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
|
|---|---|---|---|---|---|---|
SetTheory.PGame.nim_add_nim_equiv
|
Mathlib/SetTheory/Game/Nim.lean
|
theorem nim_add_nim_equiv (x y : Ordinal) :
nim x + nim y ≈ nim (toOrdinal (∗x + ∗y))
|
x y : Ordinal.{u_1}
⊢ nim x + nim y ≈ nim (toOrdinal (toNimber x + toNimber y))
|
rw [← grundyValue_eq_iff_equiv_nim, grundyValue_nim_add_nim]
|
no goals
|
014c2a7b2048de90
|
TensorAlgebra.lift_unique
|
Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
|
theorem lift_unique {A : Type*} [Semiring A] [Algebra R A] (f : M →ₗ[R] A)
(g : TensorAlgebra R M →ₐ[R] A) : g.toLinearMap.comp (ι R) = f ↔ g = lift R f
|
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : M →ₗ[R] A
g : TensorAlgebra R M →ₐ[R] A
⊢ g.toLinearMap ∘ₗ ι R = f ↔ (lift R).symm g = f
|
simp only [lift, Equiv.coe_fn_symm_mk]
|
no goals
|
d86969492c8a1bf3
|
ProbabilityTheory.iInf_rat_gt_defaultRatCDF
|
Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
|
lemma iInf_rat_gt_defaultRatCDF (t : ℚ) :
⨅ r : Ioi t, defaultRatCDF r = defaultRatCDF t
|
case intro
t : ℚ
y : ↑(Ioi t)
hx : (fun r => if ↑r < 0 then 0 else 1) y ∈ range fun r => if ↑r < 0 then 0 else 1
⊢ 0 ≤ (fun r => if ↑r < 0 then 0 else 1) y
|
dsimp only
|
case intro
t : ℚ
y : ↑(Ioi t)
hx : (fun r => if ↑r < 0 then 0 else 1) y ∈ range fun r => if ↑r < 0 then 0 else 1
⊢ 0 ≤ if ↑y < 0 then 0 else 1
|
470f37fe6a48e799
|
fwdDiff_iter_choose_zero
|
Mathlib/Algebra/Group/ForwardDiff.lean
|
lemma fwdDiff_iter_choose_zero (m n : ℕ) :
Δ_[1]^[n] (fun x ↦ x.choose m : ℕ → ℤ) 0 = if n = m then 1 else 0
|
case inl.intro
m k : ℕ
hmn : m < m + k + 1
⊢ Δ_[1]^[m + k + 1] (fun x => ↑(x.choose m)) 0 = if m + k + 1 = m then 1 else 0
|
simp_rw [hmn.ne', if_false, (by ring : m + k + 1 = k + 1 + m), iterate_add_apply,
add_zero m ▸ fwdDiff_iter_choose 0 m, choose_zero_right, iterate_one, cast_one, fwdDiff_const,
fwdDiff_iter_eq_sum_shift, smul_zero, sum_const_zero]
|
no goals
|
3b96d64e98a2a661
|
hasStrictDerivAt_zpow
|
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
theorem hasStrictDerivAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
HasStrictDerivAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) x
|
case intro
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m✝ : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m✝
m : ℕ
hm : 0 < ↑m
⊢ HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (↑m - 1)) x
|
convert hasStrictDerivAt_pow m x using 2
|
case h.e'_9.h.e'_6
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m✝ : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m✝
m : ℕ
hm : 0 < ↑m
⊢ x ^ (↑m - 1) = x ^ (m - 1)
|
c222b9aa15688dc1
|
Polynomial.eq_X_sub_C_of_separable_of_root_eq
|
Mathlib/FieldTheory/Separable.lean
|
theorem eq_X_sub_C_of_separable_of_root_eq {x : F} {h : F[X]} (h_sep : h.Separable)
(h_root : h.eval x = 0) (h_splits : Splits i h) (h_roots : ∀ y ∈ (h.map i).roots, y = i x) :
h = C (leadingCoeff h) * (X - C x)
|
F : Type u
inst✝¹ : Field F
K : Type v
inst✝ : Field K
i : F →+* K
x : F
h : F[X]
h_sep : h.Separable
h_root : eval x h = 0
h_splits : Splits i h
h_roots : ∀ y ∈ (map i h).roots, y = i x
h_ne_zero : h ≠ 0
⊢ h = C h.leadingCoeff * (X - C x)
|
apply Polynomial.eq_X_sub_C_of_splits_of_single_root i h_splits
|
F : Type u
inst✝¹ : Field F
K : Type v
inst✝ : Field K
i : F →+* K
x : F
h : F[X]
h_sep : h.Separable
h_root : eval x h = 0
h_splits : Splits i h
h_roots : ∀ y ∈ (map i h).roots, y = i x
h_ne_zero : h ≠ 0
⊢ (map i h).roots = {i x}
|
367aa3f70723825a
|
Seminorm.continuous_from_bounded
|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
theorem continuous_from_bounded {p : SeminormFamily 𝕝 E ι} {q : SeminormFamily 𝕝₂ F ι'}
{_ : TopologicalSpace E} (hp : WithSeminorms p) {_ : TopologicalSpace F} (hq : WithSeminorms q)
(f : E →ₛₗ[τ₁₂] F) (hf : Seminorm.IsBounded p q f) : Continuous f
|
case intro.intro
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
ι : Type u_8
ι' : Type u_9
inst✝⁸ : AddCommGroup E
inst✝⁷ : NormedField 𝕝
inst✝⁶ : Module 𝕝 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : NormedField 𝕝₂
inst✝³ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝² : RingHomIsometric τ₁₂
inst✝¹ : Nonempty ι
inst✝ : Nonempty ι'
p : SeminormFamily 𝕝 E ι
q : SeminormFamily 𝕝₂ F ι'
x✝¹ : TopologicalSpace E
hp : WithSeminorms p
x✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : IsBounded p q f
this : IsTopologicalAddGroup E
i : ι'
s : Finset ι
C : ℝ≥0
hC : (q i).comp f ≤ s.sup (C • p)
⊢ Continuous fun x => (coeFnAddMonoidHom 𝕝 E) (∑ i ∈ s, C • p i) x
|
simp_rw [map_sum, Finset.sum_apply]
|
case intro.intro
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
ι : Type u_8
ι' : Type u_9
inst✝⁸ : AddCommGroup E
inst✝⁷ : NormedField 𝕝
inst✝⁶ : Module 𝕝 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : NormedField 𝕝₂
inst✝³ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝² : RingHomIsometric τ₁₂
inst✝¹ : Nonempty ι
inst✝ : Nonempty ι'
p : SeminormFamily 𝕝 E ι
q : SeminormFamily 𝕝₂ F ι'
x✝¹ : TopologicalSpace E
hp : WithSeminorms p
x✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : IsBounded p q f
this : IsTopologicalAddGroup E
i : ι'
s : Finset ι
C : ℝ≥0
hC : (q i).comp f ≤ s.sup (C • p)
⊢ Continuous fun x => ∑ c ∈ s, (coeFnAddMonoidHom 𝕝 E) (C • p c) x
|
0044c0551bf1b760
|
norm_combo_lt_of_ne
|
Mathlib/Analysis/Convex/StrictConvexSpace.lean
|
theorem norm_combo_lt_of_ne (hx : ‖x‖ ≤ r) (hy : ‖y‖ ≤ r) (hne : x ≠ y) (ha : 0 < a) (hb : 0 < b)
(hab : a + b = 1) : ‖a • x + b • y‖ < r
|
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
a b r : ℝ
hne : x ≠ y
ha : 0 < a
hb : 0 < b
hab : a + b = 1
hx : x ∈ closedBall 0 r
hy : y ∈ closedBall 0 r
⊢ a • x + b • y ∈ ball 0 r
|
exact combo_mem_ball_of_ne hx hy hne ha hb hab
|
no goals
|
7813d89c83271a4b
|
Ideal.Quotient.index_eq_zero
|
Mathlib/Algebra/CharP/Quotient.lean
|
theorem Ideal.Quotient.index_eq_zero {R : Type*} [CommRing R] (I : Ideal R) :
(↑I.toAddSubgroup.index : R ⧸ I) = 0
|
R : Type u_1
inst✝ : CommRing R
I : Ideal R
⊢ ↑(Submodule.toAddSubgroup I).index = 0
|
rw [AddSubgroup.index, Nat.card_eq]
|
R : Type u_1
inst✝ : CommRing R
I : Ideal R
⊢ ↑(if x : Finite (R ⧸ Submodule.toAddSubgroup I) then Fintype.card (R ⧸ Submodule.toAddSubgroup I) else 0) = 0
|
64a79a192da7e338
|
Submodule.FG.pow
|
Mathlib/RingTheory/Finiteness/Subalgebra.lean
|
theorem FG.pow (h : M.FG) (n : ℕ) : (M ^ n).FG :=
Nat.recOn n ⟨{1}, by simp [one_eq_span]⟩ fun n ih => by simpa [pow_succ] using ih.mul h
|
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
M : Submodule R A
h : M.FG
n✝ n : ℕ
ih : (M ^ n).FG
⊢ (M ^ n.succ).FG
|
simpa [pow_succ] using ih.mul h
|
no goals
|
2974428bbe74428e
|
Computation.destruct_eq_pure
|
Mathlib/Data/Seq/Computation.lean
|
theorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a
|
case some.a
α : Type u
s : Computation α
a val✝ : α
f0 : ↑s 0 = some val✝
h :
(match some val✝ with
| none => Sum.inr s.tail
| some a => Sum.inl a) =
Sum.inl a
⊢ ↑s = ↑(pure a)
|
funext n
|
case some.a.h
α : Type u
s : Computation α
a val✝ : α
f0 : ↑s 0 = some val✝
h :
(match some val✝ with
| none => Sum.inr s.tail
| some a => Sum.inl a) =
Sum.inl a
n : ℕ
⊢ ↑s n = ↑(pure a) n
|
d7b26a3ee2818517
|
geom_sum_eq_zero_iff_neg_one
|
Mathlib/Algebra/GeomSum.lean
|
theorem geom_sum_eq_zero_iff_neg_one [LinearOrderedRing α] (hn : n ≠ 0) :
∑ i ∈ range n, x ^ i = 0 ↔ x = -1 ∧ Even n
|
case inl
α : Type u
n : ℕ
x : α
inst✝ : LinearOrderedRing α
hn : n ≠ 0
h : x = -1 → ¬Even n
hx : x = -1
⊢ ∑ i ∈ range n, x ^ i ≠ 0
|
rw [hx, neg_one_geom_sum]
|
case inl
α : Type u
n : ℕ
x : α
inst✝ : LinearOrderedRing α
hn : n ≠ 0
h : x = -1 → ¬Even n
hx : x = -1
⊢ (if Even n then 0 else 1) ≠ 0
|
b77f74310391edb9
|
SzemerediRegularity.edgeDensity_star_not_uniform
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
|
theorem edgeDensity_star_not_uniform [Nonempty α]
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5)
(hε₁ : ε ≤ 1) {hU : U ∈ P.parts} {hV : V ∈ P.parts} (hUVne : U ≠ V) (hUV : ¬G.IsUniform ε U V) :
↑3 / ↑4 * ε ≤
|(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), (G.edgeDensity ab.1 ab.2 : ℝ)) /
(#(star hP G ε hU V) * #(star hP G ε hV U)) -
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts,
(G.edgeDensity ab.1 ab.2 : ℝ)) / (16 : ℝ) ^ #P.parts|
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUVne : U ≠ V
hUV : ¬G.IsUniform ε U V
p : ℝ :=
(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) /
(↑(#(star hP G ε hU V)) * ↑(#(star hP G ε hV U)))
q : ℝ :=
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) /
(4 ^ #P.parts * 4 ^ #P.parts)
r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))
s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))
t : ℝ := ↑(G.edgeDensity U V)
hrs : |r - s| ≤ ε / 5
hst : ε ≤ |s - t|
hpr : |p - r| ≤ ε ^ 5 / 49
⊢ |q - t| ≤ ε ^ 5 / 49
|
have := average_density_near_total_density hPα hPε hε₁
(Subset.refl (chunk hP G ε hU).parts) (Subset.refl (chunk hP G ε hV).parts)
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUVne : U ≠ V
hUV : ¬G.IsUniform ε U V
p : ℝ :=
(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) /
(↑(#(star hP G ε hU V)) * ↑(#(star hP G ε hV U)))
q : ℝ :=
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) /
(4 ^ #P.parts * 4 ^ #P.parts)
r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))
s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))
t : ℝ := ↑(G.edgeDensity U V)
hrs : |r - s| ≤ ε / 5
hst : ε ≤ |s - t|
hpr : |p - r| ≤ ε ^ 5 / 49
this :
|(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) /
(↑(#(chunk hP G ε hU).parts) * ↑(#(chunk hP G ε hV).parts)) -
↑(G.edgeDensity ((chunk hP G ε hU).parts.biUnion id) ((chunk hP G ε hV).parts.biUnion id))| ≤
ε ^ 5 / 49
⊢ |q - t| ≤ ε ^ 5 / 49
|
96200eb905ae7b75
|
MeasurableSet.baireMeasurableSet
|
Mathlib/Topology/Baire/BaireMeasurable.lean
|
theorem _root_.MeasurableSet.baireMeasurableSet [MeasurableSpace α] [BorelSpace α]
(h : MeasurableSet s) : BaireMeasurableSet s
|
α : Type u_1
inst✝¹ : TopologicalSpace α
s : Set α
inst✝ : BorelSpace α
h : MeasurableSet s
this✝ : MeasurableSpace α := borel α
⊢ BaireMeasurableSet s
|
exact h.eventuallyMeasurableSet
|
no goals
|
6c46bad40395a38b
|
coe_lt_enorm
|
Mathlib/Analysis/Normed/Group/Basic.lean
|
@[simp, norm_cast] lemma coe_lt_enorm : r < ‖x‖ₑ ↔ r < ‖x‖₊
|
E : Type u_8
inst✝ : NNNorm E
x : E
r : ℝ≥0
⊢ ↑r < ‖x‖ₑ ↔ r < ‖x‖₊
|
simp [enorm]
|
no goals
|
f87fd36369fbe207
|
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
⊢ (replicate n a).eraseIdx k = replicate (n - 1) a
|
rw [eq_replicate_iff, length_eraseIdx_of_lt (by simpa using h)]
|
case isTrue
α : Type u_1
n : Nat
a : α
k : Nat
h : k < n
⊢ (replicate n a).length - 1 = n - 1 ∧ ∀ (b : α), b ∈ (replicate n a).eraseIdx k → b = a
|
eaeb466d41ba696c
|
mdifferentiableWithinAt_totalSpace
|
Mathlib/Geometry/Manifold/VectorBundle/MDifferentiable.lean
|
theorem mdifferentiableWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x₀ : M} :
MDifferentiableWithinAt IM (IB.prod 𝓘(𝕜, F)) f s x₀ ↔
MDifferentiableWithinAt IM IB (fun x => (f x).proj) s x₀ ∧
MDifferentiableWithinAt IM 𝓘(𝕜, F)
(fun x ↦ (trivializationAt F E (f x₀).proj (f x)).2) s x₀
|
case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ (fun x => ↑(extChartAt IB (f x₀).proj) (f x).proj) x =
(Prod.fst ∘ fun x =>
(↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).1,
(↑(trivializationAt F E (f x₀).proj).toPartialEquiv (f x)).2))
x
|
simp_rw [Function.comp, PartialHomeomorph.coe_coe, Trivialization.coe_coe]
|
case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace 𝕜 F
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝¹⁰ : NormedAddCommGroup EB
inst✝⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
EM : Type u_10
inst✝⁷ : NormedAddCommGroup EM
inst✝⁶ : NormedSpace 𝕜 EM
HM : Type u_11
inst✝⁵ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace HM M
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
f : M → TotalSpace F E
s : Set M
x₀ : M
hf : ContinuousWithinAt f s x₀
h1 : (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet ∈ 𝓝[s] x₀
x : M
hx : x ∈ (fun x => (f x).proj) ⁻¹' (trivializationAt F E (f x₀).proj).baseSet
⊢ ↑(extChartAt IB (f x₀).proj) (f x).proj = ↑(extChartAt IB (f x₀).proj) (↑(trivializationAt F E (f x₀).proj) (f x)).1
|
496d5b669ff12a5e
|
Finset.attach_affineCombination_of_injective
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
theorem attach_affineCombination_of_injective [DecidableEq P] (s : Finset P) (w : P → k) (f : s → P)
(hf : Function.Injective f) :
s.attach.affineCombination k f (w ∘ f) = (image f univ).affineCombination k id w
|
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
S : AffineSpace V P
inst✝ : DecidableEq P
s : Finset P
w : P → k
f : { x // x ∈ s } → P
hf : Function.Injective f
g₁ : { x // x ∈ s } → V := fun i => w (f i) • (f i -ᵥ Classical.choice ⋯)
g₂ : P → V := fun i => w i • (i -ᵥ Classical.choice ⋯)
⊢ univ.sum g₁ = (image f univ).sum g₂
|
have hgf : g₁ = g₂ ∘ f := by
ext
simp [g₁, g₂]
|
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
S : AffineSpace V P
inst✝ : DecidableEq P
s : Finset P
w : P → k
f : { x // x ∈ s } → P
hf : Function.Injective f
g₁ : { x // x ∈ s } → V := fun i => w (f i) • (f i -ᵥ Classical.choice ⋯)
g₂ : P → V := fun i => w i • (i -ᵥ Classical.choice ⋯)
hgf : g₁ = g₂ ∘ f
⊢ univ.sum g₁ = (image f univ).sum g₂
|
d5e9d609b170565a
|
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean
|
theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1
|
case arithShiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.size),
(go aig rhs).val.aig.decls[idx] = aig.decls[idx]
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (lhs.arithShiftRight rhs)).val.aig.decls.size
⊢ (blastArithShiftRight (go (go aig lhs).1.aig rhs).1.aig
{ n := n✝, target := (go aig lhs).1.vec.cast ⋯,
distance := (go (go aig lhs).1.aig rhs).1.vec }).aig.decls[idx] =
aig.decls[idx]
|
have := (bitblast.go aig lhs).property
|
case arithShiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.size),
(go aig rhs).val.aig.decls[idx] = aig.decls[idx]
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (lhs.arithShiftRight rhs)).val.aig.decls.size
this : aig.decls.size ≤ (go aig lhs).val.aig.decls.size
⊢ (blastArithShiftRight (go (go aig lhs).1.aig rhs).1.aig
{ n := n✝, target := (go aig lhs).1.vec.cast ⋯,
distance := (go (go aig lhs).1.aig rhs).1.vec }).aig.decls[idx] =
aig.decls[idx]
|
f92988d5a1595b39
|
DyckWord.monotone_semilength
|
Mathlib/Combinatorics/Enumerative/DyckWord.lean
|
lemma monotone_semilength : Monotone semilength := fun p q pq ↦ by
induction pq with
| refl => rfl
| tail _ mq ih =>
rename_i m r _
rcases eq_or_ne r 0 with rfl | hr
· rw [insidePart_zero, outsidePart_zero, or_self] at mq
rwa [mq] at ih
· rcases mq with hm | hm
· exact ih.trans (hm ▸ semilength_insidePart_lt hr).le
· exact ih.trans (hm ▸ semilength_outsidePart_lt hr).le
|
case tail
p q m r : DyckWord
a✝ : Relation.ReflTransGen (fun p q => p = q.insidePart ∨ p = q.outsidePart) p m
mq : m = r.insidePart ∨ m = r.outsidePart
ih : p.semilength ≤ m.semilength
⊢ p.semilength ≤ r.semilength
|
rcases eq_or_ne r 0 with rfl | hr
|
case tail.inl
p q m : DyckWord
a✝ : Relation.ReflTransGen (fun p q => p = q.insidePart ∨ p = q.outsidePart) p m
ih : p.semilength ≤ m.semilength
mq : m = insidePart 0 ∨ m = outsidePart 0
⊢ p.semilength ≤ semilength 0
case tail.inr
p q m r : DyckWord
a✝ : Relation.ReflTransGen (fun p q => p = q.insidePart ∨ p = q.outsidePart) p m
mq : m = r.insidePart ∨ m = r.outsidePart
ih : p.semilength ≤ m.semilength
hr : r ≠ 0
⊢ p.semilength ≤ r.semilength
|
a9bc9900c089d242
|
Nat.prime_of_pow_sub_one_prime
|
Mathlib/NumberTheory/Fermat.lean
|
theorem prime_of_pow_sub_one_prime {a n : ℕ} (hn1 : n ≠ 1) (hP : (a ^ n - 1).Prime) :
a = 2 ∧ n.Prime
|
a n : ℕ
hn1 : n ≠ 1
hP : Prime (a ^ n - 1)
⊢ a = 2 ∧ Prime n
|
have han1 : 1 < a ^ n := tsub_pos_iff_lt.mp hP.pos
|
a n : ℕ
hn1 : n ≠ 1
hP : Prime (a ^ n - 1)
han1 : 1 < a ^ n
⊢ a = 2 ∧ Prime n
|
c66e0e0b6f579aaa
|
Equiv.Perm.Disjoint.isConj_mul
|
Mathlib/GroupTheory/Perm/Finite.lean
|
theorem Disjoint.isConj_mul [Finite α] {σ τ π ρ : Perm α} (hc1 : IsConj σ π)
(hc2 : IsConj τ ρ) (hd1 : Disjoint σ τ) (hd2 : Disjoint π ρ) : IsConj (σ * τ) (π * ρ)
|
case intro.intro.intro.refine_2.inl.refine_1
α : Type u
inst✝ : Finite α
σ τ : Perm α
hd1 : σ.Disjoint τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : (f * σ * f⁻¹).Disjoint (g * τ * g⁻¹)
hd1' : ↑(σ * τ).support = ↑σ.support ∪ ↑τ.support
hd2' : ↑(f * σ * f⁻¹ * (g * τ * g⁻¹)).support = ↑(f * σ * f⁻¹).support ∪ ↑(g * τ * g⁻¹).support
hd1'' : _root_.Disjoint ↑σ.support ↑τ.support
hd2'' : _root_.Disjoint ↑(f * σ * f⁻¹).support ↑(g * τ * g⁻¹).support
x : α
hx : x ∈ ↑(σ * τ).support
hxσ : σ x ≠ x
⊢ ¬(f * σ * f⁻¹) (f x) = f x
|
rwa [mul_apply, mul_apply, inv_apply_self, apply_eq_iff_eq]
|
no goals
|
c57d6049be02c9ef
|
Polynomial.Chebyshev.U_two
|
Mathlib/RingTheory/Polynomial/Chebyshev.lean
|
theorem U_two : U R 2 = 4 * X ^ 2 - 1
|
R : Type u_1
inst✝ : CommRing R
this : U R 2 = 2 * X * (2 * X) - 1
⊢ U R 2 = 4 * X ^ 2 - 1
|
linear_combination this
|
no goals
|
487a90767cd81216
|
IsAntichain.volume_eq_zero
|
Mathlib/MeasureTheory/Order/UpperLower.lean
|
theorem IsAntichain.volume_eq_zero [Nonempty ι] (hs : IsAntichain (· ≤ ·) s) : volume s = 0
|
ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 => x1 ≤ x2) s
⊢ volume s = 0
|
refine measure_mono_null ?_ hs.ordConnected.null_frontier
|
ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 => x1 ≤ x2) s
⊢ s ⊆ frontier s
|
e697af406be31b26
|
CategoryTheory.Biprod.ofComponents_fst
|
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
theorem Biprod.ofComponents_fst :
Biprod.ofComponents f₁₁ f₁₂ f₂₁ f₂₂ ≫ biprod.fst = biprod.fst ≫ f₁₁ + biprod.snd ≫ f₂₁
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
X₁ X₂ Y₁ Y₂ : C
f₁₁ : X₁ ⟶ Y₁
f₁₂ : X₁ ⟶ Y₂
f₂₁ : X₂ ⟶ Y₁
f₂₂ : X₂ ⟶ Y₂
⊢ ofComponents f₁₁ f₁₂ f₂₁ f₂₂ ≫ biprod.fst = biprod.fst ≫ f₁₁ + biprod.snd ≫ f₂₁
|
simp [Biprod.ofComponents]
|
no goals
|
6a8e69c7d2206d3f
|
MeasureTheory.Integrable.smul_essSup
|
Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
|
theorem Integrable.smul_essSup {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
{f : α → 𝕜} (hf : Integrable f μ) {g : α → β}
(g_aestronglyMeasurable : AEStronglyMeasurable g μ) (ess_sup_g : essSup (‖g ·‖ₑ) μ ≠ ∞) :
Integrable (fun x : α => f x • g x) μ
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_6
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 β
inst✝ : BoundedSMul 𝕜 β
f : α → 𝕜
hf : Integrable f μ
g : α → β
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x => ‖g x‖ₑ) μ ≠ ⊤
⊢ Integrable (fun x => f x • g x) μ
|
rw [← memLp_one_iff_integrable] at *
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup β
𝕜 : Type u_6
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 β
inst✝ : BoundedSMul 𝕜 β
f : α → 𝕜
hf : MemLp f 1 μ
g : α → β
g_aestronglyMeasurable : AEStronglyMeasurable g μ
ess_sup_g : essSup (fun x => ‖g x‖ₑ) μ ≠ ⊤
⊢ MemLp (fun x => f x • g x) 1 μ
|
a29141a482079b8c
|
SimpleGraph.Walk.IsTrail.length_le_card_edgeFinset
|
Mathlib/Combinatorics/SimpleGraph/Path.lean
|
theorem IsTrail.length_le_card_edgeFinset [Fintype G.edgeSet] {u v : V}
{w : G.Walk u v} (h : w.IsTrail) : w.length ≤ G.edgeFinset.card
|
V : Type u
G : SimpleGraph V
inst✝ : Fintype ↑G.edgeSet
u v : V
w : G.Walk u v
h : w.IsTrail
edges : Finset (Sym2 V) := w.edges.toFinset
⊢ w.length ≤ G.edgeFinset.card
|
have : edges.card = w.length := length_edges _ ▸ List.toFinset_card_of_nodup h.edges_nodup
|
V : Type u
G : SimpleGraph V
inst✝ : Fintype ↑G.edgeSet
u v : V
w : G.Walk u v
h : w.IsTrail
edges : Finset (Sym2 V) := w.edges.toFinset
this : edges.card = w.length
⊢ w.length ≤ G.edgeFinset.card
|
117de40399459014
|
collinear_iff_exists_forall_eq_smul_vadd
|
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
theorem collinear_iff_exists_forall_eq_smul_vadd (s : Set P) :
Collinear k s ↔ ∃ (p₀ : P) (v : V), ∀ p ∈ s, ∃ r : k, p = r • v +ᵥ p₀
|
case inr.intro.mp
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
hp₁ : p₁ ∈ s
⊢ (∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁) → ∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
|
exact fun h => ⟨p₁, h⟩
|
no goals
|
5cfbd34dabed0b2c
|
Ideal.dvd_iff_le
|
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
⟨Ideal.le_of_dvd, fun h => by
by_cases hI : I = ⊥
· have hJ : J = ⊥
|
case neg
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
⊢ I ∣ J
|
have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 := by
rw [← inv_mul_cancel₀ hI']
exact mul_left_mono _ ((coeIdeal_le_coeIdeal _).mpr h)
|
case neg
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
h : J ≤ I
hI : ¬I = ⊥
hI' : ↑I ≠ 0
this : (↑I)⁻¹ * ↑J ≤ 1
⊢ I ∣ J
|
47e45faf0c157a11
|
LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton'
|
Mathlib/Algebra/Lie/Weights/Killing.lean
|
lemma coe_corootSpace_eq_span_singleton' (α : Weight K H L) :
(corootSpace α).toSubmodule = K ∙ (cartanEquivDual H).symm α
|
case refine_2
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : PerfectField K
α : Weight K (↥H) L
⊢ (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) ∈ corootSpace ⇑α
|
exact cartanEquivDual_symm_apply_mem_corootSpace α
|
no goals
|
23e360d5eb243ad0
|
FormalMultilinearSeries.derivSeries_eq_zero
|
Mathlib/Analysis/Analytic/ChangeOrigin.lean
|
theorem derivSeries_eq_zero {n : ℕ} (hp : p (n + 1) = 0) : p.derivSeries n = 0
|
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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hp : p (n + 1) = 0
s : { s // s.card = n }
hs : s ∈ Finset.univ
⊢ p.changeOriginSeriesTerm 1 n ↑s ⋯ = 0
|
ext v
|
case H.H
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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hp : p (n + 1) = 0
s : { s // s.card = n }
hs : s ∈ Finset.univ
v : Fin n → E
x✝ : Fin 1 → E
⊢ ((p.changeOriginSeriesTerm 1 n ↑s ⋯) v) x✝ = (0 v) x✝
|
e92bd3437a4bd85b
|
MeasureTheory.OuterMeasure.isCaratheodory_union
|
Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
|
theorem isCaratheodory_union (h₁ : IsCaratheodory m s₁) (h₂ : IsCaratheodory m s₂) :
IsCaratheodory m (s₁ ∪ s₂) := fun t => by
rw [h₁ t, h₂ (t ∩ s₁), h₂ (t \ s₁), h₁ (t ∩ (s₁ ∪ s₂)), inter_diff_assoc _ _ s₁,
Set.inter_assoc _ _ s₁, inter_eq_self_of_subset_right Set.subset_union_left,
union_diff_left, h₂ (t ∩ s₁)]
simp [diff_eq, add_assoc]
|
α : Type u
m : OuterMeasure α
s₁ s₂ : Set α
h₁ : m.IsCaratheodory s₁
h₂ : m.IsCaratheodory s₂
t : Set α
⊢ m t = m (t ∩ (s₁ ∪ s₂)) + m (t \ (s₁ ∪ s₂))
|
rw [h₁ t, h₂ (t ∩ s₁), h₂ (t \ s₁), h₁ (t ∩ (s₁ ∪ s₂)), inter_diff_assoc _ _ s₁,
Set.inter_assoc _ _ s₁, inter_eq_self_of_subset_right Set.subset_union_left,
union_diff_left, h₂ (t ∩ s₁)]
|
α : Type u
m : OuterMeasure α
s₁ s₂ : Set α
h₁ : m.IsCaratheodory s₁
h₂ : m.IsCaratheodory s₂
t : Set α
⊢ m (t ∩ s₁ ∩ s₂) + m ((t ∩ s₁) \ s₂) + (m (t \ s₁ ∩ s₂) + m ((t \ s₁) \ s₂)) =
m (t ∩ s₁ ∩ s₂) + m ((t ∩ s₁) \ s₂) + m (t ∩ (s₂ \ s₁)) + m (t \ (s₁ ∪ s₂))
|
8dc8dfd995295de1
|
ExceptT.bind_pure_comp
|
Mathlib/.lake/packages/lean4/src/lean/Init/Control/Lawful/Instances.lean
|
theorem bind_pure_comp [Monad m] (f : α → β) (x : ExceptT ε m α) : x >>= pure ∘ f = f <$> x
|
m : Type u_1 → Type u_2
α β ε : Type u_1
inst✝ : Monad m
f : α → β
x : ExceptT ε m α
⊢ x >>= pure ∘ f = f <$> x
|
rfl
|
no goals
|
a10b306d43aa950d
|
MeasureTheory.crossing_pos_eq
|
Mathlib/Probability/Martingale/Upcrossing.lean
|
theorem crossing_pos_eq (hab : a < b) :
upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧
lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n
|
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
⊢ ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω
|
intro i ω
|
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
hab : a < b
hab' : 0 < b - a
i : Ω
ω : ℕ
⊢ b - a ≤ (f ω i - a)⁺ ↔ b ≤ f ω i
|
3016c9a152ee0783
|
Module.End.isNilpotent_restrict_genEigenspace_top
|
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
/-- The restriction of `f - μ • 1` to the generalized `μ`-eigenspace is nilpotent. -/
lemma isNilpotent_restrict_genEigenspace_top [IsNoetherian R M] (f : End R M) (μ : R)
(h : MapsTo (f - μ • (1 : End R M))
(f.genEigenspace μ ⊤) (f.genEigenspace μ ⊤) :=
mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ _) :
IsNilpotent ((f - μ • 1).restrict h)
|
case hf
R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsNoetherian R M
f : End R M
μ : R
h : optParam (MapsTo ⇑(f - μ • 1) ↑((f.genEigenspace μ) ⊤) ↑((f.genEigenspace μ) ⊤)) ⋯
⊢ IsNilpotent (LinearMap.restrict (f - μ • 1) ?hq)
|
apply isNilpotent_restrict_genEigenspace_nat f μ (maxUnifEigenspaceIndex f μ)
|
no goals
|
e1a3726aa4d97e52
|
FermatLastTheoremForThreeGen.Solution.formula2
|
Mathlib/NumberTheory/FLT/Three.lean
|
private lemma formula2 :
S.Y ^ 3 + S.u₄ * S.Z ^ 3 = S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3
|
case refine_1
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : (S.multiplicity - 1) * 3 + 1 = 3 * S.multiplicity - 2
⊢ ↑(η ^ 2 * FermatLastTheoremForThreeGen.Solution.u₁ S * (FermatLastTheoremForThreeGen.Solution.u₂ S)⁻¹) *
(λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S) ^ 3 *
(λ * ↑(FermatLastTheoremForThreeGen.Solution.u₂ S) * ↑η) =
FermatLastTheoremForThreeGen.Solution.X S ^ 3 * ↑(FermatLastTheoremForThreeGen.Solution.u₁ S) *
λ ^ (3 * S.multiplicity - 2)
|
calc _ = S.X^3 *(S.u₂*S.u₂⁻¹)*(η^3*S.u₁)*(λ^((S.multiplicity-1)*3)*λ):= by push_cast; ring
_ = S.X^3*S.u₁*λ^(3*S.multiplicity-2) := by simp [hζ.toInteger_cube_eq_one, ← pow_succ, this]
|
no goals
|
291915db94e2c072
|
Nat.Prime.emultiplicity_factorial_mul_succ
|
Mathlib/Data/Nat/Multiplicity.lean
|
theorem emultiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
emultiplicity p (p * (n + 1))! = emultiplicity p (p * n)! + emultiplicity p (n + 1) + 1
|
n p : ℕ
hp : Prime p
hp' : _root_.Prime p
h0 : 2 ≤ p
h1 : 1 ≤ p * n + 1
h2 : p * n + 1 ≤ p * (n + 1)
h3 : p * n + 1 ≤ p * (n + 1) + 1
m : ℕ
hm : p * n + 1 ≤ m ∧ m < p * (n + 1)
⊢ ∃ k, p * k < m ∧ m < p * (k + 1)
|
exact ⟨n, lt_of_succ_le hm.1, hm.2⟩
|
no goals
|
097a3d59a78795b9
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastZeroExtend.go_le_size
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Operations/ZeroExtend.lean
|
theorem go_le_size (aig : AIG α) (w : Nat) (input : AIG.RefVec aig w) (newWidth curr : Nat)
(hcurr : curr ≤ newWidth) (s : AIG.RefVec aig curr) :
aig.decls.size ≤ (go aig w input newWidth curr hcurr s).aig.decls.size
|
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
h✝¹ : curr < newWidth
h✝ : curr < w
⊢ ?m.14001 ≤ (go aig w input newWidth (curr + 1) ⋯ (s.push (input.get curr h✝))).aig.decls.size
|
apply go_le_size
|
no goals
|
5b8a3de9428e3a64
|
IndexedPartition.piecewise_bij
|
Mathlib/Data/Setoid/Partition.lean
|
theorem piecewise_bij {β : Type*} {f : ι → α → β}
{t : ι → Set β} (ht : IndexedPartition t)
(hf : ∀ i, BijOn (f i) (s i) (t i)) :
Bijective (piecewise hs f)
|
ι : Type u_1
α : Type u_2
s : ι → Set α
hs : IndexedPartition s
β : Type u_3
f : ι → α → β
t : ι → Set β
ht : IndexedPartition t
hf : ∀ (i : ι), BijOn (f i) (s i) (t i)
g : α → β := hs.piecewise f
hg : g = hs.piecewise f
i : ι
⊢ BijOn g (s i) (t i)
|
refine BijOn.congr (hf i) ?_
|
ι : Type u_1
α : Type u_2
s : ι → Set α
hs : IndexedPartition s
β : Type u_3
f : ι → α → β
t : ι → Set β
ht : IndexedPartition t
hf : ∀ (i : ι), BijOn (f i) (s i) (t i)
g : α → β := hs.piecewise f
hg : g = hs.piecewise f
i : ι
⊢ EqOn (f i) g (s i)
|
c0eea0bbe5e450f8
|
AlgebraicGeometry.Scheme.fromSpecStalk_app
|
Mathlib/AlgebraicGeometry/Stalk.lean
|
lemma fromSpecStalk_app {x : X} (hxU : x ∈ U) :
(X.fromSpecStalk x).app U =
X.presheaf.germ U x hxU ≫
(ΓSpecIso (X.presheaf.stalk x)).inv ≫
(Spec (X.presheaf.stalk x)).presheaf.map (homOfLE le_top).op
|
case intro.intro.intro.intro.intro
X : Scheme
U : X.Opens
x : ↑↑X.toPresheafedSpace
hxU : x ∈ U
V : X.Opens
hV : V ∈ X.affineOpens
hxV : x ∈ ↑V
hVU : ↑V ⊆ ↑U
⊢ (X.presheaf.map (homOfLE hVU).op ≫ (ΓSpecIso Γ(X, V)).inv ≫ (Spec Γ(X, V)).presheaf.map (homOfLE ⋯).op) ≫
Hom.app (Spec.map (X.presheaf.germ V x hxV)) (IsAffineOpen.fromSpec hV ⁻¹ᵁ U) =
(X.presheaf.map (homOfLE hVU).op ≫ X.presheaf.germ V x hxV) ≫
(ΓSpecIso (X.presheaf.stalk x)).inv ≫ (Spec (X.presheaf.stalk x)).presheaf.map (homOfLE ⋯).op
|
simp [Category.assoc, ← ΓSpecIso_inv_naturality_assoc]
|
no goals
|
9e796c10438ef7f2
|
AlgebraicGeometry.Scheme.smallGrothendieckTopologyOfLE_eq_toGrothendieck_smallPretopology
|
Mathlib/AlgebraicGeometry/Sites/Small.lean
|
lemma smallGrothendieckTopologyOfLE_eq_toGrothendieck_smallPretopology (hPQ : P ≤ Q) :
S.smallGrothendieckTopologyOfLE hPQ = (S.smallPretopology P Q).toGrothendieck
|
case h.h.h
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁶ : P.IsMultiplicative
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.IsStableUnderBaseChange
inst✝³ : IsJointlySurjectivePreserving P
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
⊢ (∃ 𝒰 x, 𝒰.toPresieveOver ≤ (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R).arrows) ↔
∃ R_1 ∈ (smallPretopology P Q).coverings X, R_1 ≤ R.arrows
|
constructor
|
case h.h.h.mp
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁶ : P.IsMultiplicative
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.IsStableUnderBaseChange
inst✝³ : IsJointlySurjectivePreserving P
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
⊢ (∃ 𝒰 x, 𝒰.toPresieveOver ≤ (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R).arrows) →
∃ R_1 ∈ (smallPretopology P Q).coverings X, R_1 ≤ R.arrows
case h.h.h.mpr
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁶ : P.IsMultiplicative
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.IsStableUnderBaseChange
inst✝³ : IsJointlySurjectivePreserving P
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
⊢ (∃ R_1 ∈ (smallPretopology P Q).coverings X, R_1 ≤ R.arrows) →
∃ 𝒰 x, 𝒰.toPresieveOver ≤ (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R).arrows
|
d52b87d79c719fdc
|
HolderOnWith.hausdorffMeasure_image_le
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem hausdorffMeasure_image_le (h : HolderOnWith C r f s) (hr : 0 < r) {d : ℝ} (hd : 0 ≤ d) :
μH[d] (f '' s) ≤ (C : ℝ≥0∞) ^ d * μH[r * d] s
|
case inr.intro.intro.refine_1
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ⊤
R : ℝ≥0∞
hR : 0 < R
this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0)
δ : ℝ≥0∞
δ0 : 0 < δ
H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R
t : ℕ → Set X
hst : s ⊆ ⋃ n, t n
⊢ f '' s ⊆ ⋃ n, (fun n => f '' (t n ∩ s)) n
|
rw [← image_iUnion, ← iUnion_inter]
|
case inr.intro.intro.refine_1
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
C r : ℝ≥0
f : X → Y
s : Set X
h : HolderOnWith C r f s
hr : 0 < r
d : ℝ
hd : 0 ≤ d
hC0 : 0 < C
hCd0 : ↑C ^ d ≠ 0
hCd : ↑C ^ d ≠ ⊤
R : ℝ≥0∞
hR : 0 < R
this : Tendsto (fun d => ↑C * d ^ ↑r) (𝓝 0) (𝓝 0)
δ : ℝ≥0∞
δ0 : 0 < δ
H : ∀ ⦃x : ℝ≥0∞⦄, x ∈ Iic δ → ↑C * x ^ ↑r < R
t : ℕ → Set X
hst : s ⊆ ⋃ n, t n
⊢ f '' s ⊆ f '' ((⋃ i, t i) ∩ s)
|
cf145f55a786d8cb
|
Nat.fib_succ_eq_succ_sum
|
Mathlib/Data/Nat/Fib/Basic.lean
|
theorem fib_succ_eq_succ_sum (n : ℕ) : fib (n + 1) = (∑ k ∈ Finset.range n, fib k) + 1
|
n : ℕ
ih : fib (n + 1) = ∑ k ∈ Finset.range n, fib k + 1
⊢ fib n + ∑ k ∈ Finset.range n, fib k + 1 = ∑ k ∈ Finset.range (n + 1), fib k + 1
|
simp [Finset.range_add_one]
|
no goals
|
a427d39ffc0db3ae
|
Int.fmod_eq_emod
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem fmod_eq_emod (a : Int) {b : Int} (hb : 0 ≤ b) : fmod a b = a % b
|
a b : Int
hb : 0 ≤ b
⊢ a.fmod b = a % b
|
simp [fmod_def, emod_def, fdiv_eq_ediv _ hb]
|
no goals
|
f4ea0ff416263a8e
|
BoxIntegral.Prepartition.biUnion_le_iff
|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
theorem biUnion_le_iff {πi : ∀ J, Prepartition J} {π' : Prepartition I} :
π.biUnion πi ≤ π' ↔ ∀ J ∈ π, πi J ≤ π'.restrict J
|
case mpr.intro.intro.intro.intro.intro.intro
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
π' : Prepartition I
H✝ : ∀ J ∈ π, πi J ≤ π'.restrict J
J J₁ : Box ι
h₁ : J₁ ∈ π
hJ : J ∈ πi J₁
J₂ : Box ι
h₂ : J₂ ∈ π'.restrict J₁
Hle : J ≤ J₂
J₃ : Box ι
h₃ : J₃ ∈ π'
H : ↑J₂ = ↑J₁ ⊓ ↑J₃
⊢ ∃ I' ∈ π', J ≤ I'
|
exact ⟨J₃, h₃, Hle.trans <| WithBot.coe_le_coe.1 <| H.trans_le inf_le_right⟩
|
no goals
|
d064e02582f4c476
|
IsPrimitiveRoot.arg
|
Mathlib/RingTheory/RootsOfUnity/Complex.lean
|
theorem IsPrimitiveRoot.arg {n : ℕ} {ζ : ℂ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :
∃ i : ℤ, ζ.arg = i / n * (2 * Real.pi) ∧ IsCoprime i n ∧ i.natAbs < n
|
case neg
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ (Complex.cos (↑i / ↑n * (2 * ↑Real.pi)) + Complex.sin (↑i / ↑n * (2 * ↑Real.pi)) * Complex.I).arg =
↑(↑i - ↑n) / ↑n * (2 * Real.pi)
|
rw [← Complex.cos_sub_two_pi, ← Complex.sin_sub_two_pi]
|
case neg
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ (Complex.cos (↑i / ↑n * (2 * ↑Real.pi) - 2 * ↑Real.pi) +
Complex.sin (↑i / ↑n * (2 * ↑Real.pi) - 2 * ↑Real.pi) * Complex.I).arg =
↑(↑i - ↑n) / ↑n * (2 * Real.pi)
|
27e9c2564262c07b
|
IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow
|
Mathlib/NumberTheory/Cyclotomic/Rat.lean
|
theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K
|
case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ k)
x : K
h : IsIntegral ℤ x
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ k} ℚ K
H : Algebra.discr ℚ ⇑B.basis • x ∈ adjoin ℤ {B.gen}
⊢ x ∈ adjoin ℤ {ζ}
|
obtain ⟨u, n, hun⟩ := discr_prime_pow_eq_unit_mul_pow' hζ
|
case refine_1.intro.intro
p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ k)
x : K
h : IsIntegral ℤ x
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K := finiteDimensional {p ^ k} ℚ K
H : Algebra.discr ℚ ⇑B.basis • x ∈ adjoin ℤ {B.gen}
u : ℤˣ
n : ℕ
hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n
⊢ x ∈ adjoin ℤ {ζ}
|
076c1160c95c322f
|
List.isPrefix_replicate_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem isPrefix_replicate_iff {n} {a : α} {l : List α} :
l <+: List.replicate n a ↔ l.length ≤ n ∧ l = List.replicate l.length a
|
α : Type u_1
n : Nat
a : α
l : List α
⊢ (∃ t, l ++ t = replicate n a) ↔ l.length ≤ n ∧ l = replicate l.length a
|
simp only [append_eq_replicate_iff]
|
α : Type u_1
n : Nat
a : α
l : List α
⊢ (∃ t, l.length + t.length = n ∧ l = replicate l.length a ∧ t = replicate t.length a) ↔
l.length ≤ n ∧ l = replicate l.length a
|
d81e01f4aefca35b
|
EisensteinSeries.div_linear_zpow_differentiableOn
|
Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean
|
/-- Auxiliary lemma showing that for any `k : ℤ` the function `z → 1/(c*z+d)^k` is
differentiable on `{z : ℂ | 0 < z.im}`. -/
lemma div_linear_zpow_differentiableOn (k : ℤ) (a : Fin 2 → ℤ) :
DifferentiableOn ℂ (fun z : ℂ => (a 0 * z + a 1) ^ (-k)) {z : ℂ | 0 < z.im}
|
case inr
k : ℤ
⊢ DifferentiableOn ℂ (fun z => 0 ^ (-k)) {z | 0 < z.im}
|
apply differentiableOn_const
|
no goals
|
b522b463994e780a
|
Polynomial.natDegree_expand
|
Mathlib/Algebra/Polynomial/Expand.lean
|
theorem natDegree_expand (p : ℕ) (f : R[X]) : (expand R p f).natDegree = f.natDegree * p
|
case neg
R : Type u
inst✝ : CommSemiring R
p : ℕ
f : R[X]
hp : p > 0
hf : ¬f = 0
hf1 : (expand R p) f ≠ 0
⊢ ((expand R p) f).natDegree = f.natDegree * p
|
rw [← Nat.cast_inj (R := WithBot ℕ), ← degree_eq_natDegree hf1]
|
case neg
R : Type u
inst✝ : CommSemiring R
p : ℕ
f : R[X]
hp : p > 0
hf : ¬f = 0
hf1 : (expand R p) f ≠ 0
⊢ ((expand R p) f).degree = ↑(f.natDegree * p)
|
eb7f41a4dec073d7
|
Subfield.relrank_mul_relrank
|
Mathlib/FieldTheory/Relrank.lean
|
theorem relrank_mul_relrank (h1 : A ≤ B) (h2 : B ≤ C) :
relrank A B * relrank B C = relrank A C
|
E : Type v
inst✝ : Field E
A B C : Subfield E
h1 : A ≤ B
h2 : B ≤ C
h3 : A ≤ C
⊢ A.relrank B * B.relrank C = A.relrank C
|
rw [relrank_eq_rank_of_le h1, relrank_eq_rank_of_le h2, relrank_eq_rank_of_le h3]
|
E : Type v
inst✝ : Field E
A B C : Subfield E
h1 : A ≤ B
h2 : B ≤ C
h3 : A ≤ C
⊢ Module.rank ↥A ↥(extendScalars h1) * Module.rank ↥B ↥(extendScalars h2) = Module.rank ↥A ↥(extendScalars h3)
|
3c081786cfcd4655
|
Set.not_equitableOn
|
Mathlib/Data/Set/Equitable.lean
|
@[simp]
lemma not_equitableOn : ¬s.EquitableOn f ↔ ∃ a ∈ s, ∃ b ∈ s, f b + 1 < f a
|
α : Type u_1
β : Type u_2
inst✝² : LinearOrder β
inst✝¹ : Add β
inst✝ : One β
s : Set α
f : α → β
⊢ ¬s.EquitableOn f ↔ ∃ a ∈ s, ∃ b ∈ s, f b + 1 < f a
|
simp [EquitableOn]
|
no goals
|
af64e873021c1c5c
|
Submodule.LinearDisjoint.of_basis_left'
|
Mathlib/LinearAlgebra/LinearDisjoint.lean
|
theorem of_basis_left' {ι : Type*} (m : Basis ι R M)
(H : Function.Injective (mulLeftMap N m)) : M.LinearDisjoint N
|
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
ι : Type u_1
m : Basis ι R ↥M
H : Function.Injective ⇑(M.mulMap N)
⊢ M.LinearDisjoint N
|
exact ⟨H⟩
|
no goals
|
6c450f6f7004a3ca
|
hasFDerivAt_integral_of_dominated_loc_of_lip'
|
Mathlib/Analysis/Calculus/ParametricIntegral.lean
|
theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε)
(hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
(bound_integrable : Integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a) x₀) :
Integrable F' μ ∧ HasFDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀
|
case h
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
ε : ℝ
F' : α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a => |bound a|
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
∀ᶠ (x : H) in 𝓝 x₀,
‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
a✝ : H
x_in : a✝ ∈ ball x₀ ε
⊢ AEStronglyMeasurable (fun a => ‖a✝ - x₀‖⁻¹ • (F a✝ a - F x₀ a - (F' a) (a✝ - x₀))) μ
|
apply AEStronglyMeasurable.const_smul
|
case h.hf
α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
ε : ℝ
F' : α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a => |bound a|
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
∀ᶠ (x : H) in 𝓝 x₀,
‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
a✝ : H
x_in : a✝ ∈ ball x₀ ε
⊢ AEStronglyMeasurable (fun a => F a✝ a - F x₀ a - (F' a) (a✝ - x₀)) μ
|
a8f4bcb6e23b7fde
|
AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self
|
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
theorem fromSpec_preimage_self :
hU.fromSpec ⁻¹ᵁ U = ⊤
|
case h
X : Scheme
U : X.Opens
hU : IsAffineOpen U
⊢ ⇑(ConcreteCategory.hom hU.fromSpec.base) ⁻¹' (⇑(ConcreteCategory.hom hU.fromSpec.base) '' Set.univ) = Set.univ
|
exact Set.preimage_image_eq _ PresheafedSpace.IsOpenImmersion.base_open.injective
|
no goals
|
13fff602d49d1869
|
BitVec.getMsbD_rotateRight_of_lt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem getMsbD_rotateRight_of_lt {w n m : Nat} {x : BitVec w} (hr : m < w) :
(x.rotateRight m).getMsbD n = (decide (n < w) && (if (n < m % w)
then x.getMsbD ((w + n - m % w) % w) else x.getMsbD (n - m % w)))
|
n m w : Nat
x : BitVec (w + 1)
hr : m < w + 1
h : n < m
⊢ n < w + 1
|
omega
|
no goals
|
446d3dd4304c7f2b
|
LinearOrderedAddCommGroupWithTop.add_eq_top
|
Mathlib/Algebra/Order/AddGroupWithTop.lean
|
@[simp]
lemma add_eq_top : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤
|
α : Type u_1
inst✝ : LinearOrderedAddCommGroupWithTop α
a b : α
⊢ a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤
|
constructor
|
case mp
α : Type u_1
inst✝ : LinearOrderedAddCommGroupWithTop α
a b : α
⊢ a + b = ⊤ → a = ⊤ ∨ b = ⊤
case mpr
α : Type u_1
inst✝ : LinearOrderedAddCommGroupWithTop α
a b : α
⊢ a = ⊤ ∨ b = ⊤ → a + b = ⊤
|
46f63359352460d5
|
Real.rpow_le_rpow_of_exponent_nonpos
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
theorem rpow_le_rpow_of_exponent_nonpos {x y : ℝ} (hy : 0 < y) (hxy : y ≤ x) (hz : z ≤ 0) :
x ^ z ≤ y ^ z
|
case inl
z x y : ℝ
hy : 0 < y
hxy : y ≤ x
hz : z ≤ 0
hz_zero : z ≠ 0
⊢ x ^ z ≤ y ^ z
case inr
x y : ℝ
hy : 0 < y
hxy : y ≤ x
hz : 0 ≤ 0
⊢ x ^ 0 ≤ y ^ 0
|
case inl =>
rcases ne_or_eq x y with hxy' | rfl
case inl =>
exact le_of_lt <| rpow_lt_rpow_of_exponent_neg hy (Ne.lt_of_le (id (Ne.symm hxy')) hxy)
(Ne.lt_of_le hz_zero hz)
case inr => simp
|
case inr
x y : ℝ
hy : 0 < y
hxy : y ≤ x
hz : 0 ≤ 0
⊢ x ^ 0 ≤ y ^ 0
|
5347c035e5dc89b4
|
Submodule.mem_sSup_of_mem
|
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
theorem mem_sSup_of_mem {S : Set (Submodule R M)} {s : Submodule R M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S
|
R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set (Submodule R M)
s : Submodule R M
hs : s ∈ S
this : Preorder.toLE.1 s (sSup S)
⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S
|
exact this
|
no goals
|
97d38de760144cae
|
alternatingGroup.normalClosure_swap_mul_swap_five
|
Mathlib/GroupTheory/SpecificGroups/Alternating.lean
|
theorem normalClosure_swap_mul_swap_five :
normalClosure
({⟨swap 0 4 * swap 1 3, mem_alternatingGroup.2 (by decide)⟩} :
Set (alternatingGroup (Fin 5))) =
⊤
|
⊢ sign (swap 0 2 * swap 0 1) = 1
|
decide
|
no goals
|
ff4623cb61a07e92
|
FermatLastTheoremForThreeGen.Solution.lambda_pow_dvd_a_add_b
|
Mathlib/NumberTheory/FLT/Three.lean
|
/-- We have that `λ ^ (3*S.multiplicity-2)` divides `S.a + S.b`. -/
private lemma lambda_pow_dvd_a_add_b : λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b
|
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h : λ ^ S.multiplicity ∣ S.c
⊢ (λ ^ S.multiplicity) ^ 3 ∣ ↑S.u * S.c ^ 3
|
simp [h]
|
no goals
|
ca8f590be4efd444
|
PrimeSpectrum.toPiLocalization_surjective_of_discreteTopology
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
theorem toPiLocalization_surjective_of_discreteTopology :
Function.Surjective (toPiLocalization R) := fun x ↦ by
have (p : PrimeSpectrum R) : ∃ f, (basicOpen f : Set _) = {p} :=
have ⟨_, ⟨f, rfl⟩, hpf, hfp⟩ := isTopologicalBasis_basic_opens.isOpen_iff.mp
(isOpen_discrete {p}) p rfl
⟨f, hfp.antisymm <| Set.singleton_subset_iff.mpr hpf⟩
choose f hf using this
let e := Equiv.ofInjective f fun p q eq ↦ Set.singleton_injective (hf p ▸ eq ▸ hf q)
have loc a : IsLocalization.AtPrime (Localization.Away a.1) (e.symm a).1 :=
(isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton <| hf _).mp <| by
simp_rw [e, Equiv.apply_ofInjective_symm]; infer_instance
let algE a := IsLocalization.algEquiv (e.symm a).1.primeCompl
(Localization.AtPrime (e.symm a).1) (Localization.Away a.1)
have span_eq : Ideal.span (Set.range f) = ⊤ := iSup_basicOpen_eq_top_iff.mp <| top_unique
fun p _ ↦ TopologicalSpace.Opens.mem_iSup.mpr ⟨p, (hf p).ge rfl⟩
replace hf a : (basicOpen a.1 : Set _) = {e.symm a}
|
R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
a : ↑(Set.range f)
⊢ IsLocalization.Away (f (e.symm a)) (Localization.Away ↑a)
|
simp_rw [e, Equiv.apply_ofInjective_symm]
|
R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
a : ↑(Set.range f)
⊢ IsLocalization.Away (↑a) (Localization.Away ↑a)
|
680284c76fd78d8c
|
FormalMultilinearSeries.comp_coeff_one
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem comp_coeff_one (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(v : Fin 1 → E) : (q.comp p) 1 v = q 1 fun _i => p 1 v
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
v : Fin 1 → E
⊢ {Composition.ones 1}.card = Fintype.card (Composition 1)
|
simp [composition_card]
|
no goals
|
2ac1709b5f4213f3
|
CategoryTheory.MorphismProperty.IsInvertedBy.leftOp
|
Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean
|
theorem leftOp {W : MorphismProperty C} {L : C ⥤ Dᵒᵖ} (h : W.IsInvertedBy L) :
W.op.IsInvertedBy L.leftOp := fun X Y f hf => by
haveI := h f.unop hf
dsimp
infer_instance
|
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
L : C ⥤ Dᵒᵖ
h : W.IsInvertedBy L
X Y : Cᵒᵖ
f : X ⟶ Y
hf : W.op f
⊢ IsIso (L.leftOp.map f)
|
haveI := h f.unop hf
|
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
W : MorphismProperty C
L : C ⥤ Dᵒᵖ
h : W.IsInvertedBy L
X Y : Cᵒᵖ
f : X ⟶ Y
hf : W.op f
this : IsIso (L.map f.unop)
⊢ IsIso (L.leftOp.map f)
|
9cb9a5e070efd211
|
WeierstrassCurve.coeff_preΨ'_ne_zero
|
Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
|
lemma coeff_preΨ'_ne_zero {n : ℕ} (h : (n : R) ≠ 0) :
(W.preΨ' n).coeff ((n ^ 2 - if Even n then 4 else 1) / 2) ≠ 0
|
case intro.inl
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
h : ↑(2 * n) ≠ 0
⊢ (W.preΨ' (2 * n)).coeff (((2 * n) ^ 2 - if Even (2 * n) then 4 else 1) / 2) ≠ 0
|
rw [coeff_preΨ', if_pos <| even_two_mul n, n.mul_div_cancel_left two_pos]
|
case intro.inl
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
h : ↑(2 * n) ≠ 0
⊢ ↑n ≠ 0
|
fccb7dd4d924abaf
|
CategoryTheory.Pretriangulated.Triangle.coyoneda_exact₂
|
Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
|
lemma coyoneda_exact₂ {X : C} (f : X ⟶ T.obj₂) (hf : f ≫ T.mor₂ = 0) :
∃ (g : X ⟶ T.obj₁), f = g ≫ T.mor₁
|
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
X : C
f : X ⟶ T.obj₂
hf : f ≫ T.mor₂ = 0
a : (contractibleTriangle X).obj₁ ⟶ T.obj₁
ha₁ : (contractibleTriangle X).mor₁ ≫ f = a ≫ T.mor₁
right✝ : (contractibleTriangle X).mor₃ ≫ (CategoryTheory.shiftFunctor C 1).map a = 0 ≫ T.mor₃
⊢ f = a ≫ T.mor₁
|
simpa using ha₁
|
no goals
|
bd5002c2de82ef50
|
IsOfFinOrder.of_mem_zpowers
|
Mathlib/GroupTheory/OrderOfElement.lean
|
theorem IsOfFinOrder.of_mem_zpowers (h : IsOfFinOrder x) (h' : y ∈ Subgroup.zpowers x) :
IsOfFinOrder y
|
case intro
G : Type u_1
inst✝ : Group G
x : G
h : IsOfFinOrder x
k : ℤ
h' : x ^ k ∈ zpowers x
⊢ IsOfFinOrder (x ^ k)
|
exact h.zpow
|
no goals
|
bb35899fd4351fd8
|
CliffordAlgebra.toEven_ι
|
Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean
|
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m
|
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m = e0 Q * (v Q) m
|
rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.mulLeft_apply]
|
no goals
|
4b729990bfda8273
|
ComplexShape.Embedding.homRestrict.f_eq
|
Mathlib/Algebra/Homology/Embedding/HomEquiv.lean
|
lemma f_eq {i : ι} {i' : ι'} (h : e.f i = i') :
f ψ i = (K.restrictionXIso e h).hom ≫ ψ.f i' ≫ (L.extendXIso e h).hom
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i : ι
⊢ f ψ i = (K.restrictionXIso e ⋯).hom ≫ ψ.f (e.f i) ≫ (L.extendXIso e ⋯).hom
|
simp [f, restrictionXIso]
|
no goals
|
f5405c6a7b0e3ab8
|
Multiset.map_eq_map_of_bij_of_nodup
|
Mathlib/Data/Multiset/Nodup.lean
|
theorem map_eq_map_of_bij_of_nodup (f : α → γ) (g : β → γ) {s : Multiset α} {t : Multiset β}
(hs : s.Nodup) (ht : t.Nodup) (i : ∀ a ∈ s, β) (hi : ∀ a ha, i a ha ∈ t)
(i_inj : ∀ a₁ ha₁ a₂ ha₂, i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂)
(i_surj : ∀ b ∈ t, ∃ a ha, i a ha = b) (h : ∀ a ha, f a = g (i a ha)) : s.map f = t.map g
|
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
s : Multiset α
t : Multiset β
hs : s.Nodup
ht : t.Nodup
i : (a : α) → a ∈ s → β
hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t
i_inj : ∀ (a₁ : α) (ha₁ : a₁ ∈ s) (a₂ : α) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂
i_surj : ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), i a ha = b
h : ∀ (a : α) (ha : a ∈ s), f a = g (i a ha)
this : t = map (fun x => i ↑x ⋯) s.attach
⊢ map (fun x => f ↑x) s.attach = map (g ∘ fun x => i ↑x ⋯) s.attach
|
exact map_congr rfl fun x _ => h _ _
|
no goals
|
75d9895300027737
|
Real.differentiableAt_Gamma
|
Mathlib/Analysis/SpecialFunctions/Gamma/Deriv.lean
|
theorem differentiableAt_Gamma {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : DifferentiableAt ℝ Gamma s
|
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ ∀ (m : ℕ), ¬s = -↑m
|
exact hs
|
no goals
|
35506c9ecfa9da1d
|
Fin.map_valEmbedding_Iic
|
Mathlib/Order/Interval/Finset/Fin.lean
|
theorem map_valEmbedding_Iic : (Iic b).map Fin.valEmbedding = Iic ↑b
|
n : ℕ
b : Fin n
⊢ map valEmbedding (Iic b) = Iic ↑b
|
simp [Iic_eq_finset_subtype, Finset.fin, Finset.map_map, Iic_filter_lt_of_lt_right]
|
no goals
|
94722a0b3074eb1d
|
Ordinal.mem_closure_tfae
|
Mathlib/SetTheory/Ordinal/Topology.lean
|
theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
(∀ x hx, f x hx ∈ s) ∧ bsup.{u, u} o f = a,
∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ ⨆ i, f i = a]
|
case inl
a : Ordinal.{u}
s : Set Ordinal.{u}
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
x✝ : a ∈ closure (s ∩ Iic a)
h : a ∈ closure (s ∩ Iic a) := x✝
he : s ∩ Iic a = ∅
⊢ (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a
|
simp [he] at h
|
no goals
|
f361ec1995f661dc
|
Orientation.oangle_rotation_self_right
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
|
theorem oangle_rotation_self_right {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
o.oangle x (o.rotation θ x) = θ
|
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
hx : x ≠ 0
θ : Real.Angle
⊢ o.oangle x ((o.rotation θ) x) = θ
|
simp [hx]
|
no goals
|
21c9a49259599093
|
PadicSeq.norm_eq_zpow_neg_valuation
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem norm_eq_zpow_neg_valuation {f : PadicSeq p} (hf : ¬f ≈ 0) :
f.norm = (p : ℚ) ^ (-f.valuation : ℤ)
|
case hnc
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
H : ↑f (stationaryPoint hf) = 0
⊢ False
|
apply CauSeq.not_limZero_of_not_congr_zero hf
|
case hnc
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
H : ↑f (stationaryPoint hf) = 0
⊢ LimZero f
|
45ff701a2243d145
|
TruncatedWittVector.eq_of_le_of_cast_pow_eq_zero
|
Mathlib/RingTheory/WittVector/Compare.lean
|
theorem eq_of_le_of_cast_pow_eq_zero [CharP R p] (i : ℕ) (hin : i ≤ n)
(hpi : (p : TruncatedWittVector p n R) ^ i = 0) : i = n
|
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hin : i < n
this : ↑p ^ i = (WittVector.truncate n) (↑p ^ i)
⊢ ↑p ^ i ≠ 0
|
rw [this, ne_eq, TruncatedWittVector.ext_iff, not_forall]
|
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hin : i < n
this : ↑p ^ i = (WittVector.truncate n) (↑p ^ i)
⊢ ∃ x, ¬coeff x ((WittVector.truncate n) (↑p ^ i)) = coeff x 0
|
37d90faaabbdab7c
|
SimpleGraph.IsClique.of_induce
|
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
theorem IsClique.of_induce {S : Subgraph G} {F : Set α} {A : Set F}
(c : (S.induce F).coe.IsClique A) : G.IsClique (Subtype.val '' A)
|
α : Type u_1
G : SimpleGraph α
S : G.Subgraph
F : Set α
A : Set ↑F
c : (S.induce F).coe.IsClique A
⊢ ∀ ⦃x : α⦄, (∃ (x_1 : x ∈ F), ⟨x, ⋯⟩ ∈ A) → ∀ ⦃y : α⦄, (∃ (x : y ∈ F), ⟨y, ⋯⟩ ∈ A) → x ≠ y → G.Adj x y
|
intro _ ⟨_, ainA⟩ _ ⟨_, binA⟩ anb
|
α : Type u_1
G : SimpleGraph α
S : G.Subgraph
F : Set α
A : Set ↑F
c : (S.induce F).coe.IsClique A
x✝ : α
w✝¹ : x✝ ∈ F
ainA : ⟨x✝, ⋯⟩ ∈ A
y✝ : α
w✝ : y✝ ∈ F
binA : ⟨y✝, ⋯⟩ ∈ A
anb : x✝ ≠ y✝
⊢ G.Adj x✝ y✝
|
e25505f81b0d4e77
|
Turing.ToPartrec.Code.exists_code
|
Mathlib/Computability/TMConfig.lean
|
theorem exists_code {n} {f : List.Vector ℕ n →. ℕ} (hf : Nat.Partrec' f) :
∃ c : Code, ∀ v : List.Vector ℕ n, c.eval v.1 = pure <$> f v
|
case rfind.intro.mp
n✝ : ℕ
f✝ : List.Vector ℕ n✝ →. ℕ
n : ℕ
f : List.Vector ℕ (n + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
x : List ℕ
⊢ (∃
a ∈
PFun.fix
(fun v =>
(cf.eval v).bind fun y =>
Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
(0 :: ↑v),
x = [a.headI.pred]) →
∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ pure a = x
|
rintro ⟨v', h1, rfl⟩
|
case rfind.intro.mp.intro.intro
n✝ : ℕ
f✝ : List.Vector ℕ n✝ →. ℕ
n : ℕ
f : List.Vector ℕ (n + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' : List ℕ
h1 :
v' ∈
PFun.fix
(fun v =>
(cf.eval v).bind fun y =>
Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
(0 :: ↑v)
⊢ ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ pure a = [v'.headI.pred]
|
01823c87e1740c3b
|
WType.cardinalMk_le_max_aleph0_of_finite'
|
Mathlib/Data/W/Cardinal.lean
|
theorem cardinalMk_le_max_aleph0_of_finite' [∀ a, Finite (β a)] :
#(WType β) ≤ max (lift.{v} #α) ℵ₀ :=
(isEmpty_or_nonempty α).elim
(by
intro h
rw [Cardinal.mk_eq_zero (WType β)]
exact zero_le _)
fun hn =>
let m := max (lift.{v} #α) ℵ₀
cardinalMk_le_of_le' <|
calc
(Cardinal.sum fun a => m ^ lift.{u} #(β a)) ≤ lift.{v} #α * ⨆ a, m ^ lift.{u} #(β a) :=
Cardinal.sum_le_iSup_lift _
_ ≤ m * ⨆ a, m ^ lift.{u} #(β a) := mul_le_mul' (le_max_left _ _) le_rfl
_ = m :=
mul_eq_left (le_max_right _ _)
(ciSup_le' fun _ => pow_le (le_max_right _ _) (lt_aleph0_of_finite _)) <|
pos_iff_ne_zero.1 <|
Order.succ_le_iff.1
(by
rw [succ_zero]
obtain ⟨a⟩ : Nonempty α := hn
refine le_trans ?_ (le_ciSup (bddAbove_range _) a)
rw [← power_zero]
exact
power_le_power_left
(pos_iff_ne_zero.1 (aleph0_pos.trans_le (le_max_right _ _))) (zero_le _))
|
α : Type u
β : α → Type v
inst✝ : ∀ (a : α), Finite (β a)
hn : Nonempty α
m : Cardinal.{max u v} := lift.{v, u} #α ⊔ ℵ₀
⊢ Order.succ 0 ≤ ⨆ a, m ^ lift.{u, v} #(β a)
|
rw [succ_zero]
|
α : Type u
β : α → Type v
inst✝ : ∀ (a : α), Finite (β a)
hn : Nonempty α
m : Cardinal.{max u v} := lift.{v, u} #α ⊔ ℵ₀
⊢ 1 ≤ ⨆ a, m ^ lift.{u, v} #(β a)
|
943db473760739a3
|
FormalMultilinearSeries.derivSeries_eq_zero
|
Mathlib/Analysis/Analytic/ChangeOrigin.lean
|
theorem derivSeries_eq_zero {n : ℕ} (hp : p (n + 1) = 0) : p.derivSeries n = 0
|
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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hp : p (n + 1) = 0
this : p.changeOriginSeries 1 n = 0
⊢ p.derivSeries n = 0
|
ext v
|
case H.h
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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hp : p (n + 1) = 0
this : p.changeOriginSeries 1 n = 0
v : Fin n → E
x✝ : E
⊢ ((p.derivSeries n) v) x✝ = (0 v) x✝
|
e92bd3437a4bd85b
|
MeasureTheory.IntegrableOn.hasBoxIntegral
|
Mathlib/Analysis/BoxIntegral/Integrability.lean
|
theorem IntegrableOn.hasBoxIntegral [CompleteSpace E] {f : (ι → ℝ) → E} {μ : Measure (ι → ℝ)}
[IsLocallyFiniteMeasure μ] {I : Box ι} (hf : IntegrableOn f I μ) (l : IntegrationParams)
(hl : l.bRiemann = false) :
HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul (∫ x in I, f x ∂μ)
|
case intro.intro.intro.intro.intro.intro.intro.intro.refine_3
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤ ↑ε
|
refine le_trans ?_ hN₀
|
case intro.intro.intro.intro.intro.intro.intro.intro.refine_3
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ
|
1cd867d174f2adc3
|
Polynomial.scaleRoots_eval₂_mul_of_commute
|
Mathlib/RingTheory/Polynomial/ScaleRoots.lean
|
theorem scaleRoots_eval₂_mul_of_commute {p : S[X]} (f : S →+* A) (a : A) (s : S)
(hsa : Commute (f s) a) (hf : ∀ s₁ s₂, Commute (f s₁) (f s₂)) :
eval₂ f (f s * a) (scaleRoots p s) = f s ^ p.natDegree * eval₂ f a p
|
S : Type u_2
A : Type u_3
inst✝¹ : Semiring S
inst✝ : Semiring A
p : S[X]
f : S →+* A
a : A
s : S
hsa : Commute (f s) a
hf : ∀ (s₁ s₂ : S), Commute (f s₁) (f s₂)
⊢ ∑ i ∈ p.support, f s ^ p.natDegree * (f (p.coeff i) * a ^ i) = f s ^ p.natDegree * eval₂ f a p
|
simp [← Finset.mul_sum, eval₂_eq_sum, sum_def]
|
no goals
|
72127d2a3a809905
|
Mathlib.Tactic.Abel.term_smul
|
Mathlib/Tactic/Abel.lean
|
theorem term_smul {α} [AddCommMonoid α] (c n x a n' a')
(h₁ : c * n = n') (h₂ : smul c a = a') :
smul c (@term α _ n x a) = term n' x a'
|
α : Type u_1
inst✝ : AddCommMonoid α
c n : ℕ
x a : α
n' : ℕ
a' : α
h₁ : c * n = n'
h₂ : smul c a = a'
⊢ smul c (term n x a) = term n' x a'
|
simp [h₂.symm, h₁.symm, term, smul, nsmul_add, mul_nsmul']
|
no goals
|
459cd529084993cd
|
Stream'.Seq.terminatedAt_zero_iff
|
Mathlib/Data/Seq/Seq.lean
|
theorem terminatedAt_zero_iff {s : Seq α} : s.TerminatedAt 0 ↔ s = nil
|
case refine_1
α : Type u
s : Seq α
h : s.TerminatedAt 0
⊢ s = nil
|
ext n
|
case refine_1.h.a
α : Type u
s : Seq α
h : s.TerminatedAt 0
n : ℕ
a✝ : α
⊢ a✝ ∈ s.get? n ↔ a✝ ∈ nil.get? n
|
10aa63b0fda6161a
|
ax_grothendieck_of_locally_finite
|
Mathlib/FieldTheory/AxGrothendieck.lean
|
theorem ax_grothendieck_of_locally_finite {ι K R : Type*} [Field K] [Finite K] [CommRing R]
[Finite ι] [Algebra K R] [alg : Algebra.IsAlgebraic K R] (ps : ι → MvPolynomial ι R)
(S : Set (ι → R))
(hm : S.MapsTo (fun v i => eval v (ps i)) S)
(hinj : S.InjOn (fun v i => eval v (ps i))) :
S.SurjOn (fun v i => eval v (ps i)) S
|
case intro.intro.mk
ι : Type u_1
K : Type u_2
R : Type u_3
inst✝⁴ : Field K
inst✝³ : Finite K
inst✝² : CommRing R
inst✝¹ : Finite ι
inst✝ : Algebra K R
alg : Algebra.IsAlgebraic K R
ps : ι → MvPolynomial ι R
S : Set (ι → R)
hm : Set.MapsTo (fun v i => (eval v) (ps i)) S S
hinj : Set.InjOn (fun v i => (eval v) (ps i)) S
is_int : ∀ (x : R), IsIntegral K x
v : ι → R
hvS : v ∈ S
val✝ : Fintype ι
s : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) ∪ image v univ
hv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s
hs₁ : ∀ (i : ι), ∀ k ∈ (ps i).support, coeff k (ps i) ∈ Algebra.adjoin K ↑s
this✝¹ : IsNoetherian K ↥(Algebra.adjoin K ↑s)
this✝ : Module.Finite K ↥(Algebra.adjoin K ↑s)
this : Finite ↥(Algebra.adjoin K ↑s)
S' : Set (ι → ↥(Algebra.adjoin K ↑s)) := (fun v => Subtype.val ∘ v) ⁻¹' S
res : ↑S' → ↑S' := fun x => ⟨fun i => ⟨(eval fun j => ↑(↑x j)) (ps i), ⋯⟩, ⋯⟩
hres_surj : Function.Surjective res
w : ι → ↥(Algebra.adjoin K ↑s)
hwS' : w ∈ S'
hw : res ⟨w, hwS'⟩ = ⟨fun i => ⟨v i, ⋯⟩, hvS⟩
⊢ ((fun v i => (eval v) (ps i)) fun i => ↑(w i)) = v
|
simpa [Subtype.ext_iff, funext_iff] using hw
|
no goals
|
6b85e3de256b7145
|
Filter.Realizer.mem_sets
|
Mathlib/Data/Analysis/Filter.lean
|
theorem mem_sets {f : Filter α} (F : f.Realizer) {a : Set α} : a ∈ f ↔ ∃ b, F.F b ⊆ a
|
case mk
α : Type u_1
a : Set α
σ✝ : Type u_5
F✝ : CFilter (Set α) σ✝
⊢ a ∈ F✝.toFilter ↔ ∃ b, { σ := σ✝, F := F✝, eq := ⋯ }.F.f b ⊆ a
|
rfl
|
no goals
|
18165e4fada01f24
|
Std.DHashMap.Internal.List.Const.getKey?_modifyKey
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem getKey?_modifyKey [EquivBEq α] {k k' : α} {f : β → β} (l : List ((_ : α) × β))
(hl : DistinctKeys l) :
getKey? k' (modifyKey k f l) =
if k == k' then
if containsKey k l then some k else none
else
getKey? k' l
|
α : Type u
β : Type v
inst✝¹ : BEq α
inst✝ : EquivBEq α
k k' : α
f : β → β
l : List ((_ : α) × β)
hl : DistinctKeys l
⊢ getKey? k' (modifyKey k f l) =
if (k == k') = true then if containsKey k l = true then some k else none else getKey? k' l
|
simp [modifyKey_eq_alterKey, getKey?_alterKey, containsKey_eq_isSome_getValue?, hl]
|
no goals
|
678544991fa81927
|
MeasureTheory.AEEqFun.inf_le_right
|
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
theorem inf_le_right (f g : α →ₘ[μ] β) : f ⊓ g ≤ g
|
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ ↑(f ⊓ g) ≤ᶠ[ae μ] ↑g
|
filter_upwards [coeFn_inf f g] with _ ha
|
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝
|
4d1b7a227e1bb493
|
Ring.choose_succ_succ
|
Mathlib/RingTheory/Binomial.lean
|
theorem choose_succ_succ [NatPowAssoc R] (r : R) (k : ℕ) :
choose (r + 1) (k + 1) = choose r k + choose r (k + 1)
|
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
k : ℕ
⊢ choose (r + 1) (k + 1) = choose r k + choose r (k + 1)
|
rw [← nsmul_right_inj (Nat.factorial_ne_zero (k + 1))]
|
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
k : ℕ
⊢ (k + 1).factorial • choose (r + 1) (k + 1) = (k + 1).factorial • (choose r k + choose r (k + 1))
|
db680efa3366c602
|
MeasureTheory.addHaar_image_le_mul_of_det_lt
|
Mathlib/MeasureTheory/Function/Jacobian.lean
|
theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
(hm : ENNReal.ofReal |A.det| < m) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ m * μ s
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this✝ : Iio ⟨ε, ⋯⟩ ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
I : ∀ (x : E) (r : ℝ), x ∈ s → 0 ≤ r → μ (f '' (s ∩ closedBall x r)) ≤ ↑m * μ (closedBall x r)
a : ℝ≥0∞
ha : 0 < a
t : Set E
r : E → ℝ
t_count : t.Countable
ts : t ⊆ s
rpos : ∀ x ∈ t, 0 < r x
st : s ⊆ ⋃ x ∈ t, closedBall x (r x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ ∑' (x : ↑t), ↑m * μ (closedBall (↑x) (r ↑x)) ≤ ↑m * (μ s + a)
|
rw [ENNReal.tsum_mul_left]
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ENNReal.ofReal |A.det| < ↑m
d : ℝ≥0∞ := ENNReal.ofReal |A.det|
ε : ℝ
hε : μ (closedBall 0 ε + ⇑A '' closedBall 0 1) < ↑m * μ (closedBall 0 1)
εpos : 0 < ε
this✝ : Iio ⟨ε, ⋯⟩ ∈ 𝓝 0
δ : ℝ≥0
s : Set E
f : E → E
hf : ApproximatesLinearOn f A s δ
hδ : ↑δ < ε
I : ∀ (x : E) (r : ℝ), x ∈ s → 0 ≤ r → μ (f '' (s ∩ closedBall x r)) ≤ ↑m * μ (closedBall x r)
a : ℝ≥0∞
ha : 0 < a
t : Set E
r : E → ℝ
t_count : t.Countable
ts : t ⊆ s
rpos : ∀ x ∈ t, 0 < r x
st : s ⊆ ⋃ x ∈ t, closedBall x (r x)
μt : ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + a
this : Encodable ↑t
⊢ ↑m * ∑' (i : ↑t), μ (closedBall (↑i) (r ↑i)) ≤ ↑m * (μ s + a)
|
35c8118ffea35652
|
Polynomial.Splits.comp_of_map_degree_le_one
|
Mathlib/Algebra/Polynomial/Splits.lean
|
theorem Splits.comp_of_map_degree_le_one {f : K[X]} {p : K[X]} (hd : (p.map i).degree ≤ 1)
(h : f.Splits i) : (f.comp p).Splits i
|
case neg
K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
f p : K[X]
hd : (map i p).degree ≤ 1
h : Splits i f
hzero : ¬map i (f.comp p) = 0
⊢ Splits i (f.comp p)
|
cases h with
| inl h0 =>
exact Or.inl <| map_comp i _ _ ▸ h0.symm ▸ zero_comp
| inr h =>
right
intro g irr dvd
rw [map_comp] at dvd hzero
cases lt_or_eq_of_le hd with
| inl hd =>
rw [eq_C_of_degree_le_zero (Nat.WithBot.lt_one_iff_le_zero.mp hd), comp_C] at dvd hzero
refine False.elim (irr.1 (isUnit_of_dvd_unit dvd ?_))
simpa using hzero
| inr hd =>
let _ := invertibleOfNonzero (leadingCoeff_ne_zero.mpr
(ne_zero_of_degree_gt (n := ⊥) (by rw [hd]; decide)))
rw [eq_X_add_C_of_degree_eq_one hd, dvd_comp_C_mul_X_add_C_iff _ _] at dvd
have := h (irr.map (algEquivCMulXAddC _ _).symm) dvd
rw [degree_eq_natDegree irr.ne_zero]
rwa [algEquivCMulXAddC_symm_apply, ← comp_eq_aeval,
degree_eq_natDegree (fun h => WithBot.bot_ne_one (h ▸ this)),
natDegree_comp, natDegree_C_mul (invertibleInvOf.ne_zero),
natDegree_X_sub_C, mul_one] at this
|
no goals
|
7862ae74e5525276
|
derivWithin_finset_prod
|
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
theorem derivWithin_finset_prod
(hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x) :
derivWithin (∏ i ∈ u, f i ·) s x =
∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x
|
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
inst✝² : DecidableEq ι
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
⊢ derivWithin (fun x => ∏ i ∈ u, f i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x
|
rcases uniqueDiffWithinAt_or_nhdsWithin_eq_bot s x with hxs | hxs
|
case inl
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
inst✝² : DecidableEq ι
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x => ∏ i ∈ u, f i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x
case inr
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_2
inst✝² : DecidableEq ι
𝔸' : Type u_3
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → 𝕜 → 𝔸'
hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x
hxs : 𝓝[s \ {x}] x = ⊥
⊢ derivWithin (fun x => ∏ i ∈ u, f i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x
|
569cd40d28616fff
|
CliffordAlgebra.lift_unique
|
Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean
|
theorem lift_unique (f : M →ₗ[R] A) (cond : ∀ m : M, f m * f m = algebraMap _ _ (Q m))
(g : CliffordAlgebra Q →ₐ[R] A) : g.toLinearMap.comp (ι Q) = f ↔ g = lift Q ⟨f, cond⟩
|
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : M →ₗ[R] A
cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)
g : CliffordAlgebra Q →ₐ[R] A
⊢ g.toLinearMap ∘ₗ ι Q = f ↔ g = (lift Q) ⟨f, cond⟩
|
convert (lift Q : _ ≃ (CliffordAlgebra Q →ₐ[R] A)).symm_apply_eq
|
case h.e'_1.a
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : M →ₗ[R] A
cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)
g : CliffordAlgebra Q →ₐ[R] A
⊢ g.toLinearMap ∘ₗ ι Q = f ↔ (lift Q).symm g = ⟨f, cond⟩
|
ae71a02960c21142
|
Topology.IsInducing.frechetUrysohnSpace
|
Mathlib/Topology/Sequences.lean
|
theorem Topology.IsInducing.frechetUrysohnSpace [FrechetUrysohnSpace Y] {f : X → Y}
(hf : IsInducing f) : FrechetUrysohnSpace X
|
case intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : FrechetUrysohnSpace Y
f : X → Y
hf : IsInducing f
s : Set X
x : X
u : ℕ → Y
hus : ∀ (n : ℕ), u n ∈ f '' s
hu : Tendsto u atTop (𝓝 (f x))
⊢ x ∈ seqClosure s
|
choose v hv hvu using hus
|
case intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : FrechetUrysohnSpace Y
f : X → Y
hf : IsInducing f
s : Set X
x : X
u : ℕ → Y
hu : Tendsto u atTop (𝓝 (f x))
v : ℕ → X
hv : ∀ (n : ℕ), v n ∈ s
hvu : ∀ (n : ℕ), f (v n) = u n
⊢ x ∈ seqClosure s
|
cd788f2aa1246489
|
Complex.HadamardThreeLines.norm_le_sSupNormIm
|
Mathlib/Analysis/Complex/Hadamard.lean
|
/-- If `f` is bounded on the unit vertical strip, then `f` is bounded by `sSupNormIm` there. -/
lemma norm_le_sSupNormIm (f : ℂ → E) (z : ℂ) (hD : z ∈ verticalClosedStrip 0 1)
(hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1)) :
‖f z‖ ≤ sSupNormIm f (z.re)
|
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
hD : z ∈ verticalClosedStrip 0 1
hB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)
⊢ re ⁻¹' {z.re} ⊆ verticalClosedStrip 0 1
|
exact preimage_mono (singleton_subset_iff.mpr hD)
|
no goals
|
c110645de8d374d2
|
Set.iUnion_range_eq_iUnion
|
Mathlib/Data/Set/Lattice.lean
|
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x
|
case h.mpr.intro
α : Type u_1
β : Type u_2
ι : Sort u_5
C : ι → Set α
f : (x : ι) → β → ↑(C x)
hf : ∀ (x : ι), Surjective (f x)
x : α
i : ι
hx : x ∈ C i
⊢ ∃ i, x ∈ range fun x => ↑(f x i)
|
obtain ⟨y, hy⟩ := hf i ⟨x, hx⟩
|
case h.mpr.intro.intro
α : Type u_1
β : Type u_2
ι : Sort u_5
C : ι → Set α
f : (x : ι) → β → ↑(C x)
hf : ∀ (x : ι), Surjective (f x)
x : α
i : ι
hx : x ∈ C i
y : β
hy : f i y = ⟨x, hx⟩
⊢ ∃ i, x ∈ range fun x => ↑(f x i)
|
9d4f3f5bc914a4d2
|
IsTopologicalSemiring.continuousNeg_of_mul
|
Mathlib/Topology/Algebra/Ring/Basic.lean
|
theorem IsTopologicalSemiring.continuousNeg_of_mul [TopologicalSpace α] [NonAssocRing α]
[ContinuousMul α] : ContinuousNeg α where
continuous_neg
|
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : NonAssocRing α
inst✝ : ContinuousMul α
⊢ Continuous fun a => -a
|
simpa using (continuous_const.mul continuous_id : Continuous fun x : α => -1 * x)
|
no goals
|
268e24d94763f555
|
VitaliFamily.exists_measurable_supersets_limRatio
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I✝ : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L : Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x))
I : ∀ᶠ (b : Set α) in v.filterAt x, ρ b / μ b < ↑p
a : Set α
ha : ρ a / μ a < ↑p
⊢ μ a ≠ 0 ∨ ↑p ≠ ⊤
|
simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
|
no goals
|
f2acb3dc8c3a0a55
|
Finite.exists_univ_list
|
Mathlib/Data/Fintype/Card.lean
|
theorem Finite.exists_univ_list (α) [Finite α] : ∃ l : List α, l.Nodup ∧ ∀ x : α, x ∈ l
|
α : Type u_4
inst✝ : Finite α
⊢ ∃ l, l.Nodup ∧ ∀ (x : α), x ∈ l
|
cases nonempty_fintype α
|
case intro
α : Type u_4
inst✝ : Finite α
val✝ : Fintype α
⊢ ∃ l, l.Nodup ∧ ∀ (x : α), x ∈ l
|
3ac20f8d6bf74439
|
MvPFunctor.w_map_wMk
|
Mathlib/Data/PFunctor/Multivariate/W.lean
|
theorem w_map_wMk {α β : TypeVec n} (g : α ⟹ β) (a : P.A) (f' : P.drop.B a ⟹ α)
(f : P.last.B a → P.W α) : g <$$> P.wMk a f' f = P.wMk a (g ⊚ f') fun i => g <$$> f i
|
n : ℕ
P : MvPFunctor.{u} (n + 1)
α β : TypeVec.{u} n
g : α ⟹ β
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
this✝ : MvFunctor.map g ∘ f = fun i => ⟨(f i).fst, g ⊚ (f i).snd⟩
this : f = fun i => ⟨(f i).fst, (f i).snd⟩
⊢ (g <$$> P.wMk a f' fun i => ⟨(f i).fst, (f i).snd⟩) =
P.wMk a (g ⊚ f') fun i => ⟨((fun i => ⟨(f i).fst, (f i).snd⟩) i).fst, g ⊚ ((fun i => ⟨(f i).fst, (f i).snd⟩) i).snd⟩
|
dsimp
|
n : ℕ
P : MvPFunctor.{u} (n + 1)
α β : TypeVec.{u} n
g : α ⟹ β
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
this✝ : MvFunctor.map g ∘ f = fun i => ⟨(f i).fst, g ⊚ (f i).snd⟩
this : f = fun i => ⟨(f i).fst, (f i).snd⟩
⊢ (g <$$> P.wMk a f' fun i => ⟨(f i).fst, (f i).snd⟩) = P.wMk a (g ⊚ f') fun i => ⟨(f i).fst, g ⊚ (f i).snd⟩
|
2aebefc7b173d248
|
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