Raw.WFImp (m.insertₘ a b).1 | case isTrue
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
h✝ : m.containsₘ a = true
⊢ Raw.WFImp (m.replaceₘ a b).val | apply wfImp_replaceₘ _ h | no goals | 6484ce9677e56de0 |
Asymptotics.IsBigO.sum | Mathlib/Analysis/Asymptotics/Defs.lean | theorem IsBigO.sum (h : ∀ i ∈ s, A i =O[l] g) : (fun x => ∑ i ∈ s, A i x) =O[l] g | α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
l : Filter α
ι : Type u_17
A : ι → α → E'
s : Finset ι
h : ∀ i ∈ s, ∃ c, IsBigOWith c l (A i) g
⊢ ∃ c, IsBigOWith c l (fun x => ∑ i ∈ s, A i x) g | choose! C hC using h | α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
l : Filter α
ι : Type u_17
A : ι → α → E'
s : Finset ι
C : ι → ℝ
hC : ∀ i ∈ s, IsBigOWith (C i) l (A i) g
⊢ ∃ c, IsBigOWith c l (fun x => ∑ i ∈ s, A i x) g | 04f489bd1f07a4c4 |
irrational_pi | Mathlib/Data/Real/Pi/Irrational.lean | theorem irrational_pi : Irrational π | case intro.intro.intro
h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
ha : 0 < ↑a
k : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !
⊢ False | have j : ∀ᶠ n : ℕ in atTop, (a : ℝ) ^ (2 * n + 1) / n ! * I n (π / 2) < 1 := by
have := (tendsto_pow_div_factorial_at_top_aux a).eventually_lt_const
(show (0 : ℝ) < 1 / 2 by norm_num)
filter_upwards [this] with n hn
rw [lt_div_iff₀ (zero_lt_two : (0 : ℝ) < 2)] at hn
exact hn.trans_le' (mul_le_mul_of_nonneg_... | case intro.intro.intro
h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
ha : 0 < ↑a
k : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !
j : ∀ᶠ (n : ℕ) in atTop, ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1
⊢ False | 0d163f7bb7f07f4a |
Finset.prod_nat_mod | Mathlib/Algebra/BigOperators/Group/Finset/Defs.lean | theorem prod_nat_mod (s : Finset α) (n : ℕ) (f : α → ℕ) :
(∏ i ∈ s, f i) % n = (∏ i ∈ s, f i % n) % n :=
(Multiset.prod_nat_mod _ _).trans <| by rw [Finset.prod, Multiset.map_map]; rfl
| α : Type u_3
s : Finset α
n : ℕ
f : α → ℕ
⊢ (Multiset.map ((fun x => x % n) ∘ fun i => f i) s.val).prod % n = (Multiset.map (fun i => f i % n) s.val).prod % n | rfl | no goals | fd9c3d5518b7775b |
CategoryTheory.simple_of_cosimple | Mathlib/CategoryTheory/Simple.lean | theorem simple_of_cosimple (X : C) (h : ∀ {Z : C} (f : X ⟶ Z) [Epi f], IsIso f ↔ f ≠ 0) :
Simple X :=
⟨fun {Y} f I => by
classical
fconstructor
· intros
have hx := cokernel.π_of_epi f
by_contra h
subst h
exact (h _).mp (cokernel.π_of_zero _ _) hx
· intro hf
... | case mpr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
h : ∀ {Z : C} (f : X ⟶ Z) [inst : Epi f], IsIso f ↔ f ≠ 0
Y : C
f : Y ⟶ X
I : Mono f
⊢ f ≠ 0 → IsIso f | intro hf | case mpr
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
h : ∀ {Z : C} (f : X ⟶ Z) [inst : Epi f], IsIso f ↔ f ≠ 0
Y : C
f : Y ⟶ X
I : Mono f
hf : f ≠ 0
⊢ IsIso f | aea3a25865e3da1b |
SimplexCategory.eq_σ_of_epi | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem eq_σ_of_epi {n : ℕ} (θ : mk (n + 1) ⟶ mk n) [Epi θ] : ∃ i : Fin (n + 1), θ = σ i | n : ℕ
θ : ⦋n + 1⦌ ⟶ ⦋n⦌
inst✝ : Epi θ
i : Fin (n + 1)
θ' : ⦋n⦌ ⟶ ⦋n⦌
h : θ = σ i ≫ θ'
⊢ Epi θ | infer_instance | no goals | 8332dba6c017b8b2 |
factorization_eq_count | Mathlib/RingTheory/UniqueFactorizationDomain/Finsupp.lean | theorem factorization_eq_count {n p : α} :
factorization n p = Multiset.count p (normalizedFactors n) | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
n p : α
⊢ (factorization n) p = Multiset.count p (normalizedFactors n) | simp [factorization] | no goals | baae5c8c4c3735f0 |
Set.surjective_iff_surjective_of_iUnion_eq_univ | Mathlib/Data/Set/Lattice.lean | theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) | α : Type u_1
β : Type u_2
ι : Sort u_5
f : α → β
U : ι → Set β
hU : iUnion U = univ
H : ∀ (i : ι), Surjective ((U i).restrictPreimage f)
x : β
⊢ ∃ a, f a = x | obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; trivial) | case intro
α : Type u_1
β : Type u_2
ι : Sort u_5
f : α → β
U : ι → Set β
hU : iUnion U = univ
H : ∀ (i : ι), Surjective ((U i).restrictPreimage f)
x : β
i : ι
hi : x ∈ U i
⊢ ∃ a, f a = x | 7c358e04f2b1a1c0 |
Profinite.exists_isClopen_of_cofiltered | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V | case refine_3.intro.intro
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS :
S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑(ConcreteCategory.hom ((toTopCat.mapCone C).π.app j))... | have hUo : ∀ (i : ↑S), IsOpen ((fun s ↦ (C.π.app (j s)) ⁻¹' V s) i) := by
intro s
exact (hV s).1.2.preimage (C.π.app (j s)).hom.continuous | case refine_3.intro.intro
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
U : Set ↑C.pt.toTop
hC : IsLimit C
hU : IsClopen U
S : Set (Set ↑(toTopCat.mapCone C).pt)
hS :
S ⊆ {U | ∃ j, ∃ V ∈ (fun j => {W | IsClopen W}) j, U = ⇑(ConcreteCategory.hom ((toTopCat.mapCone C).π.app j))... | 0fb8cdb8a6309b67 |
Acc.ndrecOn_eq_ndrecOnC | Mathlib/.lake/packages/batteries/Batteries/WF.lean | theorem ndrecOn_eq_ndrecOnC : @Acc.ndrecOn = @Acc.ndrecOnC | ⊢ @ndrecOn = @Acc.ndrecOnC | funext α r motive intro a t | case h.h.h.h.h.h
α : Sort u_1
r : α → α → Prop
motive : α → Sort u_2
intro : α
a : Acc r intro
t : (x : α) → (∀ (y : α), r y x → Acc r y) → ((y : α) → r y x → motive y) → motive x
⊢ a.ndrecOn t = Acc.ndrecOnC a t | 42cb232bc3cb0dbf |
Real.geom_mean_eq_arith_mean_weighted_iff | Mathlib/Analysis/MeanInequalities.lean | theorem geom_mean_eq_arith_mean_weighted_iff (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i ↔ ∀ j ∈ s, w j ≠ 0 → z j = ∑ i ∈ s, w i * z i | ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∀ i ∈ s, w i * z i ≠ 0 → w i ≠ 0
h' : ∀ i ∈ s, z i ^ w i ≠ 1 → w i ≠ 0
⊢ ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i ↔ ∀ j ∈ s, w j ≠ 0 → z j = ∑ i ∈ s, w i * z i | rw [← sum_filter_of_ne h, ← prod_filter_of_ne h', geom_mean_eq_arith_mean_weighted_iff'] | ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∀ i ∈ s, w i * z i ≠ 0 → w i ≠ 0
h' : ∀ i ∈ s, z i ^ w i ≠ 1 → w i ≠ 0
⊢ (∀ j ∈ filter (fun x => w x ≠ 0) s, z j = ∑ i ∈ filter (fun x => w x ≠ 0) s, w i * z i) ↔
∀ j ∈ s, w j ≠ 0 → z j = ∑ x ∈ filter (fun x =... | ba9039bdd5218450 |
AlgebraicGeometry.isCompact_iff_exists | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | lemma isCompact_iff_exists {U : X.Opens} :
IsCompact (U : Set X) ↔ ∃ R, ∃ f : Spec R ⟶ X, Set.range f.base = U | X : Scheme
U : X.Opens
x✝ : ∃ R f, Set.range ⇑(ConcreteCategory.hom f.base) = ↑U
R : CommRingCat
f : Spec R ⟶ X
hf : Set.range ⇑(ConcreteCategory.hom f.base) = ↑U
⊢ Function.Surjective ⇑(ConcreteCategory.hom (IsOpenImmersion.lift U.ι f ⋯).base) | rw [← Set.range_eq_univ] | X : Scheme
U : X.Opens
x✝ : ∃ R f, Set.range ⇑(ConcreteCategory.hom f.base) = ↑U
R : CommRingCat
f : Spec R ⟶ X
hf : Set.range ⇑(ConcreteCategory.hom f.base) = ↑U
⊢ Set.range ⇑(ConcreteCategory.hom (IsOpenImmersion.lift U.ι f ⋯).base) = Set.univ | 966571c16bd3c244 |
InnerProductGeometry.sin_angle_sub_add_angle_sub_rev_eq_sin_angle | Mathlib/Geometry/Euclidean/Triangle.lean | theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) | case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
... | apply mul_right_cancel₀ hyn | case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
... | cff8220b46e80476 |
Std.DHashMap.Internal.exists_bucket' | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean | theorem exists_bucket' [BEq α] [Hashable α]
(self : Array (AssocList α β)) (i : USize) (hi : i.toNat < self.size) :
∃ l, Perm (self.toList.flatMap AssocList.toList) (self[i.toNat].toList ++ l) ∧
(∀ [LawfulHashable α], IsHashSelf self → ∀ k,
(mkIdx self.size (by omega) (hash k)).1.toNat = i.t... | α : Type u
β : α → Type v
γ : Type w
δ : α → Type w
inst✝² : BEq α
inst✝¹ : Hashable α
self : Array (AssocList α β)
i : USize
hi : i.toNat < self.size
l : List ((a : α) × β a)
inst✝ : LawfulHashable α
k : α
⊢ 0 < self.size | omega | no goals | f84b72c0203fd6d7 |
CategoryTheory.Pretriangulated.isIso₂_of_isIso₁₃ | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | lemma isIso₂_of_isIso₁₃ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)
(hT' : T' ∈ distTriang C) (h₁ : IsIso φ.hom₁) (h₃ : IsIso φ.hom₃) : IsIso φ.hom₂ | C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
f : A... | rw [← cancel_mono φ.hom₃, assoc, φ.comm₂, reassoc_of% hf, zero_comp, zero_comp] | no goals | 8a538ff25cabfee4 |
WellFoundedGT.finite_of_sSupIndep | Mathlib/Order/CompactlyGenerated/Basic.lean | theorem WellFoundedGT.finite_of_sSupIndep [WellFoundedGT α] {s : Set α}
(hs : sSupIndep s) : s.Finite | α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : WellFoundedGT α
s : Set α
hs : sSupIndep s
t : Finset α
ht₁ : ↑t ⊆ s
ht₂ : sSup s = t.sup id
x : α
hx₀ : x ∈ s
hx₁ : x ≠ ⊥
hx₂ : x ∉ t
⊢ x ∈ ↑t ∪ {x} | simp | no goals | 89115ca7270d48b8 |
NonUnitalStarAlgHom.coe_range | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | theorem coe_range (φ : F) :
((NonUnitalStarAlgHom.range φ : NonUnitalStarSubalgebra R B) : Set B) = Set.range (φ : A → B) :=
by ext; rw [SetLike.mem_coe, mem_range]; rfl
| case h
F : Type v'
R : Type u
A : Type v
B : Type w
inst✝⁹ : CommSemiring R
inst✝⁸ : NonUnitalNonAssocSemiring A
inst✝⁷ : Module R A
inst✝⁶ : Star A
inst✝⁵ : NonUnitalNonAssocSemiring B
inst✝⁴ : Module R B
inst✝³ : Star B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : StarHomClass F A B
φ : F
x✝ :... | rfl | no goals | ad5c66326651987b |
partialSups_succ | Mathlib/Order/PartialSups.lean | theorem partialSups_succ [LinearOrder ι] [LocallyFiniteOrderBot ι] [SuccOrder ι]
(f : ι → α) (i : ι) :
partialSups f (Order.succ i) = partialSups f i ⊔ f (Order.succ i) | α : Type u_1
ι : Type u_2
inst✝³ : SemilatticeSup α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
i : ι
⊢ (partialSups f) (Order.succ i) = (partialSups f) i ⊔ f (Order.succ i) | suffices Iic (Order.succ i) = Iic i ∪ {Order.succ i} by simp only [partialSups_apply, this,
sup'_union nonempty_Iic ⟨_, mem_singleton_self _⟩ f, sup'_singleton] | α : Type u_1
ι : Type u_2
inst✝³ : SemilatticeSup α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
i : ι
⊢ Iic (Order.succ i) = Iic i ∪ {Order.succ i} | d1b390e421b6e54e |
Submonoid.LocalizationMap.map_injective_of_injective | Mathlib/GroupTheory/MonoidLocalization/Basic.lean | theorem map_injective_of_injective (hg : Injective g) (k : LocalizationMap (S.map g) Q) :
Injective (map f (apply_coe_mem_map g S) k) := fun z w hizw ↦ by
set i := map f (apply_coe_mem_map g S) k
have ifkg (a : M) : i (f.toMap a) = k.toMap (g a) := map_eq f (apply_coe_mem_map g S) a
let ⟨z', w', x, hxz, hxw⟩ ... | M : Type u_1
inst✝³ : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝² : CommMonoid N
P : Type u_3
inst✝¹ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
Q : Type u_4
inst✝ : CommMonoid Q
hg : Injective ⇑g
k : (Submonoid.map g S).LocalizationMap Q
z w : N
i : N →* Q := f.map ⋯ k
hizw : i z = i w
ifkg : ∀ (a : M), i (... | obtain ⟨⟨_, c, hc, rfl⟩, eq⟩ := k.exists_of_eq _ _ this | case intro.mk.intro.intro
M : Type u_1
inst✝³ : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝² : CommMonoid N
P : Type u_3
inst✝¹ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
Q : Type u_4
inst✝ : CommMonoid Q
hg : Injective ⇑g
k : (Submonoid.map g S).LocalizationMap Q
z w : N
i : N →* Q := f.map ⋯ k
hizw : i z =... | e98ac16ea22fdabe |
MeasureTheory.tendsto_Lp_finite_of_tendsto_ae_of_meas | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem tendsto_Lp_finite_of_tendsto_ae_of_meas [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
{f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hg' : MemLp g p μ) (hui : UnifIntegrable f p μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
Tendst... | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun ... | norm_num | no goals | ef79cbe75454a3e9 |
unique_topology_of_t2 | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | theorem unique_topology_of_t2 {t : TopologicalSpace 𝕜} (h₁ : @IsTopologicalAddGroup 𝕜 t _)
(h₂ : @ContinuousSMul 𝕜 𝕜 _ hnorm.toUniformSpace.toTopologicalSpace t) (h₃ : @T2Space 𝕜 t) :
t = hnorm.toUniformSpace.toTopologicalSpace | case refine_1
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
⊢ ∀ (i' : ℝ), 0 < i' → Metric.closedBall 0 i' ∈ 𝓝 0 | intro ε hε | case refine_1
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
⊢ Metric.closedBall 0 ε ∈ 𝓝 0 | fe6a4018e5de36d9 |
ZMod.differentiableAt_LFunction | Mathlib/NumberTheory/LSeries/ZMod.lean | lemma differentiableAt_LFunction (Φ : ZMod N → ℂ) (s : ℂ) (hs : s ≠ 1 ∨ ∑ j, Φ j = 0) :
DifferentiableAt ℂ (LFunction Φ) s | case inr
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hs : 1 ≠ 1 ∨ ∑ j : ZMod N, Φ j = 0
this : DifferentiableAt ℂ (fun y => ∑ i : ZMod N, Φ i * (hurwitzZeta (toAddCircle i) y - 1 / (y - 1) / y.Gammaℝ)) 1
⊢ DifferentiableAt ℂ (fun y => ∑ j : ZMod N, Φ j * hurwitzZeta (toAddCircle j) y) 1 | simpa only [mul_sub, sum_sub_distrib, ← sum_mul, hs.neg_resolve_left rfl, zero_mul, sub_zero] | no goals | 0c2b6a66626a5923 |
SetTheory.PGame.star_fuzzy_down | Mathlib/SetTheory/Game/PGame.lean | theorem star_fuzzy_down : star ‖ down | ⊢ star ‖ up | exact star_fuzzy_up | no goals | 41ac60ba29e6dcc7 |
List.eraseIdx_replicate | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem eraseIdx_replicate {n : Nat} {a : α} {k : Nat} :
(replicate n a).eraseIdx k = if k < n then replicate (n - 1) a else replicate n a | case isTrue
α : Type u_1
n : Nat
a : α
k : Nat
h : k < n
b : α
m : b ∈ replicate n a
⊢ b = a | simp only [mem_replicate] at m | case isTrue
α : Type u_1
n : Nat
a : α
k : Nat
h : k < n
b : α
m : n ≠ 0 ∧ b = a
⊢ b = a | eaeb466d41ba696c |
ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | theorem exists_mem_subalgebra_near_continuousMap_of_separatesPoints (A : Subalgebra ℝ C(X, ℝ))
(w : A.SeparatesPoints) (f : C(X, ℝ)) (ε : ℝ) (pos : 0 < ε) :
∃ g : A, ‖(g : C(X, ℝ)) - f‖ < ε | case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
w✝ : A.SeparatesPoints
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring
g : C(X, ℝ)
H : ‖g - f‖ < ε
m : g ∈ ↑A.toSubsemiring
⊢ ∃ g, ‖↑g - f‖ < ε | exact ⟨⟨g, m⟩, H⟩ | no goals | 65a204744442be19 |
Option.isSome_ite | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean | theorem isSome_ite {p : Prop} [Decidable p] :
(if p then some b else none).isSome = true ↔ p | α✝ : Type u_1
b : α✝
p : Prop
inst✝ : Decidable p
⊢ (if p then some b else none).isSome = true ↔ p | split <;> simpa | no goals | e47e88f5521d0f0b |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | case h
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1... | exact sum_le_tsum _ (fun _ _ ↦ by positivity)
(summable_geometric_of_lt_one (by positivity) zn) | no goals | e1f8d437480849f9 |
Submodule.ClosedComplemented.of_quotient_finiteDimensional | Mathlib/Analysis/Normed/Module/Complemented.lean | theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace 𝕜]
[FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented | case intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
p : Subspace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 (E ⧸ p)
hp : IsClosed ↑p
q : Subspace 𝕜 E
hq : IsCompl p q
⊢ ClosedComplemented p | haveI : FiniteDimensional 𝕜 q := (p.quotientEquivOfIsCompl q hq).finiteDimensional | case intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : CompleteSpace E
p : Subspace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 (E ⧸ p)
hp : IsClosed ↑p
q : Subspace 𝕜 E
hq : IsCompl p q
this : FiniteDimensional 𝕜 ↥q
⊢... | e57481270e3fef5a |
pairwise_isRelPrime_iff_isRelPrime_prod | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) | case refine_1
α : Type u_2
I : Type u_1
inst✝² : CommMonoid α
inst✝¹ : DecompositionMonoid α
s : I → α
t : Finset I
inst✝ : DecidableEq I
hp : Pairwise (IsRelPrime on fun i => s ↑i)
i : I
hi : i ∈ t
j : I
hj : j ∈ t \ {i}
⊢ IsRelPrime (s i) (s j) | rw [Finset.mem_sdiff, Finset.mem_singleton] at hj | case refine_1
α : Type u_2
I : Type u_1
inst✝² : CommMonoid α
inst✝¹ : DecompositionMonoid α
s : I → α
t : Finset I
inst✝ : DecidableEq I
hp : Pairwise (IsRelPrime on fun i => s ↑i)
i : I
hi : i ∈ t
j : I
hj : j ∈ t ∧ ¬j = i
⊢ IsRelPrime (s i) (s j) | f359de6c0d389310 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastZeroExtend.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ZeroExtend.lean | theorem go_get_aux (aig : AIG α) (w : Nat) (input : AIG.RefVec aig w) (newWidth curr : Nat)
(hcurr : curr ≤ newWidth) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr) (hfoo),
(go aig w input newWidth curr hcurr s).vec.get idx (by omega)
=
(s.get idx hidx).cast hfoo | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
w : Nat
input : aig.RefVec w
newWidth curr : Nat
hcurr : curr ≤ newWidth
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α newWidth
h✝ : ¬curr < newWidth
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ curr = newWidth | omega | no goals | aa7bc21c8b54f977 |
Set.Finite.exists_injOn_of_encard_le | Mathlib/Data/Set/Card.lean | theorem Finite.exists_injOn_of_encard_le [Nonempty β] {s : Set α} {t : Set β} (hs : s.Finite)
(hle : s.encard ≤ t.encard) : ∃ (f : α → β), s ⊆ f ⁻¹' t ∧ InjOn f s | case h.refine_2
α : Type u_1
β : Type u_2
inst✝ : Nonempty β
s : Set α
t : Set β
hs : s.Finite
hle : s.encard ≤ t.encard
a : α
has : a ∈ s
b : β
hbt : b ∈ t
hle' : (s \ {a}).encard ≤ (t \ {b}).encard
f₀ : α → β
hinj : InjOn f₀ (s \ {a})
hf₀s : ∀ x ∈ s, ¬x = a → f₀ x ∈ t ∧ ¬f₀ x = b
⊢ InjOn (Function.update f₀ a b) (s \... | exact InjOn.congr hinj (fun x ⟨_, hxa⟩ ↦ by rwa [Function.update_of_ne]) | no goals | 827f94f416b0baac |
Polynomial.derivative_mul | Mathlib/Algebra/Polynomial/Derivative.lean | theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g | case h_add
R : Type u
inst✝ : Semiring R
g p✝ q✝ : R[X]
a✝¹ : derivative (p✝ * g) = derivative p✝ * g + p✝ * derivative g
a✝ : derivative (q✝ * g) = derivative q✝ * g + q✝ * derivative g
⊢ derivative ((p✝ + q✝) * g) = derivative (p✝ + q✝) * g + (p✝ + q✝) * derivative g | simp only [add_mul, map_add, add_assoc, add_left_comm, *] | no goals | a66065c70ef80a14 |
List.mapIdxMAux'_eq_mapIdxMGo | Mathlib/Data/List/Indexes.lean | theorem mapIdxMAux'_eq_mapIdxMGo {α} (f : ℕ → α → m PUnit) (as : List α) (arr : Array PUnit) :
mapIdxMAux' f arr.size as = mapIdxM.go f as arr *> pure PUnit.unit | case cons
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Type u_1
f : ℕ → α → m PUnit.{u + 1}
head : α
tail : List α
ih : ∀ (arr : Array PUnit.{u + 1}), mapIdxMAux' f arr.size tail = mapIdxM.go f tail arr *> pure PUnit.unit
arr : Array PUnit.{u + 1}
⊢ (do
f arr.size head
mapIdxMAux' f (arr.s... | generalize (f (Array.size arr) head) = head | case cons
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Type u_1
f : ℕ → α → m PUnit.{u + 1}
head✝ : α
tail : List α
ih : ∀ (arr : Array PUnit.{u + 1}), mapIdxMAux' f arr.size tail = mapIdxM.go f tail arr *> pure PUnit.unit
arr : Array PUnit.{u + 1}
head : m PUnit.{u + 1}
⊢ (do
head
mapIdxM... | 1afc045393ac85a0 |
ContinuousMultilinearMap.iteratedFDeriv_comp_diagonal | Mathlib/Analysis/Analytic/IteratedFDeriv.lean | lemma ContinuousMultilinearMap.iteratedFDeriv_comp_diagonal
{n : ℕ} (f : E [×n]→L[𝕜] F) (x : E) (v : Fin n → E) :
iteratedFDeriv 𝕜 n (fun x ↦ f (fun _ ↦ x)) x v = ∑ σ : Perm (Fin n), f (fun i ↦ v (σ i)) | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : ContinuousMultilinearMap 𝕜 (fun i => E) F
x : E
v : Fin n → E
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i => Co... | change iteratedFDeriv 𝕜 n (f ∘ g) x v = _ | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : ContinuousMultilinearMap 𝕜 (fun i => E) F
x : E
v : Fin n → E
g : E →L[𝕜] Fin n → E := ContinuousLinearMap.pi fun i => Co... | 541458533d92b1c7 |
AffineSubspace.mem_direction_iff_eq_vsub_right | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean | theorem mem_direction_iff_eq_vsub_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) :
v ∈ s.direction ↔ ∃ p₂ ∈ s, v = p₂ -ᵥ p | 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
s : AffineSubspace k P
p : P
hp : p ∈ s
v : V
⊢ v ∈ (fun x => x -ᵥ p) '' ↑s ↔ ∃ p₂ ∈ s, v = p₂ -ᵥ p | exact ⟨fun ⟨p₂, hp₂, hv⟩ => ⟨p₂, hp₂, hv.symm⟩, fun ⟨p₂, hp₂, hv⟩ => ⟨p₂, hp₂, hv.symm⟩⟩ | no goals | 6ef117614ac497e5 |
NNReal.inner_le_Lp_mul_Lq | Mathlib/Analysis/MeanInequalities.lean | theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) :
∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) | case inr.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i => g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, ... | suffices (∑ i ∈ s, f' i * g' i) ≤ 1 by
simp_rw [f', g', div_mul_div_comm, ← sum_div] at this
rwa [div_le_iff₀, one_mul] at this
exact mul_pos (rpow_pos hf) (rpow_pos hg) | case inr.inr
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i => f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i => g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ ∑ i ∈ s, f' i * g' i ≤ 1 | 58a496d0d9e63f48 |
RootPairing.pairing_smul_root_eq_of_not_linInd | Mathlib/LinearAlgebra/RootSystem/Reduced.lean | lemma pairing_smul_root_eq_of_not_linInd [NeZero (2 : R)] [NoZeroSMulDivisors R M]
(h : ¬ LinearIndependent R ![P.root i, P.root j]) :
P.pairing j i • P.root i = (2 : R) • P.root j | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
i j : ι
inst✝¹ : NeZero 2
inst✝ : NoZeroSMulDivisors R M
t : R
h₁ : 0 • P.root i + t • P.root j = 0
h₂ : 0 = 0 → t ≠ 0
⊢ t • P.root j = ... | simpa using h₁ | no goals | 1e9f3d77c0025a81 |
Module.Flat.exists_factorization_of_comp_eq_zero_of_free | Mathlib/RingTheory/Flat/EquationalCriterion.lean | theorem exists_factorization_of_comp_eq_zero_of_free [Flat R M] {K N : Type*} [AddCommGroup K]
[Module R K] [Module.Finite R K] [AddCommGroup N] [Module R N] [Free R N] [Module.Finite R N]
{f : K →ₗ[R] N} {x : N →ₗ[R] M} (h : x ∘ₗ f = 0) :
∃ (k : ℕ) (a : N →ₗ[R] (Fin k →₀ R)) (y : (Fin k →₀ R) →ₗ[R] M),
... | R : Type u_1
M : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : Flat R M
K : Type u_3
N : Type u_4
inst✝⁶ : AddCommGroup K
inst✝⁵ : Module R K
inst✝⁴ : Module.Finite R K
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : K →ₗ[R] N
x : N →ₗ[R] ... | ext | case h
R : Type u_1
M : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : Flat R M
K : Type u_3
N : Type u_4
inst✝⁶ : AddCommGroup K
inst✝⁵ : Module R K
inst✝⁴ : Module.Finite R K
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Free R N
inst✝ : Module.Finite R N
f : K →ₗ[R] N
x : N... | f413a07dc0009ca8 |
gauge_lt_eq' | Mathlib/Analysis/Convex/Gauge.lean | theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
{ x | gauge s x < a } = ⋃ (r : ℝ) (_ : 0 < r) (_ : r < a), r • s | case h
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
absorbs : Absorbent ℝ s
a : ℝ
x✝ : E
⊢ gauge s x✝ < a ↔ ∃ i, 0 < i ∧ i < a ∧ x✝ ∈ i • s | exact
⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
(gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩ | no goals | b0f0c0c69423d9cf |
CategoryTheory.OverPresheafAux.OverArrows.map_val | Mathlib/CategoryTheory/Comma/Presheaf/Basic.lean | /-- In the special case `F = yoneda.obj Y`, the element `p.val` for `p : OverArrows η s` is itself
a morphism `X ⟶ Y`. -/
@[simp]
lemma map_val {Y : C} {η : yoneda.obj Y ⟶ A} {X : C} {s : yoneda.obj X ⟶ A}
(p : OverArrows η s) : yoneda.map p.val ≫ η = s | C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
Y : C
η : yoneda.obj Y ⟶ A
X : C
s : yoneda.obj X ⟶ A
p : OverArrows η s
⊢ yoneda.map p.val ≫ η = s | rw [← yonedaEquiv.injective.eq_iff, yonedaEquiv_comp, yonedaEquiv_yoneda_map] | C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
Y : C
η : yoneda.obj Y ⟶ A
X : C
s : yoneda.obj X ⟶ A
p : OverArrows η s
⊢ η.app (op (unop (op X))) p.val = yonedaEquiv s | 68e2c43c118ae876 |
ProbabilityTheory.iteratedDeriv_two_cgf_eq_integral | Mathlib/Probability/Moments/MGFAnalytic.lean | lemma iteratedDeriv_two_cgf_eq_integral (h : v ∈ interior (integrableExpSet X μ)) :
iteratedDeriv 2 (cgf X μ) v
= μ[fun ω ↦ (X ω - deriv (cgf X μ) v)^2 * exp (v * X ω)] / mgf X μ v | case h
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
x✝ : ℝ
⊢ deriv 0 x✝ = 0 x✝ | exact deriv_const _ 0 | no goals | f81a041fb4b6f1f5 |
preNormEDS_even | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | lemma preNormEDS_even (m : ℤ) : preNormEDS b c d (2 * m) =
preNormEDS b c d (m - 1) ^ 2 * preNormEDS b c d m * preNormEDS b c d (m + 2) -
preNormEDS b c d (m - 2) * preNormEDS b c d m * preNormEDS b c d (m + 1) ^ 2 | case nat.succ.zero
R : Type u
inst✝ : CommRing R
b c d : R
⊢ preNormEDS b c d (2 * ↑(0 + 1)) =
preNormEDS b c d (↑(0 + 1) - 1) ^ 2 * preNormEDS b c d ↑(0 + 1) * preNormEDS b c d (↑(0 + 1) + 2) -
preNormEDS b c d (↑(0 + 1) - 2) * preNormEDS b c d ↑(0 + 1) * preNormEDS b c d (↑(0 + 1) + 1) ^ 2 | simp | no goals | 04b24e60b18bcf70 |
Fintype.linearIndependent_iff | Mathlib/LinearAlgebra/LinearIndependent/Defs.lean | theorem Fintype.linearIndependent_iff [Fintype ι] :
LinearIndependent R v ↔ ∀ g : ι → R, ∑ i, g i • v i = 0 → ∀ i, g i = 0 | ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Fintype ι
H : LinearIndependent R v
g : ι → R
⊢ ∑ i : ι, g i • v i = 0 → ∀ (i : ι), g i = 0 | simpa using linearIndependent_iff'.1 H Finset.univ g | no goals | b1aaadece3457b97 |
isCompact_of_finite_subcover | Mathlib/Topology/Compactness/Compact.lean | theorem isCompact_of_finite_subcover
(h : ∀ {ι : Type u} (U : ι → Set X), (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) →
∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i) :
IsCompact s := fun f hf hfs => by
contrapose! h
simp only [ClusterPt, not_neBot, ← disjoint_iff, SetCoe.forall',
(nhds_basis_opens _).disjoint_iff_le... | X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
t : Finset ↑s
ht : s ⊆ ⋃ i ∈ t, U i
⊢ False | refine compl_not_mem (le_principal_iff.1 hfs) ?_ | X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
t : Finset ↑s
ht : s ⊆ ⋃ i ∈ t, U i
⊢ sᶜ ∈ f | cb8cf4f8b7fb7a73 |
List.forall_getElem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem forall_getElem {l : List α} {p : α → Prop} :
(∀ (i : Nat) h, p (l[i]'h)) ↔ ∀ a, a ∈ l → p a | case cons.mpr.intro.zero
α : Type u_1
p : α → Prop
a : α
l : List α
ih : (∀ (i : Nat) (h : i < l.length), p l[i]) ↔ ∀ (a : α), a ∈ l → p a
h✝ : p a
w : ∀ (a : α), a ∈ l → p a
h : 0 < l.length + 1
⊢ p (a :: l)[0] | simpa | no goals | 1581eec4ab0c2b01 |
Std.DHashMap.Raw.Const.ofList_cons | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem ofList_cons {k : α} {v : β} {tl : List (α × β)} :
ofList (⟨k, v⟩ :: tl) = insertMany ((∅ : Raw α (fun _ => β)).insert k v) tl | α : Type u
inst✝¹ : BEq α
inst✝ : Hashable α
β : Type v
k : α
v : β
tl : List (α × β)
⊢ (Raw₀.Const.insertMany Raw₀.empty ((k, v) :: tl)).val.val =
(Raw₀.Const.insertMany (Raw₀.insert ⟨Raw₀.empty.val, ⋯⟩ k v) tl).val.val | rw [Raw₀.Const.insertMany_empty_list_cons] | no goals | 68fd004a1dcb62cd |
HurwitzKernelBounds.isBigO_atTop_F_nat_one | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | lemma isBigO_atTop_F_nat_one {a : ℝ} (ha : 0 ≤ a) : ∃ p, 0 < p ∧
F_nat 1 a =O[atTop] fun t ↦ exp (-p * t) | a : ℝ
ha : 0 ≤ a
this :
∃ p,
0 < p ∧
(fun t =>
rexp (-π * (a ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ 2 +
a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop]
fun t => rexp (-p * t)
⊢ ∃ p, 0 < p ∧ F_nat 1 a =O[atTop] fun t => rexp (-p * t) | let ⟨p, hp, hp'⟩ := this | a : ℝ
ha : 0 ≤ a
this :
∃ p,
0 < p ∧
(fun t =>
rexp (-π * (a ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ 2 +
a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop]
fun t => rexp (-p * t)
p : ℝ
hp : 0 < p
hp' :
(fun t =>
rexp (-π * (a ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ ... | 5a5132f25c8455c4 |
CategoryTheory.Abelian.PreservesCoimage.hom_coimageImageComparison | Mathlib/CategoryTheory/Limits/Preserves/Shapes/AbelianImages.lean | theorem PreservesCoimage.hom_coimageImageComparison :
(PreservesCoimage.iso F f).hom ≫ coimageImageComparison (F.map f) =
F.map (coimageImageComparison f) ≫ (PreservesImage.iso F f).hom | C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasKernels C
inst✝⁷ : HasCokernels C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
inst✝⁵ : HasZeroMorphisms D
inst✝⁴ : HasKernels D
inst✝³ : HasCokernels D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : PreservesLimitsOfShape WalkingParall... | simp [← Functor.map_comp, ← Iso.eq_inv_comp, ← cancel_epi (Abelian.coimage.π (F.map f)),
← cancel_mono (Abelian.image.ι (F.map f))] | no goals | 4569b60e94e34f53 |
CategoryTheory.Limits.pullback_lift_diagonal_isPullback | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | /-- Informally, this is a special case of `pullback_map_diagonal_isPullback` for `T = X`. -/
lemma pullback_lift_diagonal_isPullback (g : Y ⟶ X) (f : X ⟶ S) :
IsPullback g (pullback.lift (𝟙 Y) g (by simp)) (diagonal f)
(pullback.map (g ≫ f) f f f g (𝟙 X) (𝟙 S) (by simp) (by simp)) | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y : C
inst✝ : HasPullbacks C
S : C
g : Y ⟶ X
f : X ⟶ S
i : pullback (g ≫ f) f ≅ pullback (g ≫ f) (𝟙 X ≫ f) := congrHom ⋯ ⋯
⊢ diagonal f ≫ 𝟙 (diagonalObj f) = 𝟙 X ≫ diagonal f | simp | no goals | 7a4ceef0e2b9343a |
Localization.mulEquivOfQuotient_monoidOf | Mathlib/GroupTheory/MonoidLocalization/Basic.lean | theorem mulEquivOfQuotient_monoidOf (x) :
mulEquivOfQuotient f ((monoidOf S).toMap x) = f.toMap x | M : Type u_1
inst✝¹ : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝ : CommMonoid N
f : S.LocalizationMap N
x : M
⊢ (mulEquivOfQuotient f) ((monoidOf S).toMap x) = f.toMap x | simp | no goals | f7db1d54d5902124 |
Stream'.Seq1.join_join | Mathlib/Data/Seq/Seq.lean | theorem join_join (SS : Seq (Seq1 (Seq1 α))) :
Seq.join (Seq.join SS) = Seq.join (Seq.map join SS) | case bisim
α : Type u
SS : Seq (Seq1 (Seq1 α))
⊢ Seq.IsBisimulation fun s1 s2 => ∃ s SS, s1 = s.append SS.join.join ∧ s2 = s.append (Seq.map join SS).join | intro s1 s2 h | case bisim
α : Type u
SS : Seq (Seq1 (Seq1 α))
s1 s2 : Seq α
h : ∃ s SS, s1 = s.append SS.join.join ∧ s2 = s.append (Seq.map join SS).join
⊢ BisimO (fun s1 s2 => ∃ s SS, s1 = s.append SS.join.join ∧ s2 = s.append (Seq.map join SS).join) s1.destruct
s2.destruct | d27084c911526143 |
QuotientAddGroup.norm_lift_apply_le | Mathlib/Analysis/Normed/Group/Quotient.lean | theorem _root_.QuotientAddGroup.norm_lift_apply_le {S : AddSubgroup M} (f : NormedAddGroupHom M N)
(hf : ∀ x ∈ S, f x = 0) (x : M ⧸ S) : ‖lift S f.toAddMonoidHom hf x‖ ≤ ‖f‖ * ‖x‖ | case inr.intro.intro
M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
S : AddSubgroup M
f : NormedAddGroupHom M N
hf : ∀ x ∈ S, f x = 0
h : 0 < ‖f‖
x : M
hx : ‖x‖ < ‖(lift S f.toAddMonoidHom hf) ↑x‖ / ‖f‖
⊢ False | exact ((lt_div_iff₀' h).1 hx).not_le (le_opNorm f x) | no goals | 492da0d4345816ea |
Complex.hasDerivAt_log_sub_logTaylor | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | lemma hasDerivAt_log_sub_logTaylor (n : ℕ) {z : ℂ} (hz : 1 + z ∈ slitPlane) :
HasDerivAt (fun z : ℂ ↦ log (1 + z) - logTaylor (n + 1) z) ((-z) ^ n * (1 + z)⁻¹) z | case h.e'_9
n : ℕ
z : ℂ
hz : 1 + z ∈ slitPlane
hz' : -z ≠ 1
⊢ (-z) ^ n * (1 + z)⁻¹ = (1 + z)⁻¹ - ∑ j ∈ Finset.range n, (-1) ^ j * z ^ j | simp_rw [← mul_pow, neg_one_mul, geom_sum_eq hz', ← neg_add', div_neg, add_comm z] | case h.e'_9
n : ℕ
z : ℂ
hz : 1 + z ∈ slitPlane
hz' : -z ≠ 1
⊢ (-z) ^ n * (1 + z)⁻¹ = (1 + z)⁻¹ - -(((-z) ^ n - 1) / (1 + z)) | 9fee484bfe2a66d5 |
Pell.exists_of_not_isSquare | Mathlib/NumberTheory/Pell.lean | theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 | d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |ξ| + |↑q| ≤ 2 * |ξ| + 1 | rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add'] | d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := √↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |↑q| - |ξ| ≤ 1 | e113d5ba2b34ead4 |
ENNReal.log_strictMono | Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean | theorem log_strictMono : StrictMono log | case inr.inr.inl
x : ℝ≥0∞
x_real : 0 < x.toReal
h : x < ⊥
⊢ False | exact not_lt_bot h | no goals | 42b90fcfa0c95640 |
Rat.fract_inv_num_lt_num_of_pos | Mathlib/Data/Rat/Floor.lean | theorem fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (fract q⁻¹).num < q.num | q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
⊢ (q_inv - ↑⌊q_inv⌋).num < q.num | suffices ((q.den - q.num * ⌊q_inv⌋ : ℚ) / q.num).num < q.num by
field_simp [q_inv, this, ne_of_gt q_num_pos] | q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
⊢ ((↑q.den - ↑q.num * ↑⌊q_inv⌋) / ↑q.num).num < q.num | d2870a8c7e295972 |
Set.preimage_eq_preimage' | Mathlib/Data/Set/Image.lean | theorem preimage_eq_preimage' {s t : Set α} {f : β → α} (hs : s ⊆ range f) (ht : t ⊆ range f) :
f ⁻¹' s = f ⁻¹' t ↔ s = t | case mp.h₂
α : Type u_1
β : Type u_2
s t : Set α
f : β → α
hs : s ⊆ range f
ht : t ⊆ range f
h : f ⁻¹' s = f ⁻¹' t
⊢ t ⊆ s | rw [← preimage_subset_preimage_iff ht, h] | no goals | f9d58e98041c47bd |
List.enumFrom_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem enumFrom_eq_append_iff {l : List α} {n : Nat} :
l.enumFrom n = l₁ ++ l₂ ↔
∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.enumFrom n ∧ l₂ = l₂'.enumFrom (n + l₁'.length) | case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
n : Nat
y z : List α
⊢ (range' (n + y.length) (y.length + z.length - y.length)).zip z = (range' (n + y.length) z.length).zip z | congr | case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.e_a.e_len
α : Type u_1
n : Nat
y z : List α
⊢ y.length + z.length - y.length = z.length | b7be04d5d60d3f47 |
Polynomial.IsSeparableContraction.natSepDegree_eq | Mathlib/FieldTheory/SeparableDegree.lean | theorem IsSeparableContraction.natSepDegree_eq {g : Polynomial F} {q : ℕ} [ExpChar F q]
(h : IsSeparableContraction q f g) : f.natSepDegree = g.natDegree | F : Type u
inst✝¹ : Field F
f g : F[X]
q : ℕ
inst✝ : ExpChar F q
h : IsSeparableContraction q f g
⊢ f.natSepDegree = g.natDegree | obtain ⟨h1, m, h2⟩ := h | case intro.intro
F : Type u
inst✝¹ : Field F
f g : F[X]
q : ℕ
inst✝ : ExpChar F q
h1 : g.Separable
m : ℕ
h2 : (expand F (q ^ m)) g = f
⊢ f.natSepDegree = g.natDegree | c644c20c6dab910a |
Ordinal.enumOrd_isNormal_iff_isClosed | Mathlib/SetTheory/Ordinal/Topology.lean | theorem enumOrd_isNormal_iff_isClosed (hs : ¬ BddAbove s) :
IsNormal (enumOrd s) ↔ IsClosed s | case refine_1.e_s.h
s : Set Ordinal.{u}
hs : ¬BddAbove s
Hs : StrictMono (enumOrd s)
h : IsNormal (enumOrd s)
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso s hs).symm ⟨f i, ⋯⟩
x : ι
⊢ enumOrd s (g x) = f x | change (enumOrdOrderIso s hs _).val = f x | case refine_1.e_s.h
s : Set Ordinal.{u}
hs : ¬BddAbove s
Hs : StrictMono (enumOrd s)
h : IsNormal (enumOrd s)
ι : Type u
hι : Nonempty ι
f : ι → Ordinal.{u}
hf : ∀ (i : ι), f i ∈ s
g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso s hs).symm ⟨f i, ⋯⟩
x : ι
⊢ ↑((enumOrdOrderIso s hs) (g x)) = f x | 0e0eef18c9d95dc4 |
finite_powers | Mathlib/GroupTheory/OrderOfElement.lean | @[to_additive (attr := simp)]
lemma finite_powers : (powers a : Set G).Finite ↔ IsOfFinOrder a | case intro.intro.intro
G : Type u_1
inst✝ : LeftCancelMonoid G
a : G
h : (↑(powers a)).Finite
m n : ℕ
hmn : m < n
ha : a ^ m = a ^ n
⊢ a ^ (n - m) = 1 | rw [← mul_left_cancel_iff (a := a ^ m), ← pow_add, add_tsub_cancel_of_le hmn.le, ha, mul_one] | no goals | 2594fbf04d0aff1b |
MeasureTheory.IsSetRing.isSetSemiring | Mathlib/MeasureTheory/SetSemiring.lean | lemma isSetSemiring (hC : IsSetRing C) : IsSetSemiring C where
empty_mem := hC.empty_mem
inter_mem := fun _ hs _ ht => hC.inter_mem hs ht
diff_eq_sUnion' | case refine_3
α : Type u_1
C : Set (Set α)
hC : IsSetRing C
s : Set α
hs : s ∈ C
t : Set α
ht : t ∈ C
⊢ s \ t = ⋃₀ ↑{s \ t} | simp only [coe_singleton, sUnion_singleton] | no goals | d2d41d4122a7b3a8 |
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