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
|
|---|---|---|---|---|---|---|
PosNum.succ_to_nat
|
Mathlib/Data/Num/Lemmas.lean
|
theorem succ_to_nat : ∀ n, (succ n : ℕ) = n + 1
| 1 => rfl
| bit0 _ => rfl
| bit1 p =>
(congr_arg (fun n ↦ n + n) (succ_to_nat p)).trans <|
show ↑p + 1 + ↑p + 1 = ↑p + ↑p + 1 + 1 by simp [add_left_comm]
|
p : PosNum
⊢ ↑p + 1 + ↑p + 1 = ↑p + ↑p + 1 + 1
|
simp [add_left_comm]
|
no goals
|
f89e8a33aaa689e3
|
MulAction.quotient_preimage_image_eq_union_mul
|
Mathlib/GroupTheory/GroupAction/Defs.lean
|
theorem quotient_preimage_image_eq_union_mul (U : Set α) :
letI := orbitRel G α
Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (g • ·) '' U
|
case h.mpr.intro.intro.intro
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
U : Set α
this : Setoid α := orbitRel G α
f : α → Quotient (orbitRel G α) := Quotient.mk'
a : α
g : G
u : α
hu₁ : u ∈ U
hu₂ : (fun x => g • x) u = a
⊢ g⁻¹ • (fun x => g • x) u ∈ U
|
convert hu₁
|
case h.e'_5
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
U : Set α
this : Setoid α := orbitRel G α
f : α → Quotient (orbitRel G α) := Quotient.mk'
a : α
g : G
u : α
hu₁ : u ∈ U
hu₂ : (fun x => g • x) u = a
⊢ g⁻¹ • (fun x => g • x) u = u
|
089d93c3a4497d0d
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.derivedLitsInvariant_confirmRupHint
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
|
theorem derivedLitsInvariant_confirmRupHint {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(rupHints : Array Nat) (i : Fin rupHints.size)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool)
(ih : ∃ hsize : acc.1.size = n, DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.1) :
let rupHint_res := (confirmRupHint f.clauses) acc rupHints[i]
∃ hsize : rupHint_res.1.size = n, DerivedLitsInvariant f f_assignments_size rupHint_res.1 hsize rupHint_res.2.1
|
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
⊢ none = none ∨ none = some none ∨ ∃ c, none = some (some c)
|
exact Or.inl rfl
|
no goals
|
0f20fdb04006be12
|
Lean.Omega.IntList.gcd_cons_div_right
|
Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean
|
theorem gcd_cons_div_right : gcd (x::xs) ∣ gcd xs
|
x : Int
xs : List Int
⊢ x.natAbs.gcd (List.foldr (fun x g => x.natAbs.gcd g) 0 xs) ∣ List.foldr (fun x g => x.natAbs.gcd g) 0 xs
|
apply Nat.gcd_dvd_right
|
no goals
|
0c75c6152b61dcde
|
Submodule.eq_top_of_nonempty_interior'
|
Mathlib/Topology/Algebra/Module/Basic.lean
|
theorem Submodule.eq_top_of_nonempty_interior' [NeBot (𝓝[{ x : R | IsUnit x }] 0)]
(s : Submodule R M) (hs : (interior (s : Set M)).Nonempty) : s = ⊤
|
case intro
R : Type u_1
M : Type u_2
inst✝⁷ : Ring R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommGroup M
inst✝³ : ContinuousAdd M
inst✝² : Module R M
inst✝¹ : ContinuousSMul R M
inst✝ : (𝓝[{x | IsUnit x}] 0).NeBot
s : Submodule R M
y : M
hy : ↑s ∈ 𝓝 y
x : M
this : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)
⊢ x ∈ s
|
obtain ⟨_, hu : y + _ • _ ∈ s, u, rfl⟩ :=
nonempty_of_mem (inter_mem (Filter.mem_map.1 (this hy)) self_mem_nhdsWithin)
|
case intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝⁷ : Ring R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommGroup M
inst✝³ : ContinuousAdd M
inst✝² : Module R M
inst✝¹ : ContinuousSMul R M
inst✝ : (𝓝[{x | IsUnit x}] 0).NeBot
s : Submodule R M
y : M
hy : ↑s ∈ 𝓝 y
x : M
this : Tendsto (fun c => y + c • x) (𝓝[{x | IsUnit x}] 0) (𝓝 y)
u : Rˣ
hu : y + ↑u • x ∈ s
⊢ x ∈ s
|
0f5438c3c47b1300
|
ProbabilityTheory.Kernel.borelMarkovFromReal_apply
|
Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean
|
lemma borelMarkovFromReal_apply (Ω : Type*) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω]
(η : Kernel α ℝ) (a : α) :
borelMarkovFromReal Ω η a
= if η a (range (embeddingReal Ω))ᶜ = 0 then (η a).comap (embeddingReal Ω)
else (Measure.dirac (range_nonempty (embeddingReal Ω)).choose).comap (embeddingReal Ω)
|
α : Type u_1
mα : MeasurableSpace α
Ω : Type u_5
inst✝² : Nonempty Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : StandardBorelSpace Ω
η : Kernel α ℝ
a : α
⊢ (borelMarkovFromReal Ω η) a =
if (η a) (range (embeddingReal Ω))ᶜ = 0 then Measure.comap (embeddingReal Ω) (η a)
else Measure.comap (embeddingReal Ω) (Measure.dirac (Exists.choose ⋯))
|
rw [borelMarkovFromReal, comapRight_apply, piecewise_apply, deterministic_apply]
|
α : Type u_1
mα : MeasurableSpace α
Ω : Type u_5
inst✝² : Nonempty Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : StandardBorelSpace Ω
η : Kernel α ℝ
a : α
⊢ Measure.comap (embeddingReal Ω)
(if a ∈ (fun a => (η a) (range (embeddingReal Ω))ᶜ) ⁻¹' {0} then η a else Measure.dirac (Exists.choose ⋯)) =
if (η a) (range (embeddingReal Ω))ᶜ = 0 then Measure.comap (embeddingReal Ω) (η a)
else Measure.comap (embeddingReal Ω) (Measure.dirac (Exists.choose ⋯))
|
0d8be79f96546288
|
PartialHomeomorph.ball_subset_univBall_target
|
Mathlib/Analysis/NormedSpace/HomeomorphBall.lean
|
theorem ball_subset_univBall_target (c : P) (r : ℝ) : ball c r ⊆ (univBall c r).target
|
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
⊢ ball c r ⊆ (univBall c r).target
|
by_cases hr : 0 < r
|
case pos
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
hr : 0 < r
⊢ ball c r ⊆ (univBall c r).target
case neg
E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
hr : ¬0 < r
⊢ ball c r ⊆ (univBall c r).target
|
38ee5f85990fea06
|
finprod_mem_union_inter
|
Mathlib/Algebra/BigOperators/Finprod.lean
|
theorem finprod_mem_union_inter (hs : s.Finite) (ht : t.Finite) :
((∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i) = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i
|
case intro.intro
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Finset α
⊢ (∏ᶠ (i : α) (_ : i ∈ ↑s ∪ ↑t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ ↑t), f i =
(∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i
|
rw [← Finset.coe_union, ← Finset.coe_inter]
|
case intro.intro
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Finset α
⊢ (∏ᶠ (i : α) (_ : i ∈ ↑(s ∪ t)), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑(s ∩ t)), f i =
(∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i
|
274cf09bf950a217
|
Filter.limsup_sdiff
|
Mathlib/Order/LiminfLimsup.lean
|
theorem limsup_sdiff (a : α) : limsup u f \ a = limsup (fun b => u b \ a) f
|
α : Type u_1
β : Type u_2
inst✝ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
a : α
⊢ (⨅ s ∈ f, ⨆ a ∈ s, u a) ⊓ aᶜ = ⨅ s ∈ f, ⨆ a_1 ∈ s, u a_1 ⊓ aᶜ
|
rw [biInf_inf (⟨univ, univ_mem⟩ : ∃ i : Set β, i ∈ f)]
|
α : Type u_1
β : Type u_2
inst✝ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
a : α
⊢ ⨅ i ∈ f, (⨆ a ∈ i, u a) ⊓ aᶜ = ⨅ s ∈ f, ⨆ a_1 ∈ s, u a_1 ⊓ aᶜ
|
9e3d0b82dbd812d0
|
CategoryTheory.MorphismProperty.transfiniteCompositionsOfShape_eq_of_orderIso
|
Mathlib/CategoryTheory/MorphismProperty/TransfiniteComposition.lean
|
lemma transfiniteCompositionsOfShape_eq_of_orderIso (e : J ≃o J') :
W.transfiniteCompositionsOfShape J =
W.transfiniteCompositionsOfShape J'
|
C : Type u
inst✝⁸ : Category.{v, u} C
W : MorphismProperty C
J : Type w
inst✝⁷ : LinearOrder J
inst✝⁶ : SuccOrder J
inst✝⁵ : OrderBot J
inst✝⁴ : WellFoundedLT J
J' : Type w'
inst✝³ : LinearOrder J'
inst✝² : SuccOrder J'
inst✝¹ : OrderBot J'
inst✝ : WellFoundedLT J'
e : J ≃o J'
⊢ W.transfiniteCompositionsOfShape J = W.transfiniteCompositionsOfShape J'
|
ext _ _ f
|
case h
C : Type u
inst✝⁸ : Category.{v, u} C
W : MorphismProperty C
J : Type w
inst✝⁷ : LinearOrder J
inst✝⁶ : SuccOrder J
inst✝⁵ : OrderBot J
inst✝⁴ : WellFoundedLT J
J' : Type w'
inst✝³ : LinearOrder J'
inst✝² : SuccOrder J'
inst✝¹ : OrderBot J'
inst✝ : WellFoundedLT J'
e : J ≃o J'
X✝ Y✝ : C
f : X✝ ⟶ Y✝
⊢ W.transfiniteCompositionsOfShape J f ↔ W.transfiniteCompositionsOfShape J' f
|
6fdc6de3075a9720
|
StieltjesFunction.outer_Ioc
|
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a)
|
f : StieltjesFunction
a b : ℝ
s : ℕ → Set ℝ
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ⊤
δ : ℝ≥0 := ε / 2
⊢ 0 < ↑δ
|
simpa [δ] using εpos.ne'
|
no goals
|
c3b0d539cbbc47aa
|
FractionalIdeal.dual_dual
|
Mathlib/RingTheory/DedekindDomain/Different.lean
|
@[simp]
lemma dual_dual :
dual A K (dual A K I) = I
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
⊢ dual A K 1 ≠ 0
|
rw [dual_ne_zero_iff]
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
⊢ 1 ≠ 0
|
8fa9ea92bb952992
|
addRothNumber_le_ruzsaSzemerediNumber
|
Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean
|
lemma addRothNumber_le_ruzsaSzemerediNumber :
card α * addRothNumber (univ : Finset α) ≤ ruzsaSzemerediNumber (Sum α (Sum α α))
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
inst✝¹ : CommRing α
inst✝ : Fact (IsUnit 2)
⊢ Fintype.card α * addRothNumber univ ≤ ruzsaSzemerediNumber (α ⊕ α ⊕ α)
|
obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)
|
case intro.intro.intro
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
inst✝¹ : CommRing α
inst✝ : Fact (IsUnit 2)
s : Finset α
hscard : #s = addRothNumber univ
hs : ThreeAPFree ↑s
⊢ Fintype.card α * addRothNumber univ ≤ ruzsaSzemerediNumber (α ⊕ α ⊕ α)
|
2f78b728f5941b04
|
minpolyDiv_spec
|
Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean
|
lemma minpolyDiv_spec :
minpolyDiv R x * (X - C x) = (minpoly R x).map (algebraMap R S)
|
R : Type u_2
S : Type u_1
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
⊢ map (algebraMap R S) (minpoly R x) /ₘ (X - C x) * (X - C x) = map (algebraMap R S) (minpoly R x)
|
rw [mul_comm, mul_divByMonic_eq_iff_isRoot, IsRoot, eval_map, ← aeval_def, minpoly.aeval]
|
no goals
|
0e3f7575f08e79f8
|
mabs_mul_le
|
Mathlib/Algebra/Order/Group/Unbundled/Abs.lean
|
/-- The absolute value satisfies the triangle inequality. -/
@[to_additive "The absolute value satisfies the triangle inequality."]
lemma mabs_mul_le (a b : α) : |a * b|ₘ ≤ |a|ₘ * |b|ₘ
|
case a
α : Type u_1
inst✝² : Lattice α
inst✝¹ : CommGroup α
inst✝ : MulLeftMono α
a b : α
⊢ a⁻¹ * b⁻¹ ≤ mabs a * mabs b
|
exact mul_le_mul' (inv_le_mabs _) (inv_le_mabs _)
|
no goals
|
89ddb6739385a93f
|
List.Nodup.rotate_congr_iff
|
Mathlib/Data/List/Rotate.lean
|
theorem Nodup.rotate_congr_iff {l : List α} (hl : l.Nodup) {i j : ℕ} :
l.rotate i = l.rotate j ↔ i % l.length = j % l.length ∨ l = []
|
case inl
α : Type u
i j : ℕ
hl : [].Nodup
⊢ [].rotate i = [].rotate j ↔ i % [].length = j % [].length ∨ [] = []
|
simp
|
no goals
|
965e918a3e97caa0
|
FDerivMeasurableAux.isOpen_A
|
Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
|
theorem isOpen_A (L : E →L[𝕜] F) (r ε : ℝ) : IsOpen (A f L r ε)
|
case intro.intro.intro.intro
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
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : E
hx' : x' ∈ ball x (r' - s)
B : ball x' s ⊆ ball x r'
⊢ ∀ y ∈ ball x' s, ∀ z ∈ ball x' s, ‖f z - f y - L (z - y)‖ < ε * r
|
intro y hy z hz
|
case intro.intro.intro.intro
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
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : E
hx' : x' ∈ ball x (r' - s)
B : ball x' s ⊆ ball x r'
y : E
hy : y ∈ ball x' s
z : E
hz : z ∈ ball x' s
⊢ ‖f z - f y - L (z - y)‖ < ε * r
|
6e3e46aca5615231
|
ZFSet.sUnion_empty
|
Mathlib/SetTheory/ZFC/Basic.lean
|
theorem sUnion_empty : ⋃₀ (∅ : ZFSet.{u}) = ∅
|
⊢ ⋃₀ ∅ = ∅
|
ext
|
case a
z✝ : ZFSet.{u}
⊢ z✝ ∈ ⋃₀ ∅ ↔ z✝ ∈ ∅
|
ab36a3c52f4f0608
|
DFinsupp.mapRange.linearMap_comp
|
Mathlib/LinearAlgebra/DFinsupp.lean
|
theorem mapRange.linearMap_comp (f : ∀ i, β₁ i →ₗ[R] β₂ i) (f₂ : ∀ i, β i →ₗ[R] β₁ i) :
(mapRange.linearMap fun i => (f i).comp (f₂ i)) =
(mapRange.linearMap f).comp (mapRange.linearMap f₂) :=
LinearMap.ext <| mapRange_comp (fun i x => f i x) (fun i x => f₂ i x)
(fun i => (f i).map_zero) (fun i => (f₂ i).map_zero) (by simp)
|
ι : Type u_1
R : Type u_2
inst✝⁶ : Semiring R
β : ι → Type u_6
β₁ : ι → Type u_7
β₂ : ι → Type u_8
inst✝⁵ : (i : ι) → AddCommMonoid (β i)
inst✝⁴ : (i : ι) → AddCommMonoid (β₁ i)
inst✝³ : (i : ι) → AddCommMonoid (β₂ i)
inst✝² : (i : ι) → Module R (β i)
inst✝¹ : (i : ι) → Module R (β₁ i)
inst✝ : (i : ι) → Module R (β₂ i)
f : (i : ι) → β₁ i →ₗ[R] β₂ i
f₂ : (i : ι) → β i →ₗ[R] β₁ i
⊢ ∀ (i : ι), ((fun i x => (f i) x) i ∘ (fun i x => (f₂ i) x) i) 0 = 0
|
simp
|
no goals
|
e578daa14cf2cc8c
|
CategoryTheory.IsHomLift.of_fac'
|
Mathlib/CategoryTheory/FiberedCategory/HomLift.lean
|
lemma of_fac' {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (ha : p.obj a = R) (hb : p.obj b = S)
(h : p.map φ = eqToHom ha ≫ f ≫ eqToHom hb.symm) : p.IsHomLift f φ
|
𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
R S : 𝒮
a b : 𝒳
f : R ⟶ S
φ : a ⟶ b
ha : p.obj a = R
hb : p.obj b = S
h : p.map φ = eqToHom ha ≫ f ≫ eqToHom ⋯
⊢ p.IsHomLift f φ
|
subst ha hb
|
𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
f : p.obj a ⟶ p.obj b
h : p.map φ = eqToHom ⋯ ≫ f ≫ eqToHom ⋯
⊢ p.IsHomLift f φ
|
092cc10978a73530
|
comap_upperCentralSeries
|
Mathlib/GroupTheory/Nilpotent.lean
|
@[simp] lemma comap_upperCentralSeries {H : Type*} [Group H] (e : H ≃* G) :
∀ n, (upperCentralSeries G n).comap e = upperCentralSeries H n
| 0 => by simpa [MonoidHom.ker_eq_bot_iff] using e.injective
| n + 1 => by
ext
simp [mem_upperCentralSeries_succ_iff, ← comap_upperCentralSeries e n,
← e.toEquiv.forall_congr_right]
|
case h
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
e : H ≃* G
n : ℕ
x✝ : H
⊢ x✝ ∈ comap (↑e) (upperCentralSeries G (n + 1)) ↔ x✝ ∈ upperCentralSeries H (n + 1)
|
simp [mem_upperCentralSeries_succ_iff, ← comap_upperCentralSeries e n,
← e.toEquiv.forall_congr_right]
|
no goals
|
0ff1007cdb39a046
|
Algebra.FinitePresentation.mvPolynomial_of_finitePresentation
|
Mathlib/RingTheory/FinitePresentation.lean
|
theorem mvPolynomial_of_finitePresentation [FinitePresentation.{w₁, w₂} R A]
(ι : Type v) [Finite ι] :
FinitePresentation.{w₁, max v w₂} R (MvPolynomial ι A)
|
case intro.intro.intro.intro.intro.refine_1
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A := (MvPolynomial.mapAlgHom f).comp ↑(MvPolynomial.sumAlgEquiv R ι ι')
val✝ : Fintype (ι ⊕ ι')
⊢ ⊥.FG
|
exact Submodule.fg_bot
|
no goals
|
2529be89e207e207
|
Turing.PartrecToTM2.pred_ok
|
Mathlib/Computability/TMToPartrec.lean
|
theorem pred_ok (q₁ q₂ s v) (c d : List Γ') : ∃ s',
Reaches₁ (TM2.step tr) ⟨some (Λ'.pred q₁ q₂), s, K'.elim (trList v) [] c d⟩
(v.headI.rec ⟨some q₁, s', K'.elim (trList v.tail) [] c d⟩ fun n _ =>
⟨some q₂, s', K'.elim (trList (n::v.tail)) [] c d⟩)
|
case nil
q₁ q₂ : Λ'
s : Option Γ'
c d : List Γ'
⊢ ∃ s',
Reaches₁ (TM2.step tr) { l := some (q₁.pred q₂), var := s, stk := elim (trList []) [] c d }
(Nat.rec { l := some q₁, var := s', stk := elim (trList [].tail) [] c d }
(fun n x => { l := some q₂, var := s', stk := elim (trList (n :: [].tail)) [] c d }) [].headI)
|
refine ⟨none, TransGen.single ?_⟩
|
case nil
q₁ q₂ : Λ'
s : Option Γ'
c d : List Γ'
⊢ Nat.rec { l := some q₁, var := none, stk := elim (trList [].tail) [] c d }
(fun n x => { l := some q₂, var := none, stk := elim (trList (n :: [].tail)) [] c d }) [].headI ∈
TM2.step tr { l := some (q₁.pred q₂), var := s, stk := elim (trList []) [] c d }
|
b57c75f21d3c3b50
|
continuous_clm_apply
|
Mathlib/Analysis/Normed/Module/FiniteDimension.lean
|
theorem continuous_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
{f : X → E →L[𝕜] F} : Continuous f ↔ ∀ y, Continuous (f · y)
|
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : FiniteDimensional 𝕜 E
f : X → E →L[𝕜] F
⊢ Continuous f ↔ ∀ (y : E), Continuous fun x => (f x) y
|
simp_rw [continuous_iff_continuousOn_univ, continuousOn_clm_apply]
|
no goals
|
e817dfd1b43b46ee
|
PadicSeq.norm_eq_of_equiv_aux
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem norm_eq_of_equiv_aux {f g : PadicSeq p} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) (hfg : f ≈ g)
(h : padicNorm p (f (stationaryPoint hf)) ≠ padicNorm p (g (stationaryPoint hg)))
(hlt : padicNorm p (g (stationaryPoint hg)) < padicNorm p (f (stationaryPoint hf))) :
False
|
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
padicNorm p (↑g (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg))) <
padicNorm p (↑f (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)))
h :
padicNorm p (↑f (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg))) ≠
padicNorm p (↑g (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)
hi : N ≤ i
hN' :
padicNorm p (↑(f - g) i) <
padicNorm p (↑f (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg))) -
padicNorm p (↑g (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)))
⊢ False
|
have hpne : padicNorm p (f i) ≠ padicNorm p (-g i) := by rwa [← padicNorm.neg (g i)] at h
|
case intro
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt :
padicNorm p (↑g (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg))) <
padicNorm p (↑f (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)))
h :
padicNorm p (↑f (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg))) ≠
padicNorm p (↑g (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)
hi : N ≤ i
hN' :
padicNorm p (↑(f - g) i) <
padicNorm p (↑f (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg))) -
padicNorm p (↑g (N ⊔ (stationaryPoint hf ⊔ stationaryPoint hg)))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
⊢ False
|
e74467e76172943d
|
Turing.ToPartrec.stepRet_eval
|
Mathlib/Computability/TMConfig.lean
|
theorem stepRet_eval {k v} : eval step (stepRet k v) = Cfg.halt <$> k.eval v
|
case fix
f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v : List ℕ
⊢ eval step (if v.headI = 0 then stepRet k v.tail else stepNormal f (Cont.fix f k) v.tail) =
Cfg.halt <$> if v.headI = 0 then k.eval v.tail else f.fix.eval v.tail >>= k.eval
|
split_ifs
|
case pos
f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v : List ℕ
h✝ : v.headI = 0
⊢ eval step (stepRet k v.tail) = Cfg.halt <$> k.eval v.tail
case neg
f : Code
k : Cont
IH : ∀ {v : List ℕ}, eval step (stepRet k v) = Cfg.halt <$> k.eval v
v : List ℕ
h✝ : ¬v.headI = 0
⊢ eval step (stepNormal f (Cont.fix f k) v.tail) = Cfg.halt <$> (f.fix.eval v.tail >>= k.eval)
|
298dd0c162c2a772
|
Array.flatten_toArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem flatten_toArray (l : List (Array α)) :
l.toArray.flatten = (l.map Array.toList).flatten.toArray
|
case h
α : Type u_1
l : List (Array α)
⊢ l.toArray.flatten.toList = (List.map toList l).flatten.toArray.toList
|
simp
|
no goals
|
c003cf217c77b854
|
Submodule.submodule_eq_sSup_le_nonzero_spans
|
Mathlib/LinearAlgebra/Span/Defs.lean
|
theorem submodule_eq_sSup_le_nonzero_spans (p : Submodule R M) :
p = sSup { T : Submodule R M | ∃ m ∈ p, m ≠ 0 ∧ T = span R {m} }
|
case a
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
S : Set (Submodule R M) := {T | ∃ m ∈ p, m ≠ 0 ∧ T = span R {m}}
⊢ sSup {T | ∃ m ∈ p, m ≠ 0 ∧ T = span R {m}} ≤ p
|
rw [sSup_le_iff]
|
case a
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
S : Set (Submodule R M) := {T | ∃ m ∈ p, m ≠ 0 ∧ T = span R {m}}
⊢ ∀ b ∈ {T | ∃ m ∈ p, m ≠ 0 ∧ T = span R {m}}, b ≤ p
|
d695d79774d3d39d
|
Directed.disjoint_iSup_right
|
Mathlib/Order/CompactlyGenerated/Basic.lean
|
theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i)
|
ι : Sort u_1
α : Type u_2
inst✝¹ : CompleteLattice α
f : ι → α
inst✝ : IsCompactlyGenerated α
a : α
h : Directed (fun x1 x2 => x1 ≤ x2) f
⊢ Disjoint a (⨆ i, f i) ↔ ∀ (i : ι), Disjoint a (f i)
|
simp_rw [disjoint_iff, h.inf_iSup_eq, iSup_eq_bot]
|
no goals
|
4f1eb4f5d3709b11
|
Computation.terminates_parallel
|
Mathlib/Data/Seq/Parallel.lean
|
theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : Terminates c] :
Terminates (parallel S)
|
case succ.inr.inr.none
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
corec parallel.aux1 (l, S) =
(corec parallel.aux1
(match S.destruct with
| none => (l', Seq.nil)
| some (none, S') => (l', S')
| some (some c, S') => (c :: l', S'))).think
TT : ∀ (l' : List (Computation α)), (corec parallel.aux1 (l', S.tail)).Terminates
e : S.get? 0 = none
D : S.destruct = none
⊢ (corec parallel.aux1
(match S.destruct with
| none => (l', Seq.nil)
| some (none, S') => (l', S')
| some (some c, S') => (c :: l', S'))).Terminates
|
rw [D]
|
case succ.inr.inr.none
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
S : Stream'.Seq (Option (Computation α))
c : Computation α
T : c.Terminates
a : some (some c) = S.get? (n + 1)
l' : List (Computation α)
h : parallel.aux2 l = Sum.inr l'
C :
corec parallel.aux1 (l, S) =
(corec parallel.aux1
(match S.destruct with
| none => (l', Seq.nil)
| some (none, S') => (l', S')
| some (some c, S') => (c :: l', S'))).think
TT : ∀ (l' : List (Computation α)), (corec parallel.aux1 (l', S.tail)).Terminates
e : S.get? 0 = none
D : S.destruct = none
⊢ (corec parallel.aux1
(match none with
| none => (l', Seq.nil)
| some (none, S') => (l', S')
| some (some c, S') => (c :: l', S'))).Terminates
|
b95c6ab968566afe
|
Finset.mem_image
|
Mathlib/Data/Finset/Image.lean
|
theorem mem_image : b ∈ s.image f ↔ ∃ a ∈ s, f a = b
|
α : Type u_1
β : Type u_2
inst✝ : DecidableEq β
f : α → β
s : Finset α
b : β
⊢ b ∈ image f s ↔ ∃ a ∈ s, f a = b
|
simp only [mem_def, image_val, mem_dedup, Multiset.mem_map, exists_prop]
|
no goals
|
4632438d56323977
|
Subalgebra.SeparatesPoints.rclike_to_real
|
Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
|
theorem Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
(hA : A.SeparatesPoints) :
((A.restrictScalars ℝ).comap
(ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints
|
case h
𝕜 : Type u_1
X : Type u_2
inst✝¹ : RCLike 𝕜
inst✝ : TopologicalSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
x₁ x₂ : X
hx : x₁ ≠ x₂
f : C(X, 𝕜)
hfA : f ∈ ↑A.toSubalgebra
hf : (fun f => ⇑f) f x₁ ≠ (fun f => ⇑f) f x₂
F : C(X, 𝕜) := f - const X (f x₂)
a✝ : X
⊢ (f x₂ • 1) a✝ = (const X (f x₂)) a✝
|
simp only [coe_smul, coe_one, smul_apply, one_apply, Algebra.id.smul_eq_mul, mul_one,
const_apply]
|
no goals
|
bd04a32b97b71309
|
IsSepClosed.exists_root_C_mul_X_pow_add_C_mul_X_add_C
|
Mathlib/FieldTheory/IsSepClosed.lean
|
theorem exists_root_C_mul_X_pow_add_C_mul_X_add_C
[IsSepClosed k] {n : ℕ} (a b c : k) (hn : (n : k) = 0) (hn' : 2 ≤ n) (hb : b ≠ 0) :
∃ x, a * x ^ n + b * x + c = 0
|
k : Type u
inst✝¹ : Field k
inst✝ : IsSepClosed k
n : ℕ
a b c : k
hn : ↑n = 0
hn' : 2 ≤ n
hb : b ≠ 0
f : k[X] := C a * X ^ n + C b * X + C c
hdeg : f.degree ≠ 0
hsep : f.Separable
⊢ ∃ x, a * x ^ n + b * x + c = 0
|
obtain ⟨x, hx⟩ := exists_root f hdeg hsep
|
case intro
k : Type u
inst✝¹ : Field k
inst✝ : IsSepClosed k
n : ℕ
a b c : k
hn : ↑n = 0
hn' : 2 ≤ n
hb : b ≠ 0
f : k[X] := C a * X ^ n + C b * X + C c
hdeg : f.degree ≠ 0
hsep : f.Separable
x : k
hx : f.IsRoot x
⊢ ∃ x, a * x ^ n + b * x + c = 0
|
1f3e118ba5e9f6ba
|
tangentBundle_model_space_coe_chartAt_symm
|
Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
|
theorem tangentBundle_model_space_coe_chartAt_symm (p : TangentBundle I H) :
((chartAt (ModelProd H E) p).symm : ModelProd H E → TangentBundle I H) =
(TotalSpace.toProd H E).symm
|
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_4
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
p : TangentBundle I H
⊢ ↑(TotalSpace.toProd H E).toPartialEquiv.symm = ⇑(TotalSpace.toProd H E).symm
|
rfl
|
no goals
|
8796e6c99e525bf9
|
Set.pi_eq_empty_iff
|
Mathlib/Data/Set/Prod.lean
|
theorem pi_eq_empty_iff : s.pi t = ∅ ↔ ∃ i, IsEmpty (α i) ∨ i ∈ s ∧ t i = ∅
|
ι : Type u_1
α : ι → Type u_2
s : Set ι
t : (i : ι) → Set (α i)
⊢ (¬∀ (i : ι), ∃ x, i ∈ s → x ∈ t i) ↔ ∃ i, IsEmpty (α i) ∨ i ∈ s ∧ t i = ∅
|
push_neg
|
ι : Type u_1
α : ι → Type u_2
s : Set ι
t : (i : ι) → Set (α i)
⊢ (∃ i, ∀ (x : α i), i ∈ s ∧ x ∉ t i) ↔ ∃ i, IsEmpty (α i) ∨ i ∈ s ∧ t i = ∅
|
80c1016a8619f3f9
|
Finset.injOn_of_surjOn_of_card_le
|
Mathlib/Data/Finset/Card.lean
|
lemma injOn_of_surjOn_of_card_le (f : α → β) (hf : Set.MapsTo f s t) (hsurj : Set.SurjOn f s t)
(hst : #s ≤ #t) : Set.InjOn f s
|
α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
f : α → β
hf : Set.MapsTo f ↑s ↑t
hsurj : Set.SurjOn f ↑s ↑t
hst : #s ≤ #t
this✝¹ : image f s = t
this✝ : #(image f s) = #t
this : #(image f s) ≤ #s
⊢ #(image f s) = #s
|
omega
|
no goals
|
64e237fbcbdd78d5
|
padicValRat.lt_sum_of_lt
|
Mathlib/NumberTheory/Padics/PadicVal/Basic.lean
|
theorem lt_sum_of_lt {p j : ℕ} [hp : Fact (Nat.Prime p)] {F : ℕ → ℚ} {S : Finset ℕ}
(hS : S.Nonempty) (hF : ∀ i, i ∈ S → padicValRat p (F j) < padicValRat p (F i))
(hn1 : ∀ i : ℕ, 0 < F i) : padicValRat p (F j) < padicValRat p (∑ i ∈ S, F i)
|
p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
s : ℕ
S' : Finset ℕ
Hnot : s ∉ S'
Hne : S'.Nonempty
Hind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)
hF : ∀ i ∈ Finset.cons s S' Hnot, padicValRat p (F j) < padicValRat p (F i)
i : ℕ
hi : i ∈ S'
⊢ i = s ∨ i ∈ S'
|
exact Or.inr hi
|
no goals
|
f5e32c7b1adff659
|
List.map_orderedInsert
|
Mathlib/Data/List/Sort.lean
|
theorem map_orderedInsert (f : α → β) (l : List α) (x : α)
(hl₁ : ∀ a ∈ l, a ≼ x ↔ f a ≼ f x) (hl₂ : ∀ a ∈ l, x ≼ a ↔ f x ≼ f a) :
(l.orderedInsert r x).map f = (l.map f).orderedInsert s (f x)
|
case cons
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : DecidableRel r
inst✝ : DecidableRel s
f : α → β
x✝ x : α
xs : List α
ih :
(∀ a ∈ xs, r a x✝ ↔ s (f a) (f x✝)) →
(∀ a ∈ xs, r x✝ a ↔ s (f x✝) (f a)) → map f (orderedInsert r x✝ xs) = orderedInsert s (f x✝) (map f xs)
hl₁ : (r x x✝ ↔ s (f x) (f x✝)) ∧ ∀ x ∈ xs, r x x✝ ↔ s (f x) (f x✝)
hl₂ : (r x✝ x ↔ s (f x✝) (f x)) ∧ ∀ x ∈ xs, r x✝ x ↔ s (f x✝) (f x)
⊢ map f (if r x✝ x then x✝ :: x :: xs else x :: orderedInsert r x✝ xs) =
if r x✝ x then f x✝ :: f x :: map f xs else f x :: orderedInsert s (f x✝) (map f xs)
|
split_ifs
|
case pos
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : DecidableRel r
inst✝ : DecidableRel s
f : α → β
x✝ x : α
xs : List α
ih :
(∀ a ∈ xs, r a x✝ ↔ s (f a) (f x✝)) →
(∀ a ∈ xs, r x✝ a ↔ s (f x✝) (f a)) → map f (orderedInsert r x✝ xs) = orderedInsert s (f x✝) (map f xs)
hl₁ : (r x x✝ ↔ s (f x) (f x✝)) ∧ ∀ x ∈ xs, r x x✝ ↔ s (f x) (f x✝)
hl₂ : (r x✝ x ↔ s (f x✝) (f x)) ∧ ∀ x ∈ xs, r x✝ x ↔ s (f x✝) (f x)
h✝ : r x✝ x
⊢ map f (x✝ :: x :: xs) = f x✝ :: f x :: map f xs
case neg
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : DecidableRel r
inst✝ : DecidableRel s
f : α → β
x✝ x : α
xs : List α
ih :
(∀ a ∈ xs, r a x✝ ↔ s (f a) (f x✝)) →
(∀ a ∈ xs, r x✝ a ↔ s (f x✝) (f a)) → map f (orderedInsert r x✝ xs) = orderedInsert s (f x✝) (map f xs)
hl₁ : (r x x✝ ↔ s (f x) (f x✝)) ∧ ∀ x ∈ xs, r x x✝ ↔ s (f x) (f x✝)
hl₂ : (r x✝ x ↔ s (f x✝) (f x)) ∧ ∀ x ∈ xs, r x✝ x ↔ s (f x✝) (f x)
h✝ : ¬r x✝ x
⊢ map f (x :: orderedInsert r x✝ xs) = f x :: orderedInsert s (f x✝) (map f xs)
|
516a054b6e3caddb
|
CategoryTheory.GrothendieckTopology.MayerVietorisSquare.SheafCondition.bijective_toPullbackObj
|
Mathlib/CategoryTheory/Sites/MayerVietorisSquare.lean
|
lemma bijective_toPullbackObj : Function.Bijective (S.toPullbackObj P)
|
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : HasWeakSheafify J (Type v)
S : J.MayerVietorisSquare
P : Cᵒᵖ ⥤ Type v'
h : S.SheafCondition P
⊢ Function.Bijective (S.toPullbackObj P)
|
rwa [← sheafCondition_iff_bijective_toPullbackObj]
|
no goals
|
9619a9c04ac7eded
|
List.Perm.dropSlice_inter
|
Mathlib/Data/List/Perm/Lattice.lean
|
theorem Perm.dropSlice_inter {xs ys : List α} (n m : ℕ) (h : xs ~ ys)
(h' : ys.Nodup) : List.dropSlice n m xs ~ ys ∩ List.dropSlice n m xs
|
α : Type u_1
inst✝ : DecidableEq α
xs ys : List α
n m : ℕ
h : xs ~ ys
h' : ys.Nodup
this : n ≤ n + m
⊢ take n xs ++ drop (n + m) xs ~ ys ∩ (take n xs ++ drop (n + m) xs)
|
have h₂ := h.nodup_iff.2 h'
|
α : Type u_1
inst✝ : DecidableEq α
xs ys : List α
n m : ℕ
h : xs ~ ys
h' : ys.Nodup
this : n ≤ n + m
h₂ : xs.Nodup
⊢ take n xs ++ drop (n + m) xs ~ ys ∩ (take n xs ++ drop (n + m) xs)
|
6eb3e2973d049b65
|
SeminormFamily.basisSets_zero
|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
theorem basisSets_zero (U) (hU : U ∈ p.basisSets) : (0 : E) ∈ U
|
case intro.intro.intro
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
U : Set E
hU✝ : U ∈ p.basisSets
ι' : Finset ι
r : ℝ
hr : 0 < r
hU : U = (ι'.sup p).ball 0 r
⊢ 0 ∈ U
|
rw [hU, mem_ball_zero, map_zero]
|
case intro.intro.intro
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
U : Set E
hU✝ : U ∈ p.basisSets
ι' : Finset ι
r : ℝ
hr : 0 < r
hU : U = (ι'.sup p).ball 0 r
⊢ 0 < r
|
ee3cc5ec86a6eab4
|
AlgebraicGeometry.germ_stalkClosedPointIso_hom
|
Mathlib/AlgebraicGeometry/Stalk.lean
|
@[reassoc (attr := simp)]
lemma germ_stalkClosedPointIso_hom :
(Spec R).presheaf.germ ⊤ (closedPoint _) trivial ≫ (stalkClosedPointIso R).hom =
(Scheme.ΓSpecIso R).hom
|
R : CommRingCat
inst✝ : IsLocalRing ↑R
⊢ (Spec R).presheaf.germ ⊤ (closedPoint ↑R) trivial ≫ (stalkClosedPointIso R).hom = (Scheme.ΓSpecIso R).hom
|
rw [← ΓSpecIso_hom_stalkClosedPointIso_inv, Category.assoc, Iso.inv_hom_id, Category.comp_id]
|
no goals
|
206b085eee6546c7
|
AddMonoidAlgebra.mul_of'_modOf
|
Mathlib/Algebra/MonoidAlgebra/Division.lean
|
theorem mul_of'_modOf (x : k[G]) (g : G) : x * of' k G g %ᵒᶠ g = 0
|
case H
k : Type u_1
G : Type u_2
inst✝¹ : Semiring k
inst✝ : AddCancelCommMonoid G
x : k[G]
g g' : G
⊢ (x * of' k G g %ᵒᶠ g) g' = 0
|
obtain ⟨d, rfl⟩ | h := em (∃ d, g' = g + d)
|
case H.inl.intro
k : Type u_1
G : Type u_2
inst✝¹ : Semiring k
inst✝ : AddCancelCommMonoid G
x : k[G]
g d : G
⊢ (x * of' k G g %ᵒᶠ g) (g + d) = 0
case H.inr
k : Type u_1
G : Type u_2
inst✝¹ : Semiring k
inst✝ : AddCancelCommMonoid G
x : k[G]
g g' : G
h : ¬∃ d, g' = g + d
⊢ (x * of' k G g %ᵒᶠ g) g' = 0
|
746a91a884eb3ea6
|
Matrix.conjTranspose_mul_self_mul_eq_zero
|
Mathlib/LinearAlgebra/Matrix/DotProduct.lean
|
lemma conjTranspose_mul_self_mul_eq_zero {p} (A : Matrix m n R) (B : Matrix n p R) :
(Aᴴ * A) * B = 0 ↔ A * B = 0
|
m : Type u_1
n : Type u_2
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : PartialOrder R
inst✝³ : NonUnitalRing R
inst✝² : StarRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
p : Type u_5
A : Matrix m n R
B : Matrix n p R
h : Bᴴ * (Aᴴ * A * B) = Bᴴ * 0
⊢ A * B = 0
|
rwa [Matrix.mul_zero, Matrix.mul_assoc, ← Matrix.mul_assoc, ← conjTranspose_mul,
conjTranspose_mul_self_eq_zero] at h
|
no goals
|
4f2a91332cdad436
|
ConvexOn.lipschitzOnWith_of_abs_le
|
Mathlib/Analysis/Convex/Continuous.lean
|
lemma ConvexOn.lipschitzOnWith_of_abs_le (hf : ConvexOn ℝ (ball x₀ r) f) (hε : 0 < ε)
(hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
a : ℝ := ε / (ε + ‖x - y‖)
b : ℝ := ‖x - y‖ / (ε + ‖x - y‖)
hab : a + b = 1
hxyz : x = a • y + b • z
⊢ 0 ≤ b
|
positivity
|
no goals
|
f794e80453a6298e
|
isLocalMax_of_deriv_Ioo
|
Mathlib/Analysis/Calculus/FirstDerivativeTest.lean
|
/-- The First-Derivative Test from calculus, maxima version.
Suppose `a < b < c`, `f : ℝ → ℝ` is continuous at `b`,
the derivative `f'` is nonnegative on `(a,b)`, and
the derivative `f'` is nonpositive on `(b,c)`. Then `f` has a local maximum at `a`. -/
lemma isLocalMax_of_deriv_Ioo {f : ℝ → ℝ} {a b c : ℝ} (g₀ : a < b) (g₁ : b < c)
(h : ContinuousAt f b)
(hd₀ : DifferentiableOn ℝ f (Ioo a b))
(hd₁ : DifferentiableOn ℝ f (Ioo b c))
(h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x)
(h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0) : IsLocalMax f b :=
have hIoc : ContinuousOn f (Ioc a b) :=
Ioo_union_right g₀ ▸ hd₀.continuousOn.union_continuousAt (isOpen_Ioo (a := a) (b := b))
(by simp_all)
have hIco : ContinuousOn f (Ico b c) :=
Ioo_union_left g₁ ▸ hd₁.continuousOn.union_continuousAt (isOpen_Ioo (a := b) (b := c))
(by simp_all)
isLocalMax_of_mono_anti g₀ g₁
(monotoneOn_of_deriv_nonneg (convex_Ioc a b) hIoc (by simp_all) (by simp_all))
(antitoneOn_of_deriv_nonpos (convex_Ico b c) hIco (by simp_all) (by simp_all))
|
f : ℝ → ℝ
a b c : ℝ
g₀ : a < b
g₁ : b < c
h : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
hIoc : ContinuousOn f (Ioc a b)
hIco : ContinuousOn f (Ico b c)
⊢ DifferentiableOn ℝ f (interior (Ioc a b))
|
simp_all
|
no goals
|
8e32ca355c9dee01
|
TopologicalSpace.nhds_generateFrom
|
Mathlib/Topology/Order.lean
|
theorem nhds_generateFrom {g : Set (Set α)} {a : α} :
@nhds α (generateFrom g) a = ⨅ s ∈ { s | a ∈ s ∧ s ∈ g }, 𝓟 s
|
α : Type u
g : Set (Set α)
a : α
this : TopologicalSpace α := generateFrom g
⊢ 𝓝 a = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s
|
rw [nhds_def]
|
α : Type u
g : Set (Set α)
a : α
this : TopologicalSpace α := generateFrom g
⊢ ⨅ s ∈ {s | a ∈ s ∧ IsOpen s}, 𝓟 s = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s
|
658fd288ee295c79
|
MeasureTheory.setLIntegral_tilted
|
Mathlib/MeasureTheory/Measure/Tilted.lean
|
lemma setLIntegral_tilted [SFinite μ] (f : α → ℝ) (g : α → ℝ≥0∞) (s : Set α) :
∫⁻ x in s, g x ∂(μ.tilted f)
= ∫⁻ x in s, ENNReal.ofReal (exp (f x) / ∫ x, exp (f x) ∂μ) * g x ∂μ
|
case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α) in s, ((fun x => ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) * g) a ∂μ =
∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ
|
simp only [Pi.mul_apply]
|
no goals
|
2d68f42b681337c9
|
AlgebraicGeometry.isIntegral_of_isOpenImmersion
|
Mathlib/AlgebraicGeometry/Properties.lean
|
theorem isIntegral_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[IsIntegral Y] [Nonempty X] : IsIntegral X
|
X Y : Scheme
f : X ⟶ Y
H : IsOpenImmersion f
inst✝¹ : IsIntegral Y
inst✝ : Nonempty ↑↑X.toPresheafedSpace
⊢ IsIntegral X
|
constructor
|
case nonempty
X Y : Scheme
f : X ⟶ Y
H : IsOpenImmersion f
inst✝¹ : IsIntegral Y
inst✝ : Nonempty ↑↑X.toPresheafedSpace
⊢ autoParam (Nonempty ↑↑X.toPresheafedSpace) _auto✝
case component_integral
X Y : Scheme
f : X ⟶ Y
H : IsOpenImmersion f
inst✝¹ : IsIntegral Y
inst✝ : Nonempty ↑↑X.toPresheafedSpace
⊢ autoParam (∀ (U : X.Opens) [inst : Nonempty ↑↑(↑U).toPresheafedSpace], IsDomain ↑Γ(X, U)) _auto✝
|
d1ee1e1bef31e6c7
|
Set.exists_superset_subset_encard_eq
|
Mathlib/Data/Set/Card.lean
|
theorem exists_superset_subset_encard_eq {k : ℕ∞}
(hst : s ⊆ t) (hsk : s.encard ≤ k) (hkt : k ≤ t.encard) :
∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k
|
case inr.intro.intro.intro.intro
α : Type u_1
s t : Set α
hst : s ⊆ t
hs : s.encard ≠ ⊤
k' : ℕ∞
r' : Set α
hr' : r' ⊆ t \ s
hsk : s.encard ≤ s.encard + r'.encard
hkt : s.encard + r'.encard ≤ t.encard
hk' : t.encard = s.encard + r'.encard + k'
hk : r'.encard ≤ (t \ s).encard
⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = s.encard + r'.encard
|
refine ⟨s ∪ r', subset_union_left, union_subset hst (hr'.trans diff_subset), ?_⟩
|
case inr.intro.intro.intro.intro
α : Type u_1
s t : Set α
hst : s ⊆ t
hs : s.encard ≠ ⊤
k' : ℕ∞
r' : Set α
hr' : r' ⊆ t \ s
hsk : s.encard ≤ s.encard + r'.encard
hkt : s.encard + r'.encard ≤ t.encard
hk' : t.encard = s.encard + r'.encard + k'
hk : r'.encard ≤ (t \ s).encard
⊢ (s ∪ r').encard = s.encard + r'.encard
|
1897db25f460fe99
|
List.dropWhile_filter
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean
|
theorem dropWhile_filter (p q : α → Bool) (l : List α) :
(l.filter p).dropWhile q = (l.dropWhile fun a => !p a || q a).filter p
|
α : Type u_1
p q : α → Bool
l : List α
⊢ dropWhile q (filter p l) = filter p (dropWhile (fun a => !p a || q a) l)
|
simp [← filterMap_eq_filter, dropWhile_filterMap]
|
no goals
|
0c44eef09025b4de
|
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 h.e'_9.e_a.e_a
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x
hm : m < 0
hx : x ≠ 0
this : HasStrictDerivAt (fun x => x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x
⊢ m - 1 = -m - 1 + (m + m)
|
abel
|
no goals
|
c222b9aa15688dc1
|
LowerSet.sdiff_sup_lowerClosure
|
Mathlib/Order/UpperLower/Basic.lean
|
lemma sdiff_sup_lowerClosure (hts : t ⊆ s) (hst : ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t) :
s.sdiff t ⊔ lowerClosure t = s
|
case inr
α : Type u_1
inst✝ : Preorder α
s : LowerSet α
t : Set α
hts : t ⊆ ↑s
hst : ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t
a : α
ha : a ∈ ↑s
hat : a ∉ t
⊢ a ∉ ↑(upperClosure t)
|
rintro ⟨b, hb, hba⟩
|
case inr.intro.intro
α : Type u_1
inst✝ : Preorder α
s : LowerSet α
t : Set α
hts : t ⊆ ↑s
hst : ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t
a : α
ha : a ∈ ↑s
hat : a ∉ t
b : α
hb : b ∈ t
hba : b ≤ a
⊢ False
|
727f3e5c7c505993
|
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_2
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
i j : J
f : i ⟶ j
V : Set ↑((F ⋙ toTopCat).obj j)
hV : IsClopen V
⊢ ⇑(ConcreteCategory.hom ((F ⋙ toTopCat).map f)) ⁻¹' V ∈ (fun j => {W | IsClopen W}) i
|
exact ⟨hV.1.preimage ((F ⋙ toTopCat).map f).hom.continuous,
hV.2.preimage ((F ⋙ toTopCat).map f).hom.continuous⟩
|
no goals
|
0fb8cdb8a6309b67
|
Set.MapsTo.restrict_inj
|
Mathlib/Data/Set/Function.lean
|
theorem MapsTo.restrict_inj (h : MapsTo f s t) : Injective (h.restrict f s t) ↔ InjOn f s
|
α : Type u_1
β : Type u_2
s : Set α
t : Set β
f : α → β
h : MapsTo f s t
⊢ Injective (restrict f s t h) ↔ InjOn f s
|
rw [h.restrict_eq_codRestrict, injective_codRestrict, injOn_iff_injective]
|
no goals
|
816cd1014af11570
|
CategoryTheory.eHom_whisker_cancel
|
Mathlib/CategoryTheory/Enriched/Ordinary/Basic.lean
|
/-- Given an isomorphism `α : Y ≅ Y₁` in C, the enriched composition map
`eComp V X Y Z : (X ⟶[V] Y) ⊗ (Y ⟶[V] Z) ⟶ (X ⟶[V] Z)` factors through the `V`
object `(X ⟶[V] Y₁) ⊗ (Y₁ ⟶[V] Z)` via the map defined by whiskering in the
middle with `α.hom` and `α.inv`. -/
@[reassoc]
lemma eHom_whisker_cancel {X Y Y₁ Z : C} (α : Y ≅ Y₁) :
eHomWhiskerLeft V X α.hom ▷ _ ≫ _ ◁ eHomWhiskerRight V α.inv Z ≫
eComp V X Y₁ Z = eComp V X Y Z
|
V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
X Y Y₁ Z : C
α : Y ≅ Y₁
⊢ (((ρ_ (EnrichedCategory.Hom X Y)).inv ≫ EnrichedCategory.Hom X Y ◁ (eHomEquiv V) α.hom ≫ eComp V X Y Y₁) ≫
(ρ_ (EnrichedCategory.Hom X Y₁)).inv ≫ EnrichedCategory.Hom X Y₁ ◁ (eHomEquiv V) α.inv ≫ eComp V X Y₁ Y) ▷
EnrichedCategory.Hom Y Z ≫
eComp V X Y Z =
eComp V X Y Z
|
change (eHomWhiskerLeft V X α.hom ≫ eHomWhiskerLeft V X α.inv) ▷ _ ≫ _ = _
|
V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
X Y Y₁ Z : C
α : Y ≅ Y₁
⊢ (eHomWhiskerLeft V X α.hom ≫ eHomWhiskerLeft V X α.inv) ▷ EnrichedCategory.Hom Y Z ≫ eComp V X Y Z = eComp V X Y Z
|
651af7bfc1e40d25
|
aemeasurable_add_measure_iff
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
theorem _root_.aemeasurable_add_measure_iff :
AEMeasurable f (μ + ν) ↔ AEMeasurable f μ ∧ AEMeasurable f ν
|
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ ν : Measure α
⊢ AEMeasurable f (μ + ν) ↔ AEMeasurable f μ ∧ AEMeasurable f ν
|
rw [← sum_cond, aemeasurable_sum_measure_iff, Bool.forall_bool, and_comm]
|
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
f : α → β
μ ν : Measure α
⊢ AEMeasurable f (bif true then μ else ν) ∧ AEMeasurable f (bif false then μ else ν) ↔
AEMeasurable f μ ∧ AEMeasurable f ν
|
a88eb6032f3277f5
|
Measurable.piecewise
|
Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
|
theorem Measurable.piecewise {_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s)
(hf : Measurable f) (hg : Measurable g) : Measurable (piecewise s f g)
|
α : Type u_1
β : Type u_2
s : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
x✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s
hf : Measurable f
hg : Measurable g
t : Set β
ht : MeasurableSet t
⊢ MeasurableSet (s.ite (f ⁻¹' t) (g ⁻¹' t))
|
exact hs.ite (hf ht) (hg ht)
|
no goals
|
8f6f4e30f872c5e0
|
Part.mem_chain_of_mem_ωSup
|
Mathlib/Order/OmegaCompletePartialOrder.lean
|
theorem mem_chain_of_mem_ωSup {c : Chain (Part α)} {a : α} (h : a ∈ Part.ωSup c) : some a ∈ c
|
α : Type u_2
c : Chain (Part α)
a : α
h : a ∈ Part.ωSup c
⊢ some a ∈ c
|
simp only [Part.ωSup] at h
|
α : Type u_2
c : Chain (Part α)
a : α
h : a ∈ if h : ∃ a, some a ∈ c then some (Classical.choose h) else none
⊢ some a ∈ c
|
3ffe6431a6dc63d9
|
MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn
|
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
(hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
(∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ)
|
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ v ∈ range u
|
rw [← hu]
|
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
x : X
hx : x ∈ s
v : Set X
hv : v ∈ T
h'v : x ∈ v
⊢ v ∈ T'
|
7d50e417c306b9b4
|
ProbabilityTheory.gaussianReal_const_mul
|
Mathlib/Probability/Distributions/Gaussian.lean
|
/-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v)
|
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
⊢ Measure.map (fun ω => c * X ω) ℙ = gaussianReal (c * μ) (⟨c ^ 2, ⋯⟩ * v)
|
have hXm : AEMeasurable X := aemeasurable_of_map_neZero (by rw [hX]; infer_instance)
|
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
hXm : AEMeasurable X ℙ
⊢ Measure.map (fun ω => c * X ω) ℙ = gaussianReal (c * μ) (⟨c ^ 2, ⋯⟩ * v)
|
a890cf9da6070524
|
CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_inv_app_unop
|
Mathlib/CategoryTheory/Triangulated/Opposite/Functor.lean
|
@[reassoc]
lemma map_opShiftFunctorEquivalence_unitIso_inv_app_unop (X : Cᵒᵖ) (n : ℤ) :
F.map ((opShiftFunctorEquivalence C n).unitIso.inv.app X).unop =
((opShiftFunctorEquivalence D n).unitIso.inv.app (op (F.obj X.unop))).unop ≫
(((F.op).commShiftIso n).hom.app X).unop⟦n⟧' ≫
((F.commShiftIso n).inv.app _)
|
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_3, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n : ℤ
⊢ F.map (𝟙 ((𝟭 Cᵒᵖ).obj X)).unop =
((opShiftFunctorEquivalence D n).unitIso.inv.app (op (F.obj (unop X)))).unop ≫
(shiftFunctor D n).map ((F.op.commShiftIso n).inv.app X ≫ (F.op.commShiftIso n).hom.app X).unop ≫
((opShiftFunctorEquivalence D n).unitIso.hom.app (op (F.obj (unop X)))).unop
|
simp
|
no goals
|
37d63bf51b1caab7
|
Order.Iic_subset_Iio_succ
|
Mathlib/Order/SuccPred/Basic.lean
|
theorem Iic_subset_Iio_succ (a : α) : Iic a ⊆ Iio (succ a)
|
α : Type u_1
inst✝² : Preorder α
inst✝¹ : SuccOrder α
inst✝ : NoMaxOrder α
a : α
⊢ Iic a ⊆ Iio (succ a)
|
simp
|
no goals
|
347e573a04a53ef2
|
LinearMap.BilinForm.toQuadraticMap_isOrtho
|
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
theorem _root_.LinearMap.BilinForm.toQuadraticMap_isOrtho [IsCancelAdd R]
[NoZeroDivisors R] [CharZero R] {B : BilinMap R M R} {x y : M} (h : B.IsSymm) :
B.toQuadraticMap.IsOrtho x y ↔ B.IsOrtho x y
|
R : Type u_3
M : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : IsCancelAdd R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : BilinMap R M R
x y : M
h : LinearMap.IsSymm B
this : AddCancelMonoid R :=
let __src := inst✝²;
let __src_1 := inferInstanceAs (AddCommMonoid R);
AddCancelMonoid.mk ⋯
⊢ B.toQuadraticMap.IsOrtho x y ↔ LinearMap.IsOrtho B x y
|
simp_rw [isOrtho_def, LinearMap.isOrtho_def, B.toQuadraticMap_apply, map_add,
LinearMap.add_apply, add_comm _ (B y y), add_add_add_comm _ _ (B y y), add_comm (B y y)]
|
R : Type u_3
M : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : IsCancelAdd R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : BilinMap R M R
x y : M
h : LinearMap.IsSymm B
this : AddCancelMonoid R :=
let __src := inst✝²;
let __src_1 := inferInstanceAs (AddCommMonoid R);
AddCancelMonoid.mk ⋯
⊢ (B x) x + (B y) y + ((B y) x + (B x) y) = (B x) x + (B y) y ↔ (B x) y = 0
|
3f0621dd0d52a0f6
|
BitVec.ofInt_mul
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem ofInt_mul {n} (x y : Int) : BitVec.ofInt n (x * y) =
BitVec.ofInt n x * BitVec.ofInt n y
|
case a
n : Nat
x y : Int
⊢ (BitVec.ofInt n (x * y)).toInt = (BitVec.ofInt n x * BitVec.ofInt n y).toInt
|
simp
|
no goals
|
4a695bf3af345d85
|
CategoryTheory.GrothendieckTopology.Plus.exists_of_sep
|
Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
|
theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
(hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y)
(X : C) (S : J.Cover X) (s : Meq (J.plusObj P) S) :
∃ t : ToType ((J.plusObj P).obj (op X)), Meq.mk S t = s
|
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ (I : S.Arrow), (ConcreteCategory.hom (P.map I.f.op)) x = (ConcreteCategory.hom (P.map I.f.op)) y) → x = y
X : C
S : J.Cover X
s : Meq (J.plusObj P) S
⊢ ∃ t, Meq.mk S t = s
|
have inj : ∀ X : C, Function.Injective ((J.toPlus P).app (op X)) := inj_of_sep _ hsep
|
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝⁴ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep :
∀ (X : C) (S : J.Cover X) (x y : ToType (P.obj (op X))),
(∀ (I : S.Arrow), (ConcreteCategory.hom (P.map I.f.op)) x = (ConcreteCategory.hom (P.map I.f.op)) y) → x = y
X : C
S : J.Cover X
s : Meq (J.plusObj P) S
inj : ∀ (X : C), Function.Injective ⇑(ConcreteCategory.hom ((J.toPlus P).app (op X)))
⊢ ∃ t, Meq.mk S t = s
|
adc5a5926d46b74e
|
Matroid.closure_insert_closure_eq_closure_insert
|
Mathlib/Data/Matroid/Closure.lean
|
@[simp] lemma closure_insert_closure_eq_closure_insert (M : Matroid α) (e : α) (X : Set α) :
M.closure (insert e (M.closure X)) = M.closure (insert e X)
|
α : Type u_2
M : Matroid α
e : α
X : Set α
⊢ M.closure (insert e (M.closure X)) = M.closure (insert e X)
|
simp_rw [← singleton_union, closure_union_closure_right_eq]
|
no goals
|
4c9deaa38576e877
|
CategoryTheory.FintypeCat.Action.pretransitive_of_isConnected
|
Mathlib/CategoryTheory/Galois/Examples.lean
|
theorem Action.pretransitive_of_isConnected (X : Action FintypeCat G)
[IsConnected X] : MulAction.IsPretransitive G X.V where
exists_smul_eq x y
|
case h
G : Type u
inst✝¹ : Group G
X : Action FintypeCat G
inst✝ : IsConnected X
x y : X.V.carrier
T : Set X.V.carrier := MulAction.orbit G x
this✝² : Fintype ↑T
this✝¹ : MulAction G (FintypeCat.of ↑T).carrier := inferInstanceAs (MulAction G ↑(MulAction.orbit G x))
T' : Action FintypeCat G := Action.FintypeCat.ofMulAction G (FintypeCat.of ↑T)
i : T' ⟶ X := { hom := Subtype.val, comm := ⋯ }
this✝ : Mono i
this : IsIso i
hb : Function.Bijective i.hom
y' : X.V.carrier
g : G
hg : g • x = y'
hy' : y' = y
⊢ g • x = y
|
exact hg.trans hy'
|
no goals
|
b8f6bd012ef672f6
|
SimpleGraph.Walk.nodup_tail_support_reverse
|
Mathlib/Combinatorics/SimpleGraph/Walk.lean
|
theorem nodup_tail_support_reverse {u : V} {p : G.Walk u u} :
p.reverse.support.tail.Nodup ↔ p.support.tail.Nodup
|
V : Type u
G : SimpleGraph V
u : V
p : G.Walk u u
⊢ 0 ≤ p.length
|
omega
|
no goals
|
afd639d7e42502db
|
Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
|
Mathlib/Order/Interval/Set/Basic.lean
|
theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) :
s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α))
|
case pos
α : Type u_1
inst✝ : PartialOrder α
a b : α
s : Set α
ho : Ioo a b ⊆ s
hc : s ⊆ Icc a b
ha : a ∉ s
hb : b ∈ s
⊢ s ∈ {Icc a b, Ico a b, Ioc a b, Ioo a b}
|
refine Or.inr <| Or.inr <| Or.inl <| Subset.antisymm ?_ ?_
|
case pos.refine_1
α : Type u_1
inst✝ : PartialOrder α
a b : α
s : Set α
ho : Ioo a b ⊆ s
hc : s ⊆ Icc a b
ha : a ∉ s
hb : b ∈ s
⊢ s ⊆ Ioc a b
case pos.refine_2
α : Type u_1
inst✝ : PartialOrder α
a b : α
s : Set α
ho : Ioo a b ⊆ s
hc : s ⊆ Icc a b
ha : a ∉ s
hb : b ∈ s
⊢ Ioc a b ⊆ s
|
fc159751b675bff0
|
Cardinal.aleph0_le
|
Mathlib/SetTheory/Cardinal/Basic.lean
|
theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
⟨fun h _ => (nat_lt_aleph0 _).le.trans h, fun h =>
le_of_not_lt fun hn => by
rcases lt_aleph0.1 hn with ⟨n, rfl⟩
exact (Nat.lt_succ_self _).not_le (Nat.cast_le.1 (h (n + 1)))⟩
|
case intro
n : ℕ
h : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n
hn : ↑n < ℵ₀
⊢ False
|
exact (Nat.lt_succ_self _).not_le (Nat.cast_le.1 (h (n + 1)))
|
no goals
|
8a8d523c3adcb040
|
minpoly.aeval
|
Mathlib/FieldTheory/Minpoly/Basic.lean
|
theorem aeval : aeval x (minpoly A x) = 0
|
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
⊢ (Polynomial.aeval x)
(if hx : IsIntegral A x then ⋯.min (fun x_1 => x_1.Monic ∧ eval₂ (algebraMap A B) x x_1 = 0) hx else 0) =
0
|
split_ifs with hx
|
case pos
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
hx : IsIntegral A x
⊢ (Polynomial.aeval x) (⋯.min (fun x_1 => x_1.Monic ∧ eval₂ (algebraMap A B) x x_1 = 0) hx) = 0
case neg
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
hx : ¬IsIntegral A x
⊢ (Polynomial.aeval x) 0 = 0
|
b712d46c9cd74d4a
|
Convex.locPathConnectedSpace
|
Mathlib/Topology/Algebra/Module/LocallyConvex.lean
|
theorem Convex.locPathConnectedSpace [Module ℝ E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E]
{S : Set E} (hS : Convex ℝ S) : LocPathConnectedSpace S
|
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
S : Set E
hS : Convex ℝ S
x : ↑S
s : Set ↑S
hs : s ∈ 𝓝 x
t : Set E
ht : t ∈ 𝓝 ↑x ∧ Subtype.val ⁻¹' t ⊆ s
t' : Set E
ht' : (t' ∈ 𝓝 ↑x ∧ Convex ℝ t') ∧ id t' ⊆ t
⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i) ∧ id i ⊆ s
|
refine ⟨(↑) ⁻¹' t', ⟨?_, ?_⟩, (preimage_mono ht'.2).trans ht.2⟩
|
case refine_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
S : Set E
hS : Convex ℝ S
x : ↑S
s : Set ↑S
hs : s ∈ 𝓝 x
t : Set E
ht : t ∈ 𝓝 ↑x ∧ Subtype.val ⁻¹' t ⊆ s
t' : Set E
ht' : (t' ∈ 𝓝 ↑x ∧ Convex ℝ t') ∧ id t' ⊆ t
⊢ Subtype.val ⁻¹' t' ∈ 𝓝 x
case refine_2
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
S : Set E
hS : Convex ℝ S
x : ↑S
s : Set ↑S
hs : s ∈ 𝓝 x
t : Set E
ht : t ∈ 𝓝 ↑x ∧ Subtype.val ⁻¹' t ⊆ s
t' : Set E
ht' : (t' ∈ 𝓝 ↑x ∧ Convex ℝ t') ∧ id t' ⊆ t
⊢ IsPathConnected (Subtype.val ⁻¹' t')
|
78fdc754cd76e5d0
|
isLocalStructomorphOn_contDiffGroupoid_iff
|
Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean
|
theorem isLocalStructomorphOn_contDiffGroupoid_iff (f : PartialHomeomorph M M') :
LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt f f.source ↔
ContMDiffOn I I n f f.source ∧ ContMDiffOn I I n f.symm f.target
|
case mpr
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : WithTop ℕ∞
inst✝² : IsManifold I n M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : PartialHomeomorph M M'
⊢ ContMDiffOn I I n (↑f) f.source ∧ ContMDiffOn I I n (↑f.symm) f.target →
LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
|
rintro ⟨h₁, h₂⟩ x hx
|
case mpr.intro
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : WithTop ℕ∞
inst✝² : IsManifold I n M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : PartialHomeomorph M M'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
⊢ LiftPropWithinAt (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source x
|
52ce8090c899c270
|
Bool.not_ite_eq_true_eq_false
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Bool.lean
|
theorem not_ite_eq_true_eq_false {p : Prop} [h : Decidable p] {b c : Bool} :
¬(ite p (b = true) (c = false)) ↔ (ite p (b = false) (c = true))
|
case isTrue
p✝ : Prop
b c : Bool
p : p✝
⊢ (¬if p✝ then b = true else c = false) ↔ if p✝ then b = false else c = true
|
simp [p]
|
no goals
|
c73c046739a7ec39
|
CategoryTheory.Functor.prod'_μ_snd
|
Mathlib/CategoryTheory/Monoidal/Functor.lean
|
@[simp] lemma prod'_μ_snd (X Y : C) : (μ (prod' F G) X Y).2 = μ G X Y
|
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : MonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : MonoidalCategory E
F : C ⥤ D
G : C ⥤ E
inst✝¹ : F.LaxMonoidal
inst✝ : G.LaxMonoidal
X Y : C
⊢ (μ (F.prod' G) X Y).2 = μ G X Y
|
change _ ≫ G.map (𝟙 _) = _
|
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
inst✝⁶ : MonoidalCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
inst✝² : MonoidalCategory E
F : C ⥤ D
G : C ⥤ E
inst✝¹ : F.LaxMonoidal
inst✝ : G.LaxMonoidal
X Y : C
⊢ (μ (F.prod G) ((diag C).obj X) ((diag C).obj Y)).2 ≫ G.map (𝟙 ((diag C).obj X ⊗ (diag C).obj Y).2) = μ G X Y
|
fbdf2f8e1e1d9731
|
LieDerivation.ad_ker_eq_center
|
Mathlib/Algebra/Lie/Derivation/AdjointAction.lean
|
/-- The kernel of the adjoint action on a Lie algebra is equal to its center. -/
lemma ad_ker_eq_center : (ad R L).ker = LieAlgebra.center R L
|
case h
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x : L
⊢ x ∈ (ad R L).ker ↔ x ∈ LieAlgebra.center R L
|
rw [← LieAlgebra.self_module_ker_eq_center, LieHom.mem_ker, LieModule.mem_ker]
|
case h
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x : L
⊢ (ad R L) x = 0 ↔ ∀ (m : L), ⁅x, m⁆ = 0
|
b20ce324fe5b3354
|
seminormFromBounded_of_mul_le
|
Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean
|
theorem seminormFromBounded_of_mul_le (f_nonneg : 0 ≤ f) {x : R}
(hx : ∀ y : R, f (x * y) ≤ f x * f y) (h_one : f 1 ≤ 1) : seminormFromBounded' f x = f x
|
case h.intro
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
⊢ (fun y => f (x * y) / f y) y ≤ f x
|
by_cases hy0 : f y = 0
|
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
hy0 : f y = 0
⊢ (fun y => f (x * y) / f y) y ≤ f x
case neg
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
hy0 : ¬f y = 0
⊢ (fun y => f (x * y) / f y) y ≤ f x
|
76683e2ec4692258
|
FormalMultilinearSeries.comp_id
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) (x : E) : p.comp (id 𝕜 E x) = p
|
case h.h₀.intro.intro.intro.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
x : E
n : ℕ
b : Composition n
a✝ : b ∈ Finset.univ
hb : b ≠ Composition.ones n
k : ℕ
hk : k ∈ b.blocks
lt_k : 1 < k
i : Fin b.blocks.length
hi : b.blocks[i] = k
j : Fin b.length := ⟨↑i, ⋯⟩
A : 1 < b.blocksFun j
v : Fin n → E
⊢ (id 𝕜 E x (b.blocksFun j)) (v ∘ ⇑(b.embedding j)) = 0
|
rw [id_apply_of_one_lt _ _ _ A, ContinuousMultilinearMap.zero_apply]
|
no goals
|
03268d569a4e7489
|
StarConvex.affine_preimage
|
Mathlib/Analysis/Convex/Star.lean
|
theorem StarConvex.affine_preimage (f : E →ᵃ[𝕜] F) {s : Set F} (hs : StarConvex 𝕜 (f x) s) :
StarConvex 𝕜 x (f ⁻¹' s)
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x : E
f : E →ᵃ[𝕜] F
s : Set F
hs : StarConvex 𝕜 (f x) s
y : E
hy : y ∈ ⇑f ⁻¹' s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x + b • f y ∈ s
|
exact hs hy ha hb hab
|
no goals
|
705edbda48559109
|
Vector.mapIdx_setIfInBounds
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/MapIdx.lean
|
theorem mapIdx_setIfInBounds {l : Vector α n} {i : Nat} {a : α} :
(l.setIfInBounds i a).mapIdx f = (l.mapIdx f).setIfInBounds i (f i a)
|
case mk
α : Type u_1
α✝ : Type u_2
f : Nat → α → α✝
i : Nat
a : α
l : Array α
⊢ mapIdx f ({ toArray := l, size_toArray := ⋯ }.setIfInBounds i a) =
(mapIdx f { toArray := l, size_toArray := ⋯ }).setIfInBounds i (f i a)
|
simp
|
no goals
|
15859b165dc8bfc9
|
InformationTheory.mul_log_le_toReal_klDiv
|
Mathlib/InformationTheory/KullbackLeibler/Basic.lean
|
lemma mul_log_le_toReal_klDiv (hμν : μ ≪ ν) (h_int : Integrable (llr μ ν) μ) :
(μ univ).toReal * log ((μ univ).toReal / (ν univ).toReal) + (ν univ).toReal - (μ univ).toReal
≤ (klDiv μ ν).toReal
|
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hμ : ¬μ = 0
hν : ¬ν = 0
⊢ (μ univ).toReal * log ((μ univ).toReal / (ν univ).toReal) + (ν univ).toReal - (μ univ).toReal ≤ (klDiv μ ν).toReal
|
refine (le_of_eq ?_).trans (mul_klFun_le_toReal_klDiv hμν h_int)
|
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
hμ : ¬μ = 0
hν : ¬ν = 0
⊢ (μ univ).toReal * log ((μ univ).toReal / (ν univ).toReal) + (ν univ).toReal - (μ univ).toReal =
(ν univ).toReal * klFun ((μ univ).toReal / (ν univ).toReal)
|
66371ee4c0033ef9
|
Wbtw.trans_left_right
|
Mathlib/Analysis/Convex/Between.lean
|
theorem Wbtw.trans_left_right {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R w x y) :
Wbtw R x y z
|
case intro.intro.intro.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
w z : P
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
ht₂ : t₂ ∈ Set.Icc 0 1
⊢ ((t₁ - t₂ * t₁) * ((1 - t₂ * t₁) / (1 - t₂ * t₁)) + t₂ * t₁) • (z -ᵥ w) = t₁ • (z -ᵥ w)
|
by_cases h : 1 - t₂ * t₁ = 0
|
case pos
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
w z : P
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
ht₂ : t₂ ∈ Set.Icc 0 1
h : 1 - t₂ * t₁ = 0
⊢ ((t₁ - t₂ * t₁) * ((1 - t₂ * t₁) / (1 - t₂ * t₁)) + t₂ * t₁) • (z -ᵥ w) = t₁ • (z -ᵥ w)
case neg
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
w z : P
t₁ : R
ht₁ : t₁ ∈ Set.Icc 0 1
t₂ : R
ht₂ : t₂ ∈ Set.Icc 0 1
h : ¬1 - t₂ * t₁ = 0
⊢ ((t₁ - t₂ * t₁) * ((1 - t₂ * t₁) / (1 - t₂ * t₁)) + t₂ * t₁) • (z -ᵥ w) = t₁ • (z -ᵥ w)
|
a064a78f0fcc4db8
|
Matrix.cramer_eq_adjugate_mulVec
|
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
theorem cramer_eq_adjugate_mulVec (A : Matrix n n α) (b : n → α) :
cramer A b = A.adjugate *ᵥ b
|
case h
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
b : n → α
this : b = ∑ i : n, b i • Pi.single i 1
k : n
⊢ A.cramer (∑ i : n, b i • Pi.single i 1) k = ((of fun i => Aᵀᵀ.cramer (Pi.single i 1))ᵀ *ᵥ b) k
|
simp [mulVec, dotProduct, mul_comm]
|
no goals
|
bdebf67a17df8200
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastRotateLeft.go_get_aux
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/RotateLeft.lean
|
theorem go_get_aux (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr),
(go input distance curr hcurr s).get idx (by omega)
=
s.get idx hidx
|
case isTrue.isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
h✝¹ : curr < w
h✝ : curr < distance % w
⊢ (go input distance (curr + 1) ⋯ (s.push (input.get (w - distance % w + curr) ⋯))).get idx ⋯ = s.get idx hidx
|
rw [go_get_aux]
|
case isTrue.isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
h✝¹ : curr < w
h✝ : curr < distance % w
⊢ (s.push (input.get (w - distance % w + curr) ⋯)).get idx ?isTrue.isTrue.hidx = s.get idx hidx
case isTrue.isTrue.hidx
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
h✝¹ : curr < w
h✝ : curr < distance % w
⊢ idx < curr + 1
|
e1ea21e0c26513f1
|
mdifferentiableWithinAt_iff_target
|
Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
|
theorem mdifferentiableWithinAt_iff_target :
MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧
MDifferentiableWithinAt I 𝓘(𝕜, E') (extChartAt I' (f x) ∘ f) s x
|
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
⊢ ContinuousWithinAt f s x ∧
DifferentiableWithinAtProp I I' (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
(↑(chartAt H x) x) ↔
(ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x) ∧
DifferentiableWithinAtProp I 𝓘(𝕜, E')
(↑(chartAt E' ((↑(extChartAt I' (f x)) ∘ f) x)) ∘ (↑(extChartAt I' (f x)) ∘ f) ∘ ↑(chartAt H x).symm)
(↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
|
have cont :
ContinuousWithinAt f s x ∧ ContinuousWithinAt (extChartAt I' (f x) ∘ f) s x ↔
ContinuousWithinAt f s x :=
and_iff_left_of_imp <| (continuousAt_extChartAt _).comp_continuousWithinAt
|
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
cont : ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x ↔ ContinuousWithinAt f s x
⊢ ContinuousWithinAt f s x ∧
DifferentiableWithinAtProp I I' (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
(↑(chartAt H x) x) ↔
(ContinuousWithinAt f s x ∧ ContinuousWithinAt (↑(extChartAt I' (f x)) ∘ f) s x) ∧
DifferentiableWithinAtProp I 𝓘(𝕜, E')
(↑(chartAt E' ((↑(extChartAt I' (f x)) ∘ f) x)) ∘ (↑(extChartAt I' (f x)) ∘ f) ∘ ↑(chartAt H x).symm)
(↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
|
a7b5d80b8ac37cc5
|
MeasureTheory.OuterMeasure.exists_measurable_superset_forall_eq_trim
|
Mathlib/MeasureTheory/OuterMeasure/Induced.lean
|
theorem exists_measurable_superset_forall_eq_trim {ι} [Countable ι] (μ : ι → OuterMeasure α)
(s : Set α) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = (μ i).trim s
|
α : Type u_1
inst✝¹ : MeasurableSpace α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
s : Set α
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), (μ i) t = (μ i).trim s
|
choose t hst ht hμt using fun i => (μ i).exists_measurable_superset_eq_trim s
|
α : Type u_1
inst✝¹ : MeasurableSpace α
ι : Sort u_2
inst✝ : Countable ι
μ : ι → OuterMeasure α
s : Set α
t : ι → Set α
hst : ∀ (i : ι), s ⊆ t i
ht : ∀ (i : ι), MeasurableSet (t i)
hμt : ∀ (i : ι), (μ i) (t i) = (μ i).trim s
⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), (μ i) t = (μ i).trim s
|
cfb6c437a6630ff3
|
IsRelPrime.prod_left_iff
|
Mathlib/RingTheory/Coprime/Lemmas.lean
|
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x
|
α : Type u_1
I : Type u_2
inst✝¹ : CommMonoid α
inst✝ : DecompositionMonoid α
x : α
s : I → α
t : Finset I
x✝ : I
⊢ x✝ ∈ ∅ → IsRelPrime (s x✝) x
|
simp
|
no goals
|
9be85dd8865208e0
|
Bool.ofNat_le_ofNat
|
Mathlib/Data/Bool/Basic.lean
|
theorem ofNat_le_ofNat {n m : Nat} (h : n ≤ m) : ofNat n ≤ ofNat m
|
case isTrue
n m : ℕ
h : n ≤ m
hn : n = 0
⊢ (!true) ≤ !decide (m = 0)
|
exact Bool.false_le _
|
no goals
|
a12bab163ba4b474
|
SimpleGraph.edgeDisjointTriangles_iff_mem_sym2_subsingleton
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
|
lemma edgeDisjointTriangles_iff_mem_sym2_subsingleton :
G.EdgeDisjointTriangles ↔
∀ ⦃e : Sym2 α⦄, ¬ e.IsDiag → {s ∈ G.cliqueSet 3 | e ∈ (s : Finset α).sym2}.Subsingleton
|
case h
α : Type u_1
G : SimpleGraph α
a b : α
hab : a ≠ b
s : Finset α
⊢ (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ a ∈ s ∧ b ∈ s ↔
G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}
|
constructor
|
case h.mp
α : Type u_1
G : SimpleGraph α
a b : α
hab : a ≠ b
s : Finset α
⊢ (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ a ∈ s ∧ b ∈ s →
G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}
case h.mpr
α : Type u_1
G : SimpleGraph α
a b : α
hab : a ≠ b
s : Finset α
⊢ (G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) →
(∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ a ∈ s ∧ b ∈ s
|
dd5ab32164f7deff
|
derived_le_lower_central
|
Mathlib/GroupTheory/Nilpotent.lean
|
theorem derived_le_lower_central (n : ℕ) : derivedSeries G n ≤ lowerCentralSeries G n
|
G : Type u_1
inst✝ : Group G
n : ℕ
⊢ derivedSeries G n ≤ lowerCentralSeries G n
|
induction' n with i ih
|
case zero
G : Type u_1
inst✝ : Group G
⊢ derivedSeries G 0 ≤ lowerCentralSeries G 0
case succ
G : Type u_1
inst✝ : Group G
i : ℕ
ih : derivedSeries G i ≤ lowerCentralSeries G i
⊢ derivedSeries G (i + 1) ≤ lowerCentralSeries G (i + 1)
|
b685699a973e345f
|
geom_sum_inv
|
Mathlib/Algebra/GeomSum.lean
|
theorem geom_sum_inv [DivisionRing α] {x : α} (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :
∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)
|
α : Type u
inst✝ : DivisionRing α
x : α
hx1 : x ≠ 1
hx0 : x ≠ 0
n : ℕ
⊢ ∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)
|
have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]
|
α : Type u
inst✝ : DivisionRing α
x : α
hx1 : x ≠ 1
hx0 : x ≠ 0
n : ℕ
h₁ : x⁻¹ ≠ 1
⊢ ∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)
|
90f187432f831566
|
Matroid.loopyOn_isBasis_iff
|
Mathlib/Data/Matroid/Constructions.lean
|
theorem loopyOn_isBasis_iff : (loopyOn E).IsBasis I X ↔ I = ∅ ∧ X ⊆ E :=
⟨fun h ↦ ⟨loopyOn_indep_iff.mp h.indep, h.subset_ground⟩,
by rintro ⟨rfl, hX⟩; rw [isBasis_iff]; simp⟩
|
α : Type u_1
E I X : Set α
⊢ I = ∅ ∧ X ⊆ E → (loopyOn E).IsBasis I X
|
rintro ⟨rfl, hX⟩
|
case intro
α : Type u_1
E X : Set α
hX : X ⊆ E
⊢ (loopyOn E).IsBasis ∅ X
|
a7e289444aeca3df
|
FirstOrder.Language.LHom.funext
|
Mathlib/ModelTheory/LanguageMap.lean
|
theorem funext {F G : L →ᴸ L'} (h_fun : F.onFunction = G.onFunction)
(h_rel : F.onRelation = G.onRelation) : F = G
|
case mk.mk
L : Language
L' : Language
Ff : ⦃n : ℕ⦄ → L.Functions n → L'.Functions n
Fr : ⦃n : ℕ⦄ → L.Relations n → L'.Relations n
Gf : ⦃n : ℕ⦄ → L.Functions n → L'.Functions n
Gr : ⦃n : ℕ⦄ → L.Relations n → L'.Relations n
h_fun : { onFunction := Ff, onRelation := Fr }.onFunction = { onFunction := Gf, onRelation := Gr }.onFunction
h_rel : { onFunction := Ff, onRelation := Fr }.onRelation = { onFunction := Gf, onRelation := Gr }.onRelation
⊢ { onFunction := Ff, onRelation := Fr } = { onFunction := Gf, onRelation := Gr }
|
simp only [mk.injEq]
|
case mk.mk
L : Language
L' : Language
Ff : ⦃n : ℕ⦄ → L.Functions n → L'.Functions n
Fr : ⦃n : ℕ⦄ → L.Relations n → L'.Relations n
Gf : ⦃n : ℕ⦄ → L.Functions n → L'.Functions n
Gr : ⦃n : ℕ⦄ → L.Relations n → L'.Relations n
h_fun : { onFunction := Ff, onRelation := Fr }.onFunction = { onFunction := Gf, onRelation := Gr }.onFunction
h_rel : { onFunction := Ff, onRelation := Fr }.onRelation = { onFunction := Gf, onRelation := Gr }.onRelation
⊢ Ff = Gf ∧ Fr = Gr
|
7d892da77329675e
|
Flow.omegaLimit_omegaLimit
|
Mathlib/Dynamics/OmegaLimit.lean
|
theorem omegaLimit_omegaLimit (hf : ∀ t, Tendsto (t + ·) f f) : ω f ϕ (ω f ϕ s) ⊆ ω f ϕ s
|
case intro.intro
τ : Type u_1
inst✝³ : TopologicalSpace τ
inst✝² : AddCommGroup τ
inst✝¹ : IsTopologicalAddGroup τ
α : Type u_2
inst✝ : TopologicalSpace α
f : Filter τ
ϕ : Flow τ α
s : Set α
hf : ∀ (t : τ), Tendsto (fun x => t + x) f f
x✝ : α
h : ∀ n ∈ 𝓝 x✝, ∀ {U : Set τ}, U ∈ f → ∃ x ∈ U, (ϕ.toFun x '' ω f ϕ.toFun s ∩ n).Nonempty
n : Set α
hn : n ∈ 𝓝 x✝
u : Set τ
hu : u ∈ f
o : Set α
ho₁ : o ⊆ n
ho₂ : IsOpen o
ho₃ : x✝ ∈ o
t : τ
_ht₁ : t ∈ u
ht₂ : (ϕ.toFun t '' ω f ϕ.toFun s ∩ o).Nonempty
l₁ : (ω f ϕ.toFun s ∩ o).Nonempty
b : α
hb₁ : b ∈ closure (image2 ϕ.toFun u s)
hb₂ : b ∈ o
⊢ (o ∩ image2 ϕ.toFun u s).Nonempty
|
exact mem_closure_iff_nhds.mp hb₁ o (IsOpen.mem_nhds ho₂ hb₂)
|
no goals
|
3464979464374ec5
|
extend_partialOrder
|
Mathlib/Order/Extension/Linear.lean
|
theorem extend_partialOrder {α : Type u} (r : α → α → Prop) [IsPartialOrder α r] :
∃ s : α → α → Prop, IsLinearOrder α s ∧ r ≤ s
|
case refine_3.intro.intro.intro.intro.inr
α : Type u
r : α → α → Prop
inst✝ : IsPartialOrder α r
S : Set (α → α → Prop) := {s | IsPartialOrder α s}
c : Set (α → α → Prop)
hc₁ : c ⊆ S
hc₂ : IsChain (fun x1 x2 => x1 ≤ x2) c
s : α → α → Prop
hs : s ∈ c
this✝¹ : IsPreorder α s
x y : α
s₁ : α → α → Prop
h₁s₁ : s₁ ∈ c
h₂s₁ : s₁ x y
s₂ : α → α → Prop
h₁s₂ : s₂ ∈ c
h₂s₂ : s₂ y x
this✝ : IsPartialOrder α s₁
this : IsPartialOrder α s₂
h : s₂ ≤ s₁
⊢ x = y
|
apply antisymm h₂s₁ (h _ _ h₂s₂)
|
no goals
|
f4cf57fbd32a5c9a
|
unitary.star_mem
|
Mathlib/Algebra/Star/Unitary.lean
|
theorem star_mem {U : R} (hU : U ∈ unitary R) : star U ∈ unitary R :=
⟨by rw [star_star, mul_star_self_of_mem hU], by rw [star_star, star_mul_self_of_mem hU]⟩
|
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U : R
hU : U ∈ unitary R
⊢ star U * star (star U) = 1
|
rw [star_star, star_mul_self_of_mem hU]
|
no goals
|
0304e8c706770550
|
Set.image_preimage_eq_range_inter
|
Mathlib/Data/Set/Image.lean
|
theorem image_preimage_eq_range_inter {f : α → β} {t : Set β} : f '' (f ⁻¹' t) = range f ∩ t :=
ext fun x =>
⟨fun ⟨_, hx, HEq⟩ => HEq ▸ ⟨mem_range_self _, hx⟩, fun ⟨⟨y, h_eq⟩, hx⟩ =>
h_eq ▸ mem_image_of_mem f <| show y ∈ f ⁻¹' t by rw [preimage, mem_setOf, h_eq]; exact hx⟩
|
α : Type u_1
β : Type u_2
f : α → β
t : Set β
x : β
x✝ : x ∈ range f ∩ t
y : α
h_eq : f y = x
hx : x ∈ t
⊢ y ∈ f ⁻¹' t
|
rw [preimage, mem_setOf, h_eq]
|
α : Type u_1
β : Type u_2
f : α → β
t : Set β
x : β
x✝ : x ∈ range f ∩ t
y : α
h_eq : f y = x
hx : x ∈ t
⊢ x ∈ t
|
63a53823e8c4689a
|
Algebra.FormallyUnramified.isField_of_isAlgClosed_of_isLocalRing
|
Mathlib/RingTheory/Unramified/Field.lean
|
theorem isField_of_isAlgClosed_of_isLocalRing
[IsAlgClosed K] [IsLocalRing A] : IsField A
|
case intro
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
x : K
hx : (algebraMap K A) x ∈ IsLocalRing.maximalIdeal A
⊢ (algebraMap K A) x ∈ ⊥
|
show _ = 0
|
case intro
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
x : K
hx : (algebraMap K A) x ∈ IsLocalRing.maximalIdeal A
⊢ (algebraMap K A) x = 0
|
5080e195e5854ef7
|
Fin.insertNth_mem_piFinset_insertNth
|
Mathlib/Data/Fin/Tuple/Finset.lean
|
lemma insertNth_mem_piFinset_insertNth {x_pivot : α p} {x_remove : ∀ i, α (succAbove p i)}
{s_pivot : Finset (α p)} {s_remove : ∀ i, Finset (α (succAbove p i))} :
insertNth p x_pivot x_remove ∈ piFinset (insertNth p s_pivot s_remove) ↔
x_pivot ∈ s_pivot ∧ x_remove ∈ piFinset s_remove
|
n : ℕ
α : Fin (n + 1) → Type u_1
p : Fin (n + 1)
x_pivot : α p
x_remove : (i : Fin n) → α (p.succAbove i)
s_pivot : Finset (α p)
s_remove : (i : Fin n) → Finset (α (p.succAbove i))
⊢ p.insertNth x_pivot x_remove ∈ piFinset (p.insertNth s_pivot s_remove) ↔
x_pivot ∈ s_pivot ∧ x_remove ∈ piFinset s_remove
|
simp [mem_piFinset_iff_pivot_removeNth p]
|
no goals
|
a6ae7a5b62a6de5e
|
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