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
|
|---|---|---|---|---|---|---|
Finset.card_le_card_shatterer
|
Mathlib/Combinatorics/SetFamily/Shatter.lean
|
/-- Pajor's variant of the **Sauer-Shelah lemma**. -/
lemma card_le_card_shatterer (𝒜 : Finset (Finset α)) : #𝒜 ≤ #𝒜.shatterer
|
case refine_3.refine_1
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ : Finset (Finset α)
a : α
𝒜 : Finset (Finset α)
ih₀ : #(nonMemberSubfamily a 𝒜) ≤ #(nonMemberSubfamily a 𝒜).shatterer
ih₁ : #(memberSubfamily a 𝒜) ≤ #(memberSubfamily a 𝒜).shatterer
ℬ : Finset (Finset α) := image (insert a) ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
⊢ (memberSubfamily a 𝒜).shatterer ⊆ 𝒜.shatterer
|
simp only [subset_iff, mem_shatterer]
|
case refine_3.refine_1
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ : Finset (Finset α)
a : α
𝒜 : Finset (Finset α)
ih₀ : #(nonMemberSubfamily a 𝒜) ≤ #(nonMemberSubfamily a 𝒜).shatterer
ih₁ : #(memberSubfamily a 𝒜) ≤ #(memberSubfamily a 𝒜).shatterer
ℬ : Finset (Finset α) := image (insert a) ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
⊢ ∀ ⦃x : Finset α⦄, (memberSubfamily a 𝒜).Shatters x → 𝒜.Shatters x
|
7b2e4a2398648e6f
|
SetTheory.PGame.birthday_def
|
Mathlib/SetTheory/Game/Birthday.lean
|
theorem birthday_def (x : PGame) :
birthday x =
max (lsub.{u, u} fun i => birthday (x.moveLeft i))
(lsub.{u, u} fun i => birthday (x.moveRight i))
|
x : PGame
⊢ x.birthday = (lsub fun i => (x.moveLeft i).birthday) ⊔ lsub fun i => (x.moveRight i).birthday
|
cases x
|
case mk
α✝ β✝ : Type u
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (mk α✝ β✝ a✝¹ a✝).birthday =
(lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) ⊔ lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday
|
0e9d2c95a121b08b
|
Std.DHashMap.Raw.size_alter_eq_add_one
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem size_alter_eq_add_one [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)}
(h : m.WF) (h₁ : k ∉ m) (h₂ : (f (m.get? k)).isSome) :
(m.alter k f).size = m.size + 1
|
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
m : Raw α β
inst✝ : LawfulBEq α
k : α
f : Option (β k) → Option (β k)
h : m.WF
⊢ m.contains k = false → (f (m.get? k)).isSome = true → (m.alter k f).size = m.size + 1
|
simp_to_raw using Raw₀.size_alter_eq_add_one
|
no goals
|
36b9c2a1dc985d01
|
Nat.rfindOpt_mono
|
Mathlib/Computability/Partrec.lean
|
theorem rfindOpt_mono {α} {f : ℕ → Option α} (H : ∀ {a m n}, m ≤ n → a ∈ f m → a ∈ f n) {a} :
a ∈ rfindOpt f ↔ ∃ n, a ∈ f n :=
⟨rfindOpt_spec, fun ⟨n, h⟩ => by
have h' := rfindOpt_dom.2 ⟨_, _, h⟩
obtain ⟨k, hk⟩ := rfindOpt_spec ⟨h', rfl⟩
have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)
simp at this; simp [this, get_mem]⟩
|
case intro
α : Type u_1
f : ℕ → Option α
H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n
a : α
x✝ : ∃ n, a ∈ f n
n : ℕ
h : a ∈ f n
h' : (rfindOpt f).Dom
k : ℕ
hk : (rfindOpt f).get h' ∈ f k
this : Option.some a = Option.some ((rfindOpt f).get h')
⊢ a ∈ rfindOpt f
|
simp at this
|
case intro
α : Type u_1
f : ℕ → Option α
H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n
a : α
x✝ : ∃ n, a ∈ f n
n : ℕ
h : a ∈ f n
h' : (rfindOpt f).Dom
k : ℕ
hk : (rfindOpt f).get h' ∈ f k
this : a = (rfindOpt f).get h'
⊢ a ∈ rfindOpt f
|
dcee59b58e9bc853
|
rank_subsingleton
|
Mathlib/LinearAlgebra/Dimension/Basic.lean
|
theorem rank_subsingleton [Subsingleton R] : Module.rank R M = 1
|
case h₂
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Subsingleton R
this✝ : Subsingleton M
this : Nonempty { s // LinearIndepOn R id s }
w : Cardinal.{v}
hw : w < 1
⊢ ∃ i, w < #↑↑i
|
exact ⟨⟨{0}, LinearIndepOn.of_subsingleton⟩, hw.trans_eq (Cardinal.mk_singleton _).symm⟩
|
no goals
|
04529e8e88b39ed6
|
List.sublist_replicate_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem sublist_replicate_iff : l <+ replicate m a ↔ ∃ n, n ≤ m ∧ l = replicate n a
|
case cons.mp.succ.cons.intro.intro
α✝ : Type u_1
b : α✝
m n : Nat
le : n ≤ m
ih : ∀ {m : Nat}, replicate n b <+ replicate m b ↔ ∃ n_1, n_1 ≤ m ∧ replicate n b = replicate n_1 b
w : b :: replicate n b <+ replicate m b
⊢ ∃ n_1, n_1 ≤ m + 1 ∧ b :: replicate n b = replicate n_1 b
|
exact ⟨n+1, Nat.add_le_add_right le 1, rfl⟩
|
no goals
|
b36e6729c3d1c9f6
|
Lean.Grind.decide_eq_false
|
Mathlib/.lake/packages/lean4/src/lean/Init/Grind/Lemmas.lean
|
theorem decide_eq_false {p : Prop} {_ : Decidable p} : p = False → decide p = false
|
p : Prop
x✝ : Decidable p
⊢ p = False → decide p = false
|
simp
|
no goals
|
31b2819c3ab52cc8
|
TensorProduct.finsuppLeft_symm_apply_single
|
Mathlib/LinearAlgebra/DirectSum/Finsupp.lean
|
@[simp]
lemma finsuppLeft_symm_apply_single (i : ι) (m : M) (n : N) :
(finsuppLeft R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) =
Finsupp.single i m ⊗ₜ[R] n
|
R : Type u_1
inst✝⁵ : CommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : DecidableEq ι
i : ι
m : M
n : N
⊢ (finsuppLeft R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) = Finsupp.single i m ⊗ₜ[R] n
|
simp [finsuppLeft, Finsupp.lsum]
|
no goals
|
f7e7ed6f2e501d86
|
PiTensorProduct.map_comp_reindex_symm
|
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
theorem map_comp_reindex_symm (f : Π i, s i →ₗ[R] t i) (e : ι ≃ ι₂) :
map f ∘ₗ (reindex R s e).symm = (reindex R t e).symm ∘ₗ map (fun i => f (e.symm i))
|
ι : Type u_1
ι₂ : Type u_2
R : Type u_4
inst✝⁴ : CommSemiring R
s : ι → Type u_7
inst✝³ : (i : ι) → AddCommMonoid (s i)
inst✝² : (i : ι) → Module R (s i)
t : ι → Type u_11
inst✝¹ : (i : ι) → AddCommMonoid (t i)
inst✝ : (i : ι) → Module R (t i)
f : (i : ι) → s i →ₗ[R] t i
e : ι ≃ ι₂
⊢ map f ∘ₗ ↑(reindex R s e).symm = ↑(reindex R t e).symm ∘ₗ map fun i => f (e.symm i)
|
ext m
|
case H.H
ι : Type u_1
ι₂ : Type u_2
R : Type u_4
inst✝⁴ : CommSemiring R
s : ι → Type u_7
inst✝³ : (i : ι) → AddCommMonoid (s i)
inst✝² : (i : ι) → Module R (s i)
t : ι → Type u_11
inst✝¹ : (i : ι) → AddCommMonoid (t i)
inst✝ : (i : ι) → Module R (t i)
f : (i : ι) → s i →ₗ[R] t i
e : ι ≃ ι₂
m : (i : ι₂) → s (e.symm i)
⊢ ((map f ∘ₗ ↑(reindex R s e).symm).compMultilinearMap (tprod R)) m =
((↑(reindex R t e).symm ∘ₗ map fun i => f (e.symm i)).compMultilinearMap (tprod R)) m
|
eb43e0ef2f5c1f07
|
TopologicalSpace.nhds_mkOfNhds_single
|
Mathlib/Topology/Order.lean
|
theorem nhds_mkOfNhds_single [DecidableEq α] {a₀ : α} {l : Filter α} (h : pure a₀ ≤ l) (b : α) :
@nhds α (TopologicalSpace.mkOfNhds (update pure a₀ l)) b =
(update pure a₀ l : α → Filter α) b
|
case h.inr
α : Type u
inst✝ : DecidableEq α
l : Filter α
b✝ a : α
s : Set α
h : pure a ≤ l
hs : s ∈ update pure a l a
b : α
hb✝ : b ∈ s
hb : b ≠ a
⊢ s ∈ update pure a l b
|
rwa [update_of_ne hb]
|
no goals
|
870f5cf084a599e2
|
Finset.centroid_mem_convexHull
|
Mathlib/Analysis/Convex/Combination.lean
|
theorem Finset.centroid_mem_convexHull (s : Finset E) (hs : s.Nonempty) :
s.centroid R id ∈ convexHull R (s : Set E)
|
case hw₀
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Finset E
hs : s.Nonempty
⊢ ∀ i ∈ s, 0 ≤ centroidWeights R s i
|
simp only [inv_nonneg, imp_true_iff, Nat.cast_nonneg, Finset.centroidWeights_apply]
|
no goals
|
4a96c136a460710a
|
strictAnti_of_succ_lt
|
Mathlib/Order/SuccPred/Archimedean.lean
|
lemma strictAnti_of_succ_lt (hf : ∀ a, ¬ IsMax a → f (succ a) < f a) : StrictAnti f
|
α : Type u_3
β : Type u_4
inst✝³ : PartialOrder α
inst✝² : Preorder β
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
f : α → β
hf : ∀ (a : α), ¬IsMax a → f (succ a) < f a
⊢ StrictAnti f
|
simpa using strictAntiOn_of_succ_lt Set.ordConnected_univ (by simpa using hf)
|
no goals
|
cac69e7fe6b4e543
|
MeasureTheory.Measure.exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
|
Mathlib/MeasureTheory/Measure/Haar/Unique.lean
|
/-- Given two left-invariant measures which are finite on
compacts, they coincide in the following sense: they give the same value to the integral of
continuous compactly supported functions, up to a multiplicative constant. -/
@[to_additive exists_integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport (μ' μ : Measure G)
[IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ)
|
case pos.intro.mk.intro.intro
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
H : LocallyCompactSpace G
g : G → ℝ
g_cont : Continuous g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
int_g_pos : 0 < ∫ (x : G), g x ∂μ
c : ℝ := (∫ (x : G), g x ∂μ)⁻¹ * ∫ (x : G), g x ∂μ'
c_nonneg : 0 ≤ c
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport f
ν : Measure G := μ.inv
A : (∫ (x : G), f x ∂μ) * (∫ (x : G), g x ∂μ)⁻¹ = ∫ (y : G), f y * (∫ (z : G), g (z⁻¹ * y) ∂ν)⁻¹ ∂ν
B : ∫ (x : G), f x ∂μ' = ((∫ (x : G), g x ∂μ)⁻¹ * ∫ (x : G), g x ∂μ') * ∫ (x : G), f x ∂μ
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂⟨c, c_nonneg⟩ • μ
|
simp [B, integral_smul_nnreal_measure, c, NNReal.smul_def]
|
no goals
|
f459b191694f5c04
|
Std.Sat.AIG.denote_mkAtom
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/Lemmas.lean
|
theorem denote_mkAtom {aig : AIG α} :
⟦(aig.mkAtom var), assign⟧ = assign var
|
case h_2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
var : α
aig : AIG α
idx✝ : α
heq✝ : (aig.mkAtom var).aig.decls[(aig.mkAtom var).ref.gate] = Decl.atom idx✝
⊢ assign idx✝ = assign var
|
next heq =>
rw [mkAtom, Array.getElem_push_eq] at heq
injection heq with heq
rw [heq]
|
no goals
|
209125ccf5e1ce3b
|
Matrix.det_updateCol_smul_left
|
Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
|
theorem det_updateCol_smul_left (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
det (updateCol (s • M) j u) = s ^ (Fintype.card n - 1) * det (updateCol M j u)
|
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
M : Matrix n n R
j : n
s : R
u : n → R
⊢ s ^ (Fintype.card n - 1) * (Mᵀ.updateRow j u).det = s ^ (Fintype.card n - 1) * (M.updateCol j u).det
|
simp [updateRow_transpose, det_transpose]
|
no goals
|
e3cf10bd7e79875e
|
LieModule.isTrivial_iff_max_triv_eq_top
|
Mathlib/Algebra/Lie/Abelian.lean
|
theorem isTrivial_iff_max_triv_eq_top : IsTrivial L M ↔ maxTrivSubmodule R L M = ⊤
|
case mpr.trivial
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
h : maxTrivSubmodule R L M = ⊤
x : L
m : M
⊢ ⁅x, m⁆ = 0
|
revert x
|
case mpr.trivial
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
h : maxTrivSubmodule R L M = ⊤
m : M
⊢ ∀ (x : L), ⁅x, m⁆ = 0
|
7b69b1178ad50453
|
gauge_smul_of_nonneg
|
Mathlib/Analysis/Convex/Gauge.lean
|
theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
(ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x
|
case inr.e_a.h.mpr.intro.intro.intro
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
α : Type u_3
inst✝⁴ : LinearOrderedField α
inst✝³ : MulActionWithZero α ℝ
inst✝² : OrderedSMul α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : 0 < r
hx : r⁻¹ • x ∈ s
⊢ 0 < a • r ∧ (a • r)⁻¹ • a • x ∈ s
|
rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
|
case inr.e_a.h.mpr.intro.intro.intro
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
α : Type u_3
inst✝⁴ : LinearOrderedField α
inst✝³ : MulActionWithZero α ℝ
inst✝² : OrderedSMul α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
hr : 0 < r
hx : x ∈ r • s
⊢ 0 < a • r ∧ (a • r)⁻¹ • a • x ∈ s
|
2224562162173f32
|
AffineBasis.convexHull_eq_nonneg_coord
|
Mathlib/Analysis/Convex/Combination.lean
|
theorem AffineBasis.convexHull_eq_nonneg_coord {ι : Type*} (b : AffineBasis ι R E) :
convexHull R (range b) = { x | ∀ i, 0 ≤ b.coord i x }
|
case h.refine_1
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
ι : Type u_8
b : AffineBasis ι R E
x : E
⊢ x ∈ {x | ∃ s w, (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧ (affineCombination R s ⇑b) w = x} →
x ∈ {x | ∀ (i : ι), 0 ≤ (b.coord i) x}
|
rintro ⟨s, w, hw₀, hw₁, rfl⟩ i
|
case h.refine_1.intro.intro.intro.intro
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
ι : Type u_8
b : AffineBasis ι R E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
i : ι
⊢ 0 ≤ (b.coord i) ((affineCombination R s ⇑b) w)
|
1fd8376dbf4dc8b4
|
clusterPt_iff_lift'_closure'
|
Mathlib/Topology/Basic.lean
|
theorem clusterPt_iff_lift'_closure' {F : Filter X} :
ClusterPt x F ↔ (F.lift' closure ⊓ pure x).NeBot
|
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
⊢ pure x ≤ F.lift' closure ↔ (pure x ⊓ F.lift' closure).NeBot
|
constructor
|
case mp
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
⊢ pure x ≤ F.lift' closure → (pure x ⊓ F.lift' closure).NeBot
case mpr
X : Type u
x : X
inst✝ : TopologicalSpace X
F : Filter X
⊢ (pure x ⊓ F.lift' closure).NeBot → pure x ≤ F.lift' closure
|
70879bd9b2cfc6e6
|
SimpleGraph.Walk.IsCycle.ncard_neighborSet_toSubgraph_eq_two
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
|
lemma ncard_neighborSet_toSubgraph_eq_two {u v} {p : G.Walk u u} (hpc : p.IsCycle)
(h : v ∈ p.support) : (p.toSubgraph.neighborSet v).ncard = 2
|
case neg
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : ¬(i = 0 ∨ i = p.length)
⊢ (p.toSubgraph.neighborSet v).ncard = 2
|
push_neg at he
|
case neg
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : i ≠ 0 ∧ i ≠ p.length
⊢ (p.toSubgraph.neighborSet v).ncard = 2
|
ab28954a43d4f7d4
|
Algebra.PreSubmersivePresentation.baseChange_jacobian
|
Mathlib/RingTheory/Smooth/StandardSmooth.lean
|
@[simp]
lemma baseChange_jacobian : (P.baseChange T).jacobian = 1 ⊗ₜ P.jacobian
|
case intro
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_1
inst✝¹ : CommRing T
inst✝ : Algebra R T
P : PreSubmersivePresentation R S
val✝ : Fintype P.rels
⊢ (baseChange T P).jacobian = 1 ⊗ₜ[R] P.jacobian
|
letI : Fintype (P.baseChange T).rels := inferInstanceAs <| Fintype P.rels
|
case intro
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_1
inst✝¹ : CommRing T
inst✝ : Algebra R T
P : PreSubmersivePresentation R S
val✝ : Fintype P.rels
this : Fintype (baseChange T P).rels := inferInstanceAs (Fintype P.rels)
⊢ (baseChange T P).jacobian = 1 ⊗ₜ[R] P.jacobian
|
8d698f05bd3b180b
|
contDiff_iff_contDiffAt
|
Mathlib/Analysis/Calculus/ContDiff/Defs.lean
|
theorem contDiff_iff_contDiffAt : ContDiff 𝕜 n f ↔ ∀ x, ContDiffAt 𝕜 n f x
|
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : WithTop ℕ∞
⊢ ContDiff 𝕜 n f ↔ ∀ (x : E), ContDiffAt 𝕜 n f x
|
simp [← contDiffOn_univ, ContDiffOn, ContDiffAt]
|
no goals
|
a1785a9a09bdd27d
|
Metric.dist_le_infDist_add_diam
|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
theorem dist_le_infDist_add_diam (hs : IsBounded s) (hy : y ∈ s) :
dist x y ≤ infDist x s + diam s
|
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x y : α
hs : Bornology.IsBounded s
hy : y ∈ s
⊢ (edist x y).toReal ≤ (infEdist x s).toReal + (EMetric.diam s).toReal
|
exact toReal_le_add (edist_le_infEdist_add_ediam hy) (infEdist_ne_top ⟨y, hy⟩) hs.ediam_ne_top
|
no goals
|
9c70edebbffebde5
|
Rel.interedges_disjoint_left
|
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
theorem interedges_disjoint_left {s s' : Finset α} (hs : Disjoint s s') (t : Finset β) :
Disjoint (interedges r s t) (interedges r s' t)
|
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s s' : Finset α
hs : ∀ ⦃a : α⦄, a ∈ s → a ∉ s'
t : Finset β
a✝ : α × β
hx : a✝ ∈ interedges r s t
hy : a✝ ∈ interedges r s' t
⊢ False
|
rw [mem_interedges_iff] at hx hy
|
α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s s' : Finset α
hs : ∀ ⦃a : α⦄, a ∈ s → a ∉ s'
t : Finset β
a✝ : α × β
hx : a✝.1 ∈ s ∧ a✝.2 ∈ t ∧ r a✝.1 a✝.2
hy : a✝.1 ∈ s' ∧ a✝.2 ∈ t ∧ r a✝.1 a✝.2
⊢ False
|
ef2275d757111a1a
|
Asymptotics.isLittleOTVS_iff_isLittleO
|
Mathlib/Analysis/Asymptotics/TVS.lean
|
lemma isLittleOTVS_iff_isLittleO {f : α → E} {g : α → F} {l : Filter α} :
f =o[𝕜;l] g ↔ f =o[l] g
|
case h₁
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
∀ (i : ℝ),
0 < i →
∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
ε δ : ℝ≥0
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 ↑δ) (g x)
hε : 0 < ε
hδ₀ : 0 < δ
x : α
hx : egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑(δ / ‖c‖₊) * egauge 𝕜 (ball 0 ↑δ) (g x)
⊢ ↑(δ / ‖c‖₊) ≤ ↑δ / ↑‖c‖₊
case h₂
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
g : α → F
l : Filter α
c : 𝕜
hc : 1 < ‖c‖₊
hc₀ : 0 < ‖c‖₊
h :
∀ (i : ℝ),
0 < i →
∃ j, 0 < j ∧ ∀ (ε : ℝ≥0), ε ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 i) (f x) ≤ ↑ε * egauge 𝕜 (ball 0 j) (g x)
ε δ : ℝ≥0
hδ : ∀ (ε_1 : ℝ≥0), ε_1 ≠ 0 → ∀ᶠ (x : α) in l, egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑ε_1 * egauge 𝕜 (ball 0 ↑δ) (g x)
hε : 0 < ε
hδ₀ : 0 < δ
x : α
hx : egauge 𝕜 (ball 0 ↑ε) (f x) ≤ ↑(δ / ‖c‖₊) * egauge 𝕜 (ball 0 ↑δ) (g x)
⊢ egauge 𝕜 (ball 0 ↑δ) (g x) ≤ ↑‖c‖₊ * ↑‖g x‖₊ / ↑δ
|
exacts [ENNReal.coe_div_le, egauge_ball_le_of_one_lt_norm hc (.inl <| ne_of_gt hδ₀)]
|
no goals
|
edd1acc6d74b4b2d
|
Set.Finite.infClosure
|
Mathlib/Order/SupClosed.lean
|
/-- The semilatice generated by a finite set is finite. -/
protected lemma Set.Finite.infClosure (hs : s.Finite) : (infClosure s).Finite
|
α : Type u_3
inst✝ : SemilatticeInf α
s : Set α
hs : s.Finite
⊢ (infClosure s).Finite
|
lift s to Finset α using hs
|
case intro
α : Type u_3
inst✝ : SemilatticeInf α
s : Finset α
⊢ (infClosure ↑s).Finite
|
107c23b3c856e816
|
ContinuousLinearMap.opNorm_extend_le
|
Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean
|
theorem opNorm_extend_le :
‖f.extend e h_dense (isUniformEmbedding_of_bound _ h_e).isUniformInducing‖ ≤ N * ‖f‖
|
case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
Fₗ : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜 Fₗ
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝¹ : CompleteSpace F
e : E →L[𝕜] Fₗ
h_dense : DenseRange ⇑e
N : ℝ≥0
h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖
inst✝ : RingHomIsometric σ₁₂
⊢ 0 ≤ ↑N * ‖f‖
|
cases le_total 0 N with
| inl hN => exact mul_nonneg hN (norm_nonneg _)
| inr hN =>
have : Unique E := ⟨⟨0⟩, fun x ↦ norm_le_zero_iff.mp <|
(h_e x).trans (mul_nonpos_of_nonpos_of_nonneg hN (norm_nonneg _))⟩
obtain rfl : f = 0 := Subsingleton.elim ..
simp
|
no goals
|
1e363b3609cf7221
|
MulAction.smul_mem_fixedBy_iff_mem_fixedBy
|
Mathlib/GroupTheory/GroupAction/FixedPoints.lean
|
theorem smul_mem_fixedBy_iff_mem_fixedBy {a : α} {g : G} :
g • a ∈ fixedBy α g ↔ a ∈ fixedBy α g
|
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
a : α
g : G
⊢ g • a = a ↔ a ∈ fixedBy α g
|
rfl
|
no goals
|
c3092fe306a81fbd
|
LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure
|
Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean
|
theorem LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure [NormedSpace 𝕜 E]
(f : E →ₗ[𝕜] E) (hf : LipschitzWith 1 f) (g : E →L[𝕜] LinearMap.eqLocus f 1)
(hg_proj : ∀ x : LinearMap.eqLocus f 1, g x = x)
(hg_ker : (LinearMap.ker g : Set E) ⊆ closure (LinearMap.range (f - 1))) (x : E) :
Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop (𝓝 (g x))
|
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : LipschitzWith 1 ⇑f
g : E →L[𝕜] ↥(eqLocus f 1)
hg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x
hg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))
y : E
hy : g y = 0
z : E
hz : IsFixedPt (⇑f) z
this✝ : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}
x : E
this : Bornology.IsBounded (Set.range fun x_1 => _root_.id ((⇑f)^[x_1] x))
H : ∀ (n : ℕ) (x y : E), (⇑f)^[n] (x - y) = (⇑f)^[n] x - (⇑f)^[n] y
⊢ (f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}
|
simpa [birkhoffAverage, birkhoffSum, Finset.sum_sub_distrib, smul_sub, H]
using tendsto_birkhoffAverage_apply_sub_birkhoffAverage 𝕜 this
|
no goals
|
ffcebdb2b0436857
|
Matroid.Indep.eq_union_image_of_disjointSum
|
Mathlib/Data/Matroid/Sum.lean
|
lemma Indep.eq_union_image_of_disjointSum {h I} (hI : (disjointSum M N h).Indep I) :
∃ IM IN, M.Indep IM ∧ N.Indep IN ∧ Disjoint IM IN ∧ I = IM ∪ IN
|
α : Type u_1
M N : Matroid α
h : Disjoint M.E N.E
I : Set α
hI : (M.disjointSum N h).Indep I
⊢ ∃ IM IN, M.Indep IM ∧ N.Indep IN ∧ Disjoint IM IN ∧ I = IM ∪ IN
|
rw [disjointSum_indep_iff] at hI
|
α : Type u_1
M N : Matroid α
h : Disjoint M.E N.E
I : Set α
hI : M.Indep (I ∩ M.E) ∧ N.Indep (I ∩ N.E) ∧ I ⊆ M.E ∪ N.E
⊢ ∃ IM IN, M.Indep IM ∧ N.Indep IN ∧ Disjoint IM IN ∧ I = IM ∪ IN
|
019f7a9decf15201
|
Int.ne_iff_lt_or_gt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem ne_iff_lt_or_gt {a b : Int} : a ≠ b ↔ a < b ∨ b < a
|
a b : Int
gt : b < a
⊢ a ≠ b
|
exact Int.ne_of_gt gt
|
no goals
|
a2564da64123f791
|
Multiset.powersetCardAux_zero
|
Mathlib/Data/Multiset/Powerset.lean
|
theorem powersetCardAux_zero (l : List α) : powersetCardAux 0 l = [0]
|
α : Type u_1
l : List α
⊢ powersetCardAux 0 l = [0]
|
simp [powersetCardAux_eq_map_coe]
|
no goals
|
348e428319e078f4
|
Array.SatisfiesM_mapM
|
Mathlib/.lake/packages/batteries/Batteries/Data/Array/Monadic.lean
|
theorem SatisfiesM_mapM [Monad m] [LawfulMonad m] (as : Array α) (f : α → m β)
(motive : Nat → Prop) (h0 : motive 0)
(p : Fin as.size → β → Prop)
(hs : ∀ i, motive i.1 → SatisfiesM (p i · ∧ motive (i + 1)) (f as[i])) :
SatisfiesM
(fun arr => motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i, h⟩ arr[i])
(Array.mapM f as)
|
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
as : Array α
f : α → m β
motive : Nat → Prop
h0 : motive 0
p : Fin as.size → β → Prop
hs : ∀ (i : Fin as.size), motive ↑i → SatisfiesM (fun x => p i x ∧ motive (↑i + 1)) (f as[i])
i : Nat
hi : i < as.size
arr : Array β
ih₁ : motive ↑⟨i, hi⟩
eq : arr.size = ↑⟨i, hi⟩
ih₂ : ∀ (i : Fin as.size) (h2 : ↑i < arr.size), p i arr[↑i]
a✝ : β
x✝ : p ⟨i, hi⟩ a✝ ∧ motive (↑⟨i, hi⟩ + 1)
h₁ : p ⟨i, hi⟩ a✝
h₂ : motive (↑⟨i, hi⟩ + 1)
j : Fin as.size
hj : ↑j < arr.size + 1
⊢ p j (if h : ↑j < arr.size then arr[↑j] else a✝)
|
split
|
case isTrue
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
as : Array α
f : α → m β
motive : Nat → Prop
h0 : motive 0
p : Fin as.size → β → Prop
hs : ∀ (i : Fin as.size), motive ↑i → SatisfiesM (fun x => p i x ∧ motive (↑i + 1)) (f as[i])
i : Nat
hi : i < as.size
arr : Array β
ih₁ : motive ↑⟨i, hi⟩
eq : arr.size = ↑⟨i, hi⟩
ih₂ : ∀ (i : Fin as.size) (h2 : ↑i < arr.size), p i arr[↑i]
a✝ : β
x✝ : p ⟨i, hi⟩ a✝ ∧ motive (↑⟨i, hi⟩ + 1)
h₁ : p ⟨i, hi⟩ a✝
h₂ : motive (↑⟨i, hi⟩ + 1)
j : Fin as.size
hj : ↑j < arr.size + 1
h✝ : ↑j < arr.size
⊢ p j arr[↑j]
case isFalse
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
as : Array α
f : α → m β
motive : Nat → Prop
h0 : motive 0
p : Fin as.size → β → Prop
hs : ∀ (i : Fin as.size), motive ↑i → SatisfiesM (fun x => p i x ∧ motive (↑i + 1)) (f as[i])
i : Nat
hi : i < as.size
arr : Array β
ih₁ : motive ↑⟨i, hi⟩
eq : arr.size = ↑⟨i, hi⟩
ih₂ : ∀ (i : Fin as.size) (h2 : ↑i < arr.size), p i arr[↑i]
a✝ : β
x✝ : p ⟨i, hi⟩ a✝ ∧ motive (↑⟨i, hi⟩ + 1)
h₁ : p ⟨i, hi⟩ a✝
h₂ : motive (↑⟨i, hi⟩ + 1)
j : Fin as.size
hj : ↑j < arr.size + 1
h✝ : ¬↑j < arr.size
⊢ p j a✝
|
2d55fc71b4122bfa
|
t1Space_TFAE
|
Mathlib/Topology/Separation/Basic.lean
|
theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :
List.TFAE [T1Space X,
∀ x, IsClosed ({ x } : Set X),
∀ x, IsOpen ({ x }ᶜ : Set X),
Continuous (@CofiniteTopology.of X),
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U,
∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
∀ ⦃x y : X⦄, x ⤳ y → x = y]
|
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous ⇑CofiniteTopology.of
tfae_4_to_2 : Continuous ⇑CofiniteTopology.of → ∀ (x : X), IsClosed {x}
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE
|
tfae_have 2 ↔ 10 := by
simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def,
mem_singleton_iff, eq_comm]
|
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous ⇑CofiniteTopology.of
tfae_4_to_2 : Continuous ⇑CofiniteTopology.of → ∀ (x : X), IsClosed {x}
tfae_2_iff_10 : (∀ (x : X), IsClosed {x}) ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE
|
92ea821073dea3c3
|
DFinsupp.Lex.acc_single
|
Mathlib/Data/DFinsupp/WellFounded.lean
|
theorem Lex.acc_single (hbot : ∀ ⦃i a⦄, ¬s i a 0) (hs : ∀ i, WellFounded (s i))
[DecidableEq ι] {i : ι} (hi : Acc (rᶜ ⊓ (· ≠ ·)) i) :
∀ a, Acc (DFinsupp.Lex r s) (single i a)
|
case intro
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0
hs : ∀ (i : ι), WellFounded (s i)
inst✝ : DecidableEq ι
i✝ i : ι
h✝ : ∀ (y : ι), (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) y i → Acc (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) y
ih : ∀ (y : ι), (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) y i → ∀ (a : α y), Acc (DFinsupp.Lex r s) (single y a)
⊢ ∀ (a : α i), Acc (DFinsupp.Lex r s) (single i a)
|
refine fun a => WellFounded.induction (hs i)
(C := fun x ↦ Acc (DFinsupp.Lex r s) (single i x)) a fun a ha ↦ ?_
|
case intro
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0
hs : ∀ (i : ι), WellFounded (s i)
inst✝ : DecidableEq ι
i✝ i : ι
h✝ : ∀ (y : ι), (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) y i → Acc (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) y
ih : ∀ (y : ι), (rᶜ ⊓ fun x1 x2 => x1 ≠ x2) y i → ∀ (a : α y), Acc (DFinsupp.Lex r s) (single y a)
a✝ a : α i
ha : ∀ (y : α i), s i y a → (fun x => Acc (DFinsupp.Lex r s) (single i x)) y
⊢ (fun x => Acc (DFinsupp.Lex r s) (single i x)) a
|
4e9f90d22d70a2fd
|
Real.deriv_qaryEntropy
|
Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean
|
lemma deriv_qaryEntropy (hp₀ : p ≠ 0) (hp₁ : p ≠ 1) :
deriv (qaryEntropy q) p = log (q - 1) + log (1 - p) - log p
|
case hf
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ DifferentiableAt ℝ (fun p => p * log ↑(↑q - 1)) p
case hg
q : ℕ
p : ℝ
hp₀ : p ≠ 0
hp₁ : p ≠ 1
⊢ DifferentiableAt ℝ binEntropy p
|
all_goals fun_prop (disch := assumption)
|
no goals
|
a8878fef28b9e7fe
|
Nat.all_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Fold.lean
|
theorem all_succ {n : Nat} (f : (i : Nat) → i < n + 1 → Bool) :
all (n + 1) f = (all n (fun i h => f i (by omega)) && f n (by omega))
|
n : Nat
f : (i : Nat) → i < n + 1 → Bool
i : Nat
h : i < n
⊢ i < n + 1
|
omega
|
no goals
|
cac12ba618906a84
|
LieSubalgebra.mem_ofLe
|
Mathlib/Algebra/Lie/Subalgebra.lean
|
theorem mem_ofLe (x : K') : x ∈ ofLe h ↔ (x : L) ∈ K
|
case mp
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
K K' : LieSubalgebra R L
h : K ≤ K'
x : ↥K'
⊢ (∃ y, ⟨↑y, ⋯⟩ = x) → ↑x ∈ K
|
rintro ⟨y, rfl⟩
|
case mp.intro
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
K K' : LieSubalgebra R L
h : K ≤ K'
y : ↥K
⊢ ↑⟨↑y, ⋯⟩ ∈ K
|
5d9bb5db043a9471
|
MeasureTheory.self_mem_ae_restrict
|
Mathlib/MeasureTheory/Measure/Restrict.lean
|
theorem self_mem_ae_restrict {s} (hs : MeasurableSet s) : s ∈ ae (μ.restrict s)
|
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
⊢ ∃ t₁ ∈ ae μ, ∃ t₂, s ⊆ t₂ ∧ s = t₁ ∩ t₂
|
exact ⟨_, univ_mem, s, Subset.rfl, (univ_inter s).symm⟩
|
no goals
|
42080b33a9f5bcb5
|
Nat.succ_sub
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean
|
theorem succ_sub {m n : Nat} (h : n ≤ m) : succ m - n = succ (m - n)
|
m n : Nat
h : n ≤ m
k : Nat
hk : n + k = m
⊢ m.succ - n = (m - n).succ
|
rw [← hk, Nat.add_sub_cancel_left, ← add_succ, Nat.add_sub_cancel_left]
|
no goals
|
286b51de4d0e7224
|
CategoryTheory.Iso.map_inv_hom_id_app
|
Mathlib/CategoryTheory/Functor/Category.lean
|
theorem map_inv_hom_id_app {X Y : C} (e : X ≅ Y) (F : C ⥤ D ⥤ E) (Z : D) :
(F.map e.inv).app Z ≫ (F.map e.hom).app Z = 𝟙 _
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
X Y : C
e : X ≅ Y
F : C ⥤ D ⥤ E
Z : D
⊢ (F.map e.inv).app Z ≫ (F.map e.hom).app Z = 𝟙 ((F.obj Y).obj Z)
|
simp [← NatTrans.comp_app, ← Functor.map_comp]
|
no goals
|
bb340065915ccdab
|
Interval.coe_sInf
|
Mathlib/Order/Interval/Basic.lean
|
theorem coe_sInf [DecidableRel (α := α) (· ≤ ·)] (S : Set (Interval α)) :
↑(sInf S) = ⋂ s ∈ S, (s : Set α)
|
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
S : Set (Interval α)
⊢ ↑(if h :
⊥ ∉ S ∧ ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ S → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ S → s.toProd.1 ≤ t.toProd.2 then
↑{ toProd := (⨆ s, ⨆ (_ : ↑s ∈ S), s.toProd.1, ⨅ s, ⨅ (_ : ↑s ∈ S), s.toProd.2), fst_le_snd := ⋯ }
else ⊥) =
⋂ s ∈ S, ↑s
|
split_ifs with h
|
case pos
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
S : Set (Interval α)
h : ⊥ ∉ S ∧ ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ S → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ S → s.toProd.1 ≤ t.toProd.2
⊢ ↑↑{ toProd := (⨆ s, ⨆ (_ : ↑s ∈ S), s.toProd.1, ⨅ s, ⨅ (_ : ↑s ∈ S), s.toProd.2), fst_le_snd := ⋯ } = ⋂ s ∈ S, ↑s
case neg
α : Type u_1
inst✝¹ : CompleteLattice α
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
S : Set (Interval α)
h : ¬(⊥ ∉ S ∧ ∀ ⦃s : NonemptyInterval α⦄, ↑s ∈ S → ∀ ⦃t : NonemptyInterval α⦄, ↑t ∈ S → s.toProd.1 ≤ t.toProd.2)
⊢ ↑⊥ = ⋂ s ∈ S, ↑s
|
432700be06931a9b
|
Metric.hausdorffDist_le_of_mem_dist
|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
theorem hausdorffDist_le_of_mem_dist {r : ℝ} (hr : 0 ≤ r) (H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r)
(H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r) : hausdorffDist s t ≤ r
|
case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
⊢ ∀ x ∈ t, infDist x s ≤ r
|
intro x xt
|
case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
x : α
xt : x ∈ t
⊢ infDist x s ≤ r
|
628fd108fcb99ee2
|
Behrend.threeAPFree_image_sphere
|
Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
|
theorem threeAPFree_image_sphere :
ThreeAPFree ((sphere n d k).image (map (2 * d - 1)) : Set ℕ)
|
n d k : ℕ
⊢ ThreeAPFree (⇑(map (2 * d - 1)) '' ↑(sphere n d k))
|
apply ThreeAPFree.image' (α := Fin n → ℕ) (β := ℕ) (s := sphere n d k) (map (2 * d - 1))
(map_injOn.mono _) threeAPFree_sphere
|
n d k : ℕ
⊢ ↑(sphere n d k) + ↑(sphere n d k) ⊆ {x | ∀ (i : Fin n), x i < 2 * d - 1}
|
e8af164492af265d
|
Real.antitone_rpow_of_base_le_one
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
lemma antitone_rpow_of_base_le_one {b : ℝ} (hb₀ : 0 < b) (hb₁ : b ≤ 1) :
Antitone (b ^ · : ℝ → ℝ)
|
case inl
b : ℝ
hb₀ : 0 < b
hb₁✝ : b ≤ 1
hb₁ : b < 1
⊢ Antitone fun x => b ^ x
case inr
hb₀ : 0 < 1
hb₁ : 1 ≤ 1
⊢ Antitone fun x => 1 ^ x
|
case inl => exact (strictAnti_rpow_of_base_lt_one hb₀ hb₁).antitone
|
case inr
hb₀ : 0 < 1
hb₁ : 1 ≤ 1
⊢ Antitone fun x => 1 ^ x
|
70788659184c90ef
|
hasSum_mellin_pi_mul
|
Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
|
/-- Shortcut version for the commonly arising special case when `p i = π * q i` for some other
sequence `q`. -/
lemma hasSum_mellin_pi_mul {a : ι → ℂ} {q : ι → ℝ} {F : ℝ → ℂ} {s : ℂ}
(hq : ∀ i, a i = 0 ∨ 0 < q i) (hs : 0 < s.re)
(hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * rexp (-π * q i * t)) (F t))
(h_sum : Summable fun i ↦ ‖a i‖ / (q i) ^ s.re) :
HasSum (fun i ↦ π ^ (-s) * Gamma s * a i / q i ^ s) (mellin F s)
|
case h.e'_5.h
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
q : ι → ℝ
F : ℝ → ℂ
s : ℂ
hq : ∀ (i : ι), a i = 0 ∨ 0 < q i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(rexp (-π * q i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / q i ^ s.re
hp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i
i : ι
this : a i / ↑(π * q i) ^ s = ↑π ^ (-s) * a i / ↑(q i) ^ s
⊢ ↑π ^ (-s) * Complex.Gamma s * (a i / ↑(q i) ^ s) = Complex.Gamma s * (↑π ^ (-s) * a i / ↑(q i) ^ s)
|
ring_nf
|
no goals
|
29b92a7c6424e7e6
|
CategoryTheory.Presheaf.isLeftKanExtension_along_yoneda_iff
|
Mathlib/CategoryTheory/Limits/Presheaf.lean
|
lemma isLeftKanExtension_along_yoneda_iff :
L.IsLeftKanExtension α ↔
(IsIso α ∧ PreservesColimitsOfSize.{v₁, max u₁ v₁} L)
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝¹ : Category.{v₁, u₂} ℰ
A : C ⥤ ℰ
inst✝ : yoneda.HasPointwiseLeftKanExtension A
L : (Cᵒᵖ ⥤ Type v₁) ⥤ ℰ
α : A ⟶ yoneda ⋙ L
⊢ L.IsLeftKanExtension α ↔ IsIso α ∧ PreservesColimitsOfSize.{v₁, max u₁ v₁, max u₁ v₁, v₁, max u₁ (v₁ + 1), u₂} L
|
constructor
|
case mp
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝¹ : Category.{v₁, u₂} ℰ
A : C ⥤ ℰ
inst✝ : yoneda.HasPointwiseLeftKanExtension A
L : (Cᵒᵖ ⥤ Type v₁) ⥤ ℰ
α : A ⟶ yoneda ⋙ L
⊢ L.IsLeftKanExtension α → IsIso α ∧ PreservesColimitsOfSize.{v₁, max u₁ v₁, max u₁ v₁, v₁, max u₁ (v₁ + 1), u₂} L
case mpr
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝¹ : Category.{v₁, u₂} ℰ
A : C ⥤ ℰ
inst✝ : yoneda.HasPointwiseLeftKanExtension A
L : (Cᵒᵖ ⥤ Type v₁) ⥤ ℰ
α : A ⟶ yoneda ⋙ L
⊢ IsIso α ∧ PreservesColimitsOfSize.{v₁, max u₁ v₁, max u₁ v₁, v₁, max u₁ (v₁ + 1), u₂} L → L.IsLeftKanExtension α
|
fb09d1319e0f9b0d
|
Finset.extract_gcd
|
Mathlib/Algebra/GCDMonoid/Finset.lean
|
theorem extract_gcd (f : β → α) (hs : s.Nonempty) :
∃ g : β → α, (∀ b ∈ s, f b = s.gcd f * g b) ∧ s.gcd g = 1
|
case neg.refine_1
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Finset β
f : β → α
hs : s.Nonempty
g' : {b : β} → b ∈ s → α
hg : ∀ {b : β} (hb : b ∈ s), f b = s.gcd f * g' hb
h : ∃ x ∈ s, f x ≠ 0
b : β
hb : b ∈ s
⊢ f b = s.gcd f * (fun b => if hb : b ∈ s then g' hb else 0) b
|
simp only [hb, hg, dite_true]
|
no goals
|
330403550ff056bb
|
traceForm_dualBasis_powerBasis_eq
|
Mathlib/RingTheory/Trace/Basic.lean
|
/--
The dual basis of a powerbasis `{1, x, x²...}` under the trace form is `aᵢ / f'(x)`,
with `f` being the minimal polynomial of `x` and `f / (X - x) = ∑ aᵢxⁱ`.
-/
lemma traceForm_dualBasis_powerBasis_eq [FiniteDimensional K L] [Algebra.IsSeparable K L]
(pb : PowerBasis K L) (i) :
(Algebra.traceForm K L).dualBasis (traceForm_nondegenerate K L) pb.basis i =
(minpolyDiv K pb.gen).coeff i / aeval pb.gen (derivative <| minpoly K pb.gen)
|
case a.a
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
pb : PowerBasis K L
i j : Fin pb.dim
this :
∑ x : L →ₐ[K] AlgebraicClosure K,
x pb.gen ^ ↑j / x ((aeval pb.gen) (derivative (minpoly K pb.gen))) * x ((minpolyDiv K pb.gen).coeff ↑i) =
if j = i then 1 else 0
⊢ ∀ x ∈ Finset.univ,
x pb.gen ^ ↑j / x ((aeval pb.gen) (derivative (minpoly K pb.gen))) * x ((minpolyDiv K pb.gen).coeff ↑i) =
x ((minpolyDiv K pb.gen).coeff ↑i / (aeval pb.gen) (derivative (minpoly K pb.gen)) * pb.gen ^ ↑j)
|
intro σ _
|
case a.a
K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
pb : PowerBasis K L
i j : Fin pb.dim
this :
∑ x : L →ₐ[K] AlgebraicClosure K,
x pb.gen ^ ↑j / x ((aeval pb.gen) (derivative (minpoly K pb.gen))) * x ((minpolyDiv K pb.gen).coeff ↑i) =
if j = i then 1 else 0
σ : L →ₐ[K] AlgebraicClosure K
a✝ : σ ∈ Finset.univ
⊢ σ pb.gen ^ ↑j / σ ((aeval pb.gen) (derivative (minpoly K pb.gen))) * σ ((minpolyDiv K pb.gen).coeff ↑i) =
σ ((minpolyDiv K pb.gen).coeff ↑i / (aeval pb.gen) (derivative (minpoly K pb.gen)) * pb.gen ^ ↑j)
|
619b8227edc12ad8
|
SeparationQuotient.tendsto_lift₂_nhds
|
Mathlib/Topology/Inseparable.lean
|
theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l
|
X : Type u_1
Y : Type u_2
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf)) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l
|
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
|
X : Type u_1
Y : Type u_2
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf) ∘ Prod.map mk mk) (𝓝 (x, y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l
|
49b4006fda81687a
|
Fintype.card_filter_piFinset_eq
|
Mathlib/Data/Fintype/BigOperators.lean
|
lemma card_filter_piFinset_eq [∀ i, DecidableEq (α i)] (s : ∀ i, Finset (α i)) (i : ι) (a : α i) :
#{f ∈ piFinset s | f i = a} = if a ∈ s i then ∏ b ∈ univ.erase i, #(s b) else 0
|
ι : Type u_4
α : ι → Type u_6
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (α i)
s : (i : ι) → Finset (α i)
i : ι
a : α i
⊢ #(filter (fun f => f i = a) (piFinset s)) = if a ∈ s i then ∏ b ∈ univ.erase i, #(s b) else 0
|
split_ifs with h
|
case pos
ι : Type u_4
α : ι → Type u_6
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (α i)
s : (i : ι) → Finset (α i)
i : ι
a : α i
h : a ∈ s i
⊢ #(filter (fun f => f i = a) (piFinset s)) = ∏ b ∈ univ.erase i, #(s b)
case neg
ι : Type u_4
α : ι → Type u_6
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : (i : ι) → DecidableEq (α i)
s : (i : ι) → Finset (α i)
i : ι
a : α i
h : a ∉ s i
⊢ #(filter (fun f => f i = a) (piFinset s)) = 0
|
2fd67ab890630265
|
FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos
|
Mathlib/Analysis/Analytic/Inverse.lean
|
theorem radius_rightInv_pos_of_radius_pos
{p : FormalMultilinearSeries 𝕜 E F} {i : E ≃L[𝕜] F} {x : E}
(hp : 0 < p.radius) : 0 < (p.rightInv i x).radius
|
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
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
hp : 0 < p.radius
C r : ℝ
Cpos : 0 < C
rpos : 0 < r
ple : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
a : ℝ
apos : 0 < a
ha1 : 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a ≤ 1
ha2 : r * (I + 1) * a ≤ 1 / 2
S : ℕ → ℝ := fun n => ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
n : ℕ
one_le_n : 1 ≤ n
hn : S n ≤ (I + 1) * a
In : 2 ≤ n + 1
rSn : r * S n ≤ 1 / 2
⊢ 1 / 2 < 1
|
norm_num
|
no goals
|
8af3e9ca55cc3d57
|
Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero
|
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0)
(him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))
|
case convert_3
z : ℂ
hre : z.re < 0
him : z.im = 0
⊢ ‖z‖ ≠ 0
|
lift z to ℝ using him
|
case convert_3.intro
z : ℝ
hre : (↑z).re < 0
⊢ ‖↑z‖ ≠ 0
|
fa827d9e0ddf2875
|
HahnModule.mul_smul'
|
Mathlib/RingTheory/HahnSeries/Multiplication.lean
|
theorem mul_smul' [Semiring R] [Module R V] (x y : HahnSeries Γ R)
(z : HahnModule Γ' R V) : (x * y) • z = x • (y • z)
|
case h.h
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : OrderedCancelAddCommMonoid Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : AddAction Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Semiring R
inst✝ : Module R V
x y : HahnSeries Γ R
z : HahnModule Γ' R V
b : Γ'
⊢ ((of R).symm ((x * y) • z)).coeff b = ((of R).symm (x • y • z)).coeff b
|
rw [coeff_smul_left (x.isPWO_support.add y.isPWO_support)
HahnSeries.support_mul_subset_add_support, coeff_smul_right
(y.isPWO_support.vadd ((of R).symm z).isPWO_support) support_smul_subset_vadd_support]
|
case h.h
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : OrderedCancelAddCommMonoid Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : AddAction Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Semiring R
inst✝ : Module R V
x y : HahnSeries Γ R
z : HahnModule Γ' R V
b : Γ'
⊢ ∑ ij ∈ VAddAntidiagonal ⋯ ⋯ b, (x * y).coeff ij.1 • ((of R).symm z).coeff ij.2 =
∑ ij ∈ VAddAntidiagonal ⋯ ⋯ b, x.coeff ij.1 • ((of R).symm (y • z)).coeff ij.2
|
948382d12ebb34ef
|
Multiset.noncommProd_coe
|
Mathlib/Data/Finset/NoncommProd.lean
|
theorem noncommProd_coe (l : List α) (comm) : noncommProd (l : Multiset α) comm = l.prod
|
case cons
α : Type u_3
inst✝ : Monoid α
hd : α
tl : List α
hl : {x | x ∈ ↑tl}.Pairwise Commute → List.foldr (fun x1 x2 => x1 * x2) 1 tl = tl.prod
comm : {x | x ∈ ↑(hd :: tl)}.Pairwise Commute
⊢ List.foldr (fun x1 x2 => x1 * x2) 1 (hd :: tl) = (hd :: tl).prod
|
rw [List.prod_cons, List.foldr, hl]
|
case cons
α : Type u_3
inst✝ : Monoid α
hd : α
tl : List α
hl : {x | x ∈ ↑tl}.Pairwise Commute → List.foldr (fun x1 x2 => x1 * x2) 1 tl = tl.prod
comm : {x | x ∈ ↑(hd :: tl)}.Pairwise Commute
⊢ {x | x ∈ ↑tl}.Pairwise Commute
|
61949c74a00986e6
|
ProbabilityTheory.gaussianReal_map_const_mul
|
Mathlib/Probability/Distributions/Gaussian.lean
|
/-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v)
|
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
⊢ ∀ (x : ℝ), HasDerivAt (fun x => c⁻¹ * x) (c⁻¹ * 1) x
|
exact fun _ ↦ HasDerivAt.const_mul _ (hasDerivAt_id _)
|
no goals
|
1ceed6d297abcf77
|
refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set
|
Mathlib/Topology/Compactness/Paracompact.lean
|
theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [WeaklyLocallyCompactSpace X]
[SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
{s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
∃ (α : Type v) (c : α → X) (r : ∀ a, ι (c a)),
(∀ a, c a ∈ s ∧ p (c a) (r a)) ∧
(s ⊆ ⋃ a, B (c a) (r a)) ∧ LocallyFinite fun a ↦ B (c a) (r a)
|
case refine_1.mk.mk
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.choice X
K : CompactExhaustion X := K'.shiftr.shiftr
Kdiff : ℕ → Set X := fun n => K (n + 1) \ interior (K n)
hKcov : ∀ (x : X), x ∈ Kdiff (K'.find x + 1)
Kdiffc : ∀ (n : ℕ), IsCompact (Kdiff n ∩ s)
this : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), (K n)ᶜ ∈ 𝓝 ↑x
r : (n : ℕ) → (x : ↑(Kdiff (n + 1) ∩ s)) → ι ↑x
hrp : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), p (↑x) (r n x)
hr : ∀ (n : ℕ) (x : ↑(Kdiff (n + 1) ∩ s)), B (↑x) (r n x) ⊆ (K n)ᶜ
hxr : ∀ (n : ℕ) (x : X) (hx : x ∈ Kdiff (n + 1) ∩ s), B x (r n ⟨x, hx⟩) ∈ 𝓝 x
T : (n : ℕ) → Finset ↑(Kdiff (n + 1) ∩ s)
hT : ∀ (n : ℕ), Kdiff (n + 1) ∩ s ⊆ ⋃ x ∈ T n, B (↑x) (r n ⟨↑x, ⋯⟩)
T' : (n : ℕ) → Set ↑(Kdiff (n + 1) ∩ s) := fun n => ↑(T n)
n : ℕ
x : ↑(Kdiff (n + 1) ∩ s)
hx : x ∈ T' n
⊢ (fun a => ↑↑a.snd) ⟨n, ⟨x, hx⟩⟩ ∈ s ∧ p ((fun a => ↑↑a.snd) ⟨n, ⟨x, hx⟩⟩) ((fun a => r a.fst ↑a.snd) ⟨n, ⟨x, hx⟩⟩)
|
exact ⟨x.2.2, hrp _ _⟩
|
no goals
|
d7d8360667f4b953
|
Real.Gamma_ne_zero
|
Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
|
theorem Gamma_ne_zero {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0
|
case zero
s : ℝ
hs✝ : ∀ (m : ℕ), s ≠ -↑m
hs : -↑0 < s
⊢ 0 < s
|
rwa [Nat.cast_zero, neg_zero] at hs
|
no goals
|
b27fef48da598597
|
GenContFract.convs_eq_convs'
|
Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
|
theorem convs_eq_convs' [LinearOrderedField K]
(s_pos : ∀ {gp : Pair K} {m : ℕ}, m < n → g.s.get? m = some gp → 0 < gp.a ∧ 0 < gp.b) :
g.convs n = g.convs' n
|
case refl.intro.intro
K : Type u_1
inst✝ : LinearOrderedField K
g : GenContFract K
gp' : Pair K
m : ℕ
IH :
∀ {g : GenContFract K},
(∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b) →
g.convs m.succ = g.convs' m.succ
s_pos : ∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ + 1 → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b
g' : GenContFract K := g.squashGCF m.succ
not_terminatedAt_n : ¬g.TerminatedAt m.succ
this : g.convs (m.succ + 1) = g'.convs m.succ
s_mth_eq' : g'.s.get? m = some gp'
gp_succ_m : Pair K
s_succ_mth_eq : g.s.get? (m + 1) = some gp_succ_m
gp_m : Pair K
mth_s_eq : g.s.get? m = some gp_m
m_lt_n : m < m.succ
⊢ 0 < gp_succ_m.a / gp_succ_m.b
|
have : 0 < gp_succ_m.a ∧ 0 < gp_succ_m.b := s_pos (lt_add_one <| m + 1) s_succ_mth_eq
|
case refl.intro.intro
K : Type u_1
inst✝ : LinearOrderedField K
g : GenContFract K
gp' : Pair K
m : ℕ
IH :
∀ {g : GenContFract K},
(∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b) →
g.convs m.succ = g.convs' m.succ
s_pos : ∀ {gp : Pair K} {m_1 : ℕ}, m_1 < m.succ + 1 → g.s.get? m_1 = some gp → 0 < gp.a ∧ 0 < gp.b
g' : GenContFract K := g.squashGCF m.succ
not_terminatedAt_n : ¬g.TerminatedAt m.succ
this✝ : g.convs (m.succ + 1) = g'.convs m.succ
s_mth_eq' : g'.s.get? m = some gp'
gp_succ_m : Pair K
s_succ_mth_eq : g.s.get? (m + 1) = some gp_succ_m
gp_m : Pair K
mth_s_eq : g.s.get? m = some gp_m
m_lt_n : m < m.succ
this : 0 < gp_succ_m.a ∧ 0 < gp_succ_m.b
⊢ 0 < gp_succ_m.a / gp_succ_m.b
|
b83d55996ff7dd7d
|
Real.Gamma_ne_zero
|
Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
|
theorem Gamma_ne_zero {s : ℝ} (hs : ∀ m : ℕ, s ≠ -m) : Gamma s ≠ 0
|
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ Gamma s ≠ 0
|
apply this
|
case a
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ -↑?n < s
case n
s : ℝ
hs : ∀ (m : ℕ), s ≠ -↑m
this : ∀ {n : ℕ}, -↑n < s → Gamma s ≠ 0
⊢ ℕ
|
b27fef48da598597
|
isBoundedUnder_le_finset_sup
|
Mathlib/Order/LiminfLimsup.lean
|
theorem isBoundedUnder_le_finset_sup [LinearOrder β] [OrderBot β] {f : Filter α} {F : ι → α → β}
{s : Finset ι} (h : ∀ i ∈ s, f.IsBoundedUnder (· ≤ ·) (F i)) :
f.IsBoundedUnder (· ≤ ·) (fun a ↦ sup s (fun i ↦ F i a))
|
case h
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : LinearOrder β
inst✝ : OrderBot β
f : Filter α
F : ι → α → β
s : Finset ι
m : ι → β
hm : ∀ i ∈ s, ∀ᶠ (x : β) in Filter.map (F i) f, (fun x1 x2 => x1 ≤ x2) x (m i)
⊢ ∀ᶠ (x : β) in Filter.map (fun a => s.sup fun i => F i a) f, (fun x1 x2 => x1 ≤ x2) x (s.sup m)
|
simp only [eventually_map] at hm ⊢
|
case h
α : Type u_1
β : Type u_2
ι : Type u_4
inst✝¹ : LinearOrder β
inst✝ : OrderBot β
f : Filter α
F : ι → α → β
s : Finset ι
m : ι → β
hm : ∀ i ∈ s, ∀ᶠ (a : α) in f, F i a ≤ m i
⊢ ∀ᶠ (a : α) in f, (s.sup fun i => F i a) ≤ s.sup m
|
2c89705f49896138
|
WittVector.iterate_frobenius_coeff
|
Mathlib/RingTheory/WittVector/Identities.lean
|
theorem iterate_frobenius_coeff (x : 𝕎 R) (i k : ℕ) :
(frobenius^[i] x).coeff k = x.coeff k ^ p ^ i
|
case zero
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
x : 𝕎 R
k : ℕ
⊢ ((⇑frobenius)^[0] x).coeff k = x.coeff k ^ p ^ 0
|
simp
|
no goals
|
c214fc875b9b7f93
|
LinearMap.charpoly_nilpotent_tfae
|
Mathlib/LinearAlgebra/Eigenspace/Zero.lean
|
lemma charpoly_nilpotent_tfae [IsNoetherian R M] (φ : Module.End R M) :
List.TFAE [
IsNilpotent φ,
φ.charpoly = X ^ finrank R M,
∀ m : M, ∃ (n : ℕ), (φ ^ n) m = 0,
natTrailingDegree φ.charpoly = finrank R M ]
|
R : Type u_1
M : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Finite R M
inst✝¹ : Free R M
inst✝ : IsNoetherian R M
φ : End R M
⊢ [IsNilpotent φ, charpoly φ = X ^ finrank R M, ∀ (m : M), ∃ n, (φ ^ n) m = 0,
(charpoly φ).natTrailingDegree = finrank R M].TFAE
|
tfae_have 1 → 2 := IsNilpotent.charpoly_eq_X_pow_finrank
|
R : Type u_1
M : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : IsDomain R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module.Finite R M
inst✝¹ : Free R M
inst✝ : IsNoetherian R M
φ : End R M
tfae_1_to_2 : IsNilpotent φ → charpoly φ = X ^ finrank R M
⊢ [IsNilpotent φ, charpoly φ = X ^ finrank R M, ∀ (m : M), ∃ n, (φ ^ n) m = 0,
(charpoly φ).natTrailingDegree = finrank R M].TFAE
|
9c8228786990f11d
|
Char.card_pow_card
|
Mathlib/NumberTheory/GaussSum.lean
|
theorem Char.card_pow_card {F : Type*} [Field F] [Fintype F] {F' : Type*} [Field F'] [Fintype F']
{χ : MulChar F F'} (hχ₁ : χ ≠ 1) (hχ₂ : IsQuadratic χ)
(hch₁ : ringChar F' ≠ ringChar F) (hch₂ : ringChar F' ≠ 2) :
(χ (-1) * Fintype.card F) ^ (Fintype.card F' / 2) = χ (Fintype.card F')
|
case intro.intro.intro.intro.a
F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁✝ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' : Nat.Prime (ringChar F')
hc' : Fintype.card F' = ringChar F' ^ ↑n'
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' hch₁✝
FF' : Type u_2 := CyclotomicField ψ.n F'
hch₁ : IsUnit ↑(ringChar FF')
hchar : ringChar F' = ringChar FF'
this✝ : Fact (Nat.Prime (ringChar F'))
this : Fact (Nat.Prime (ringChar FF'))
⊢ ((χ.ringHomComp (algebraMap F' FF')) (-1) * ↑(Fintype.card F)) ^ (ringChar FF' ^ ↑n' / 2) =
(χ.ringHomComp (algebraMap F' FF')) (↑(ringChar FF') ^ ↑n')
|
exact Char.card_pow_char_pow (hχ₂.comp _) ψ.char (ringChar FF') n' hch₁ (hchar ▸ hch₂)
(gaussSum_sq ((ringHomComp_ne_one_iff (RingHom.injective _)).mpr hχ₁) (hχ₂.comp _) ψ.prim)
|
no goals
|
e01707f133d9b7bd
|
Stream'.WSeq.exists_of_mem_join
|
Mathlib/Data/Seq/WSeq.lean
|
theorem exists_of_mem_join {a : α} : ∀ {S : WSeq (WSeq α)}, a ∈ join S → ∃ s, s ∈ S ∧ a ∈ s
|
case h2.h1.h2.h.inl
α : Type u
a : α
ss : WSeq α
h : a ∈ ss
s : WSeq α
S : WSeq (WSeq α)
IH :
∀ (s_1 : WSeq α) (S_1 : WSeq (WSeq α)),
s_1.append S_1.join = s.append S.join → a ∈ s_1.append S_1.join → a ∈ s_1 ∨ ∃ s, s ∈ S_1 ∧ a ∈ s
ej : (s.append S.join).think = (s.append S.join).think
m : a ∈ s.append S.join
as : a ∈ s
⊢ ∃ s_1, (s_1 = s ∨ s_1 ∈ S) ∧ a ∈ s_1
|
exact ⟨s, Or.inl rfl, as⟩
|
no goals
|
ec68064248c6eec8
|
Turing.ToPartrec.cont_eval_fix
|
Mathlib/Computability/TMConfig.lean
|
theorem cont_eval_fix {f k v} (fok : Code.Ok f) :
Turing.eval step (stepNormal f (Cont.fix f k) v) =
f.fix.eval v >>= fun v => Turing.eval step (Cfg.ret k v)
|
case pos
f : Code
k : Cont
v : List ℕ
fok : f.Ok
x : Cfg
this :
∀ (c : Cfg),
x ∈ eval step c →
∀ (v : List ℕ) (c' : Cfg),
c = c'.then (Cont.fix f k) →
Reaches step (stepNormal f Cont.halt v) c' →
∃ v₁ ∈ f.eval v,
∃ v₂ ∈ if v₁.headI = 0 then pure v₁.tail else f.fix.eval v₁.tail, x ∈ eval step (Cfg.ret k v₂)
h : x ∈ eval step (stepNormal f (Cont.fix f k) v)
v₁ : List ℕ
hv₁ : v₁ ∈ f.eval v
v₂ : List ℕ
h₃ : x ∈ eval step (Cfg.ret k v₂)
h✝ : v₁.headI = 0
hv₂ : v₂ ∈ pure v₁.tail
⊢ Sum.inl v₂ ∈ Part.some (Sum.inl v₁.tail) ∨
∃ a',
Sum.inr a' ∈ Part.some (Sum.inl v₁.tail) ∧
v₂ ∈
PFun.fix (fun v => Part.map (fun v => if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v)) a'
|
rw [Part.mem_some_iff.1 hv₂]
|
case pos
f : Code
k : Cont
v : List ℕ
fok : f.Ok
x : Cfg
this :
∀ (c : Cfg),
x ∈ eval step c →
∀ (v : List ℕ) (c' : Cfg),
c = c'.then (Cont.fix f k) →
Reaches step (stepNormal f Cont.halt v) c' →
∃ v₁ ∈ f.eval v,
∃ v₂ ∈ if v₁.headI = 0 then pure v₁.tail else f.fix.eval v₁.tail, x ∈ eval step (Cfg.ret k v₂)
h : x ∈ eval step (stepNormal f (Cont.fix f k) v)
v₁ : List ℕ
hv₁ : v₁ ∈ f.eval v
v₂ : List ℕ
h₃ : x ∈ eval step (Cfg.ret k v₂)
h✝ : v₁.headI = 0
hv₂ : v₂ ∈ pure v₁.tail
⊢ Sum.inl v₁.tail ∈ Part.some (Sum.inl v₁.tail) ∨
∃ a',
Sum.inr a' ∈ Part.some (Sum.inl v₁.tail) ∧
v₁.tail ∈
PFun.fix (fun v => Part.map (fun v => if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v)) a'
|
4a2318a05b9f1866
|
Stream'.WSeq.mem_rec_on
|
Mathlib/Data/Seq/WSeq.lean
|
theorem mem_rec_on {C : WSeq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s'))
(h2 : ∀ s, C s → C (think s)) : C s
|
case none.a.inl
α : Type u
C : WSeq α → Prop
a : α
s : WSeq α
M : a ∈ s
h1 : ∀ (b : α) (s' : WSeq α), a = b ∨ C s' → C (cons b s')
h2 : ∀ (s : WSeq α), C s → C s.think
s' : Seq (Option α)
h✝ : some a = none
⊢ C s'
|
contradiction
|
no goals
|
33df928e7244e2c7
|
padicValRat_two_harmonic
|
Mathlib/NumberTheory/Harmonic/Int.lean
|
theorem padicValRat_two_harmonic (n : ℕ) : padicValRat 2 (harmonic n) = -Nat.log 2 n
|
n : ℕ
ih : padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)
hn : n ≠ 0
⊢ padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹
|
rw [ih, padicValRat.inv, padicValRat.of_nat, Ne, neg_inj, Nat.cast_inj]
|
n : ℕ
ih : padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)
hn : n ≠ 0
⊢ ¬Nat.log 2 n = padicValNat 2 (n + 1)
|
058460957d5d8ddd
|
EReal.mul_eq_top
|
Mathlib/Data/Real/EReal.lean
|
lemma mul_eq_top (a b : EReal) :
a * b = ⊤ ↔ (a = ⊥ ∧ b < 0) ∨ (a < 0 ∧ b = ⊥) ∨ (a = ⊤ ∧ 0 < b) ∨ (0 < a ∧ b = ⊤)
|
case symm.refine_2.inr.inr
x✝ y✝ : EReal
h✝ : x✝ * y✝ = ⊤ ↔ x✝ = ⊥ ∧ y✝ < 0 ∨ x✝ < 0 ∧ y✝ = ⊥ ∨ x✝ = ⊤ ∧ 0 < y✝ ∨ 0 < x✝ ∧ y✝ = ⊤
h : y✝ = ⊤ ∧ 0 < x✝ ∨ 0 < y✝ ∧ x✝ = ⊤
⊢ x✝ = ⊥ ∧ y✝ < 0 ∨ x✝ < 0 ∧ y✝ = ⊥ ∨ x✝ = ⊤ ∧ 0 < y✝ ∨ 0 < x✝ ∧ y✝ = ⊤
|
cases h with
| inl h => exact Or.inr (Or.inr (Or.inr ⟨h.2, h.1⟩))
| inr h => exact Or.inr (Or.inr (Or.inl ⟨h.2, h.1⟩))
|
no goals
|
5482c4c67a9994f2
|
PNat.dvd_iff
|
Mathlib/Data/PNat/Basic.lean
|
theorem dvd_iff {k m : ℕ+} : k ∣ m ↔ (k : ℕ) ∣ (m : ℕ)
|
case mp
k m : ℕ+
h : k ∣ m
⊢ ↑k ∣ ↑m
|
rcases h with ⟨_, rfl⟩
|
case mp.intro
k w✝ : ℕ+
⊢ ↑k ∣ ↑(k * w✝)
|
35c69359256e7052
|
Function.FactorsThrough.extend_apply
|
Mathlib/Logic/Function/Basic.lean
|
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a
|
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
hf : FactorsThrough g f
e' : β → γ
a : α
⊢ extend f g e' (f a) = g a
|
simp only [extend_def, dif_pos, exists_apply_eq_apply]
|
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
hf : FactorsThrough g f
e' : β → γ
a : α
⊢ g (Classical.choose ⋯) = g a
|
c7f09213ea3f0a7a
|
Filter.comap_snd_neBot_iff
|
Mathlib/Order/Filter/Map.lean
|
theorem comap_snd_neBot_iff {f : Filter β} :
(f.comap (Prod.snd : α × β → β)).NeBot ↔ Nonempty α ∧ f.NeBot
|
case inl
α : Type u_1
β : Type u_2
f : Filter β
hα : IsEmpty α
⊢ ¬⊥.NeBot ↔ ¬(Nonempty α ∧ f.NeBot)
|
simp
|
no goals
|
af15d3b1c17460fe
|
IsometryEquiv.midpoint_fixed
|
Mathlib/Analysis/Normed/Affine/MazurUlam.lean
|
theorem midpoint_fixed {x y : PE} :
∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y
|
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z
|
set f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
|
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z
|
b718555705b87dec
|
Ring.jacobson_le_of_eq_bot
|
Mathlib/RingTheory/Jacobson/Radical.lean
|
theorem jacobson_le_of_eq_bot {I : Ideal R} [I.IsTwoSided] (h : jacobson (R ⧸ I) = ⊥) :
jacobson R ≤ I :=
Module.jacobson_le_of_eq_bot <| by
rw [← le_bot_iff, ← SetLike.coe_subset_coe] at h ⊢
rwa [← coe_jacobson_quotient]
|
R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
h : jacobson (R ⧸ I) = ⊥
⊢ Module.jacobson R (R ⧸ I) = ⊥
|
rw [← le_bot_iff, ← SetLike.coe_subset_coe] at h ⊢
|
R : Type u_1
inst✝¹ : Ring R
I : Ideal R
inst✝ : I.IsTwoSided
h : ↑(jacobson (R ⧸ I)) ⊆ ↑⊥
⊢ ↑(Module.jacobson R (R ⧸ I)) ⊆ ↑⊥
|
dbebedd335a09a0b
|
IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one
|
Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean
|
theorem bot_eq_top_of_finrank_adjoin_eq_one (h : ∀ x : E, finrank F F⟮x⟯ = 1) :
(⊥ : IntermediateField F E) = ⊤
|
case h
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : ∀ (x : E), finrank F ↥F⟮x⟯ = 1
y : E
⊢ y ∈ ⊥ ↔ y ∈ ⊤
|
rw [iff_true_right IntermediateField.mem_top]
|
case h
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : ∀ (x : E), finrank F ↥F⟮x⟯ = 1
y : E
⊢ y ∈ ⊥
|
ff249bea377d24b1
|
IsLocalization.AtPrime.comap_maximalIdeal
|
Mathlib/RingTheory/Localization/AtPrime.lean
|
theorem comap_maximalIdeal (h : IsLocalRing S := isLocalRing S I) :
(IsLocalRing.maximalIdeal S).comap (algebraMap R S) = I :=
Ideal.ext fun x => by simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
|
R : Type u_1
inst✝³ : CommSemiring R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
I : Ideal R
hI : I.IsPrime
inst✝ : IsLocalization.AtPrime S I
h : optParam (IsLocalRing S) ⋯
x : R
⊢ x ∈ Ideal.comap (algebraMap R S) (IsLocalRing.maximalIdeal S) ↔ x ∈ I
|
simpa only [Ideal.mem_comap] using to_map_mem_maximal_iff _ I x
|
no goals
|
b881c82fa6463eb9
|
MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι}
[l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞)
(hF_meas : ∀ᶠ n in l, Measurable (F n)) (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, F n a ≤ bound a)
(h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) :
Tendsto (fun n => ∫⁻ a, F n a ∂μ) l (𝓝 <| ∫⁻ a, f a ∂μ)
|
case intro.refine_5.hg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_5
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤
h_lim✝ : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n => Measurable (F n)) x} ∩ {x | (fun n => ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x}
a : α
h_lim : Tendsto (fun n => F n a) l (𝓝 (f a))
⊢ Tendsto (fun n => F n a) ?intro.refine_5.y (𝓝 (f a))
|
assumption
|
no goals
|
879da8b2b29c96c4
|
Subalgebra.isSimpleOrder_of_finrank
|
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
theorem Subalgebra.isSimpleOrder_of_finrank (hr : finrank F E = 2) :
IsSimpleOrder (Subalgebra F E) :=
let i := nontrivial_of_finrank_pos (zero_lt_two.trans_eq hr.symm)
{ toNontrivial :=
⟨⟨⊥, ⊤, fun h => by cases hr.symm.trans (Subalgebra.bot_eq_top_iff_finrank_eq_one.1 h)⟩⟩
eq_bot_or_eq_top
|
case «1».h
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Ring E
inst✝ : Algebra F E
hr : finrank F E = 2
i : Nontrivial E := nontrivial_of_finrank_pos (LT.lt.trans_eq zero_lt_two (Eq.symm hr))
S : Subalgebra F E
this✝² : FiniteDimensional F E
this✝¹ : FiniteDimensional F ↥S
h : finrank F ↥S = 1
this✝ : 1 ≤ 2
this : 0 < 1
⊢ S = ⊥
|
exact Subalgebra.eq_bot_of_finrank_one h
|
no goals
|
c7e4894f93ed9891
|
TopCat.Presheaf.stalkToFiber_injective
|
Mathlib/Topology/Sheaves/Sheafify.lean
|
theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x)
|
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
e : F.germ U' x ⋯ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU (iU x_2)) x_1 = F.germ U' ↑x_1 ⋯ gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV (iV x_2)) x_1 = F.germ V' ↑x_1 ⋯ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU (iU w) = fV (iV w)
|
have wVx := wV ⟨x, mV⟩
|
case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj (op U')
e : F.germ U' x ⋯ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU (iU x_2)) x_1 = F.germ U' ↑x_1 ⋯ gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj (op V')
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV (iV x_2)) x_1 = F.germ V' ↑x_1 ⋯ gV
wVx : (fun x_1 => fV (iV x_1)) ⟨x, mV⟩ = F.germ V' ↑⟨x, mV⟩ ⋯ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU (iU w) = fV (iV w)
|
c3bddb63a4c72896
|
Classical.em
|
Mathlib/.lake/packages/lean4/src/lean/Init/Classical.lean
|
theorem em (p : Prop) : p ∨ ¬p :=
let U (x : Prop) : Prop := x = True ∨ p
let V (x : Prop) : Prop := x = False ∨ p
have exU : ∃ x, U x := ⟨True, Or.inl rfl⟩
have exV : ∃ x, V x := ⟨False, Or.inl rfl⟩
let u : Prop := choose exU
let v : Prop := choose exV
have u_def : U u := choose_spec exU
have v_def : V v := choose_spec exV
have not_uv_or_p : u ≠ v ∨ p :=
match u_def, v_def with
| Or.inr h, _ => Or.inr h
| _, Or.inr h => Or.inr h
| Or.inl hut, Or.inl hvf =>
have hne : u ≠ v
|
p : Prop
U : Prop → Prop := fun x => x = True ∨ p
V : Prop → Prop := fun x => x = False ∨ p
exU : ∃ x, U x
exV : ∃ x, V x
u : Prop := choose exU
v : Prop := choose exV
u_def : U u
v_def : V v
not_uv_or_p : u ≠ v ∨ p
hp : p
hpred : U = V
⊢ ∀ (exU exV : ∃ x, V x), choose exU = choose exV
|
intros
|
p : Prop
U : Prop → Prop := fun x => x = True ∨ p
V : Prop → Prop := fun x => x = False ∨ p
exU : ∃ x, U x
exV : ∃ x, V x
u : Prop := choose exU
v : Prop := choose exV
u_def : U u
v_def : V v
not_uv_or_p : u ≠ v ∨ p
hp : p
hpred : U = V
exU✝ exV✝ : ∃ x, V x
⊢ choose exU✝ = choose exV✝
|
70edce7f91918fea
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastRotateRight.go_get
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/RotateRight.lean
|
theorem go_get (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
(go input distance curr hcurr s).get idx hidx1
=
if hidx3 : idx < w - distance % w then
input.get ((distance % w) + idx) (by omega)
else
input.get (idx - (w - (distance % w))) (by omega)
|
case isTrue.inl.isFalse.isFalse
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
hidx1 : idx < w
hidx2 : curr ≤ idx
h✝² : curr < w
heq : curr = idx
h✝¹ : ¬curr < w - distance % w
h✝ : ¬idx < w - distance % w
⊢ (go input distance (curr + 1) ⋯ (s.push (input.get (curr - (w - distance % w)) ⋯))).get idx hidx1 =
input.get (idx - (w - distance % w)) ⋯
|
rw [go_get_aux]
|
case isTrue.inl.isFalse.isFalse
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
hidx1 : idx < w
hidx2 : curr ≤ idx
h✝² : curr < w
heq : curr = idx
h✝¹ : ¬curr < w - distance % w
h✝ : ¬idx < w - distance % w
⊢ (s.push (input.get (curr - (w - distance % w)) ⋯)).get idx ?isTrue.inl.isFalse.isFalse.hidx =
input.get (idx - (w - distance % w)) ⋯
case isTrue.inl.isFalse.isFalse.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
hidx1 : idx < w
hidx2 : curr ≤ idx
h✝² : curr < w
heq : curr = idx
h✝¹ : ¬curr < w - distance % w
h✝ : ¬idx < w - distance % w
⊢ idx < curr + 1
|
24de4b0f184e4333
|
ProperSpace.of_nontriviallyNormedField_of_weaklyLocallyCompactSpace
|
Mathlib/Analysis/Normed/Field/ProperSpace.lean
|
/-- A weakly locally compact normed field is proper.
This is a specialization of `ProperSpace.of_locallyCompactSpace`
which holds for `NormedSpace`s but requires more imports. -/
lemma ProperSpace.of_nontriviallyNormedField_of_weaklyLocallyCompactSpace
(𝕜 : Type*) [NontriviallyNormedField 𝕜] [WeaklyLocallyCompactSpace 𝕜] :
ProperSpace 𝕜
|
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : WeaklyLocallyCompactSpace 𝕜
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
this : c ^ n ≠ 0
⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))
|
convert hr.smul (c ^ n)
|
case h.e'_3
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : WeaklyLocallyCompactSpace 𝕜
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
this : c ^ n ≠ 0
⊢ closedBall 0 (‖c‖ ^ n * r) = c ^ n • closedBall 0 r
|
095c2cdeb26cc130
|
MeasureTheory.Measure.ae_ae_comm
|
Mathlib/MeasureTheory/Measure/Prod.lean
|
theorem ae_ae_comm {p : α → β → Prop} (h : MeasurableSet {x : α × β | p x.1 x.2}) :
(∀ᵐ x ∂μ, ∀ᵐ y ∂ν, p x y) ↔ ∀ᵐ y ∂ν, ∀ᵐ x ∂μ, p x y := calc
_ ↔ ∀ᵐ x ∂μ.prod ν, p x.1 x.2 := .symm <| ae_prod_iff_ae_ae h
_ ↔ ∀ᵐ x ∂ν.prod μ, p x.2 x.1
|
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
p : α → β → Prop
h : MeasurableSet {x | p x.1 x.2}
⊢ (∀ᵐ (x : β × α) ∂ν.prod μ, p x.swap.1 x.swap.2) ↔ ∀ᵐ (x : β × α) ∂ν.prod μ, p x.2 x.1
|
simp
|
no goals
|
6daedaed0d9d67c4
|
Equiv.Perm.Disjoint.commute
|
Mathlib/GroupTheory/Perm/Support.lean
|
theorem Disjoint.commute (h : Disjoint f g) : Commute f g :=
Equiv.ext fun x =>
(h x).elim
(fun hf =>
(h (g x)).elim (fun hg => by simp [mul_apply, hf, hg]) fun hg => by
simp [mul_apply, hf, g.injective hg])
fun hg =>
(h (f x)).elim (fun hf => by simp [mul_apply, f.injective hf, hg]) fun hf => by
simp [mul_apply, hf, hg]
|
α : Type u_1
f g : Perm α
h : f.Disjoint g
x : α
hg : g x = x
hf : f (f x) = f x
⊢ (f * g) x = (g * f) x
|
simp [mul_apply, f.injective hf, hg]
|
no goals
|
6be8e54c61f441bb
|
nonempty_sections_of_finite_cofiltered_system.init
|
Mathlib/CategoryTheory/CofilteredSystem.lean
|
theorem nonempty_sections_of_finite_cofiltered_system.init {J : Type u} [SmallCategory J]
[IsCofilteredOrEmpty J] (F : J ⥤ Type u) [hf : ∀ j, Finite (F.obj j)]
[hne : ∀ j, Nonempty (F.obj j)] : F.sections.Nonempty
|
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsCofilteredOrEmpty J
F : J ⥤ Type u
hf : ∀ (j : J), Finite (F.obj j)
hne : ∀ (j : J), Nonempty (F.obj j)
⊢ F.sections.Nonempty
|
let F' : J ⥤ TopCat := F ⋙ TopCat.discrete
|
J : Type u
inst✝¹ : SmallCategory J
inst✝ : IsCofilteredOrEmpty J
F : J ⥤ Type u
hf : ∀ (j : J), Finite (F.obj j)
hne : ∀ (j : J), Nonempty (F.obj j)
F' : J ⥤ TopCat := F ⋙ TopCat.discrete
⊢ F.sections.Nonempty
|
1c6bf88426a36d9a
|
Complex.arg_mul_cos_add_sin_mul_I
|
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ
|
case pos
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Ioc (-π) π
h₁ : θ ∈ Icc (-(π / 2)) (π / 2)
⊢ (if 0 ≤ Real.cos θ then Real.arcsin (Real.sin θ)
else if 0 ≤ Real.sin θ then Real.arcsin (-Real.sin θ) + π else Real.arcsin (-Real.sin θ) - π) =
θ
|
rw [if_pos]
|
case pos
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Ioc (-π) π
h₁ : θ ∈ Icc (-(π / 2)) (π / 2)
⊢ Real.arcsin (Real.sin θ) = θ
case pos.hc
r : ℝ
hr : 0 < r
θ : ℝ
hθ : θ ∈ Ioc (-π) π
h₁ : θ ∈ Icc (-(π / 2)) (π / 2)
⊢ 0 ≤ Real.cos θ
|
b22411895133f55c
|
Dense.isSeparable_iff
|
Mathlib/Topology/Bases.lean
|
theorem _root_.Dense.isSeparable_iff (hs : Dense s) :
IsSeparable s ↔ SeparableSpace α
|
α : Type u
t : TopologicalSpace α
s : Set α
hs : Dense s
⊢ IsSeparable s ↔ SeparableSpace α
|
simp_rw [IsSeparable, separableSpace_iff, dense_iff_closure_eq, ← univ_subset_iff,
← hs.closure_eq, isClosed_closure.closure_subset_iff]
|
no goals
|
af1336bf85955924
|
hasStrictDerivAt_abs
|
Mathlib/Analysis/Calculus/Deriv/Abs.lean
|
theorem hasStrictDerivAt_abs {x : ℝ} (hx : x ≠ 0) :
HasStrictDerivAt (|·|) (SignType.sign x : ℝ) x
|
x : ℝ
hx : x ≠ 0
⊢ HasStrictDerivAt (fun x => |x|) (↑(SignType.sign x)) x
|
obtain hx | hx := hx.lt_or_lt
|
case inl
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ HasStrictDerivAt (fun x => |x|) (↑(SignType.sign x)) x
case inr
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
⊢ HasStrictDerivAt (fun x => |x|) (↑(SignType.sign x)) x
|
47b4b623c1f919c3
|
LinearPMap.sup_aux
|
Mathlib/LinearAlgebra/LinearPMap.lean
|
theorem sup_aux (f g : E →ₗ.[R] F)
(h : ∀ (x : f.domain) (y : g.domain), (x : E) = y → f x = g y) :
∃ fg : ↥(f.domain ⊔ g.domain) →ₗ[R] F,
∀ (x : f.domain) (y : g.domain) (z : ↥(f.domain ⊔ g.domain)),
(x : E) + y = ↑z → fg z = f x + g y
|
case w.refine_1.mk.mk
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f g : E →ₗ.[R] F
h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y
x : ↥(f.domain ⊔ g.domain) → E
hx : ∀ (z : ↥(f.domain ⊔ g.domain)), x z ∈ f.domain
y : ↥(f.domain ⊔ g.domain) → E
hy : ∀ (z : ↥(f.domain ⊔ g.domain)), y z ∈ g.domain
hxy : ∀ (z : ↥(f.domain ⊔ g.domain)), x z + y z = ↑z
fg : ↥(f.domain ⊔ g.domain) → F := fun z => ↑f ⟨x z, ⋯⟩ + ↑g ⟨y z, ⋯⟩
fg_eq : ∀ (x' : ↥f.domain) (y' : ↥g.domain) (z' : ↥(f.domain ⊔ g.domain)), ↑x' + ↑y' = ↑z' → fg z' = ↑f x' + ↑g y'
z₁ : E
hz₁ : z₁ ∈ f.domain ⊔ g.domain
z₂ : E
hz₂ : z₂ ∈ f.domain ⊔ g.domain
⊢ fg (⟨z₁, hz₁⟩ + ⟨z₂, hz₂⟩) = fg ⟨z₁, hz₁⟩ + fg ⟨z₂, hz₂⟩
|
rw [← add_assoc, add_right_comm (f _), ← map_add, add_assoc, ← map_add]
|
case w.refine_1.mk.mk
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f g : E →ₗ.[R] F
h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y
x : ↥(f.domain ⊔ g.domain) → E
hx : ∀ (z : ↥(f.domain ⊔ g.domain)), x z ∈ f.domain
y : ↥(f.domain ⊔ g.domain) → E
hy : ∀ (z : ↥(f.domain ⊔ g.domain)), y z ∈ g.domain
hxy : ∀ (z : ↥(f.domain ⊔ g.domain)), x z + y z = ↑z
fg : ↥(f.domain ⊔ g.domain) → F := fun z => ↑f ⟨x z, ⋯⟩ + ↑g ⟨y z, ⋯⟩
fg_eq : ∀ (x' : ↥f.domain) (y' : ↥g.domain) (z' : ↥(f.domain ⊔ g.domain)), ↑x' + ↑y' = ↑z' → fg z' = ↑f x' + ↑g y'
z₁ : E
hz₁ : z₁ ∈ f.domain ⊔ g.domain
z₂ : E
hz₂ : z₂ ∈ f.domain ⊔ g.domain
⊢ fg (⟨z₁, hz₁⟩ + ⟨z₂, hz₂⟩) = ↑f (⟨x ⟨z₁, hz₁⟩, ⋯⟩ + ⟨x ⟨z₂, hz₂⟩, ⋯⟩) + ↑g (⟨y ⟨z₁, hz₁⟩, ⋯⟩ + ⟨y ⟨z₂, hz₂⟩, ⋯⟩)
|
0acdd19d93971ad5
|
CategoryTheory.Limits.PullbackCone.mono_snd_of_is_pullback_of_mono
|
Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
|
theorem mono_snd_of_is_pullback_of_mono {t : PullbackCone f g} (ht : IsLimit t) [Mono f] :
Mono t.snd
|
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
t : PullbackCone f g
ht : IsLimit t
inst✝ : Mono f
⊢ Mono t.snd
|
refine ⟨fun {W} h k i => IsLimit.hom_ext ht ?_ i⟩
|
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
t : PullbackCone f g
ht : IsLimit t
inst✝ : Mono f
W : C
h k : W ⟶ t.pt
i : h ≫ t.snd = k ≫ t.snd
⊢ h ≫ t.fst = k ≫ t.fst
|
4a08ad6ab69bebb6
|
Nat.choose_symm_of_eq_add
|
Mathlib/Data/Nat/Choose/Basic.lean
|
theorem choose_symm_of_eq_add {n a b : ℕ} (h : n = a + b) : Nat.choose n a = Nat.choose n b
|
n a b : ℕ
h : n = a + b
this : (a + b).choose a = (a + b).choose b
⊢ n.choose a = n.choose b
|
rwa [h]
|
no goals
|
1664122da5c3e255
|
DyckWord.firstReturn_nest
|
Mathlib/Combinatorics/Enumerative/DyckWord.lean
|
@[simp]
lemma firstReturn_nest : p.nest.firstReturn = p.toList.length + 1
|
case right
p : DyckWord
u : ↑p.nest = U :: ↑p ++ [D]
j : ℕ
hj : j < (↑p).length + 1
this : count D (List.take j ↑p) ≤ count U (List.take j ↑p)
⊢ ¬count U (List.take j ↑p) + 1 = count D (List.take j ↑p)
|
omega
|
no goals
|
7eee40486e6509d1
|
sum_mul_Ico_le_integral_of_monotone_antitone
|
Mathlib/Analysis/SumIntegralComparisons.lean
|
lemma sum_mul_Ico_le_integral_of_monotone_antitone
(hab : a ≤ b) (hf : MonotoneOn f (Icc a b)) (hg : AntitoneOn g (Icc (a - 1) (b - 1)))
(fpos : 0 ≤ f a) (gpos : 0 ≤ g (b - 1)) :
∑ i ∈ Finset.Ico a b, f i * g i ≤ ∫ x in a..b, f x * g (x - 1)
|
case h.h₂
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc ↑a ↑b)
hg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑a
gpos : 0 ≤ g (↑b - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ g ↑i ≤ g (x - 1)
|
apply hg
|
case h.h₂.x
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc ↑a ↑b)
hg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑a
gpos : 0 ≤ g (↑b - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ x - 1 ∈ Icc (↑a - 1) (↑b - 1)
case h.h₂.x
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc ↑a ↑b)
hg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑a
gpos : 0 ≤ g (↑b - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ ↑i ∈ Icc (↑a - 1) (↑b - 1)
case h.h₂.a
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc ↑a ↑b)
hg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑a
gpos : 0 ≤ g (↑b - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ x - 1 ≤ ↑i
|
cf3bc679c1354b33
|
Module.Relations.map_single
|
Mathlib/Algebra/Module/Presentation/Basic.lean
|
@[simp]
lemma map_single (r : relations.R) :
relations.map (Finsupp.single r 1) = relations.relation r
|
A : Type u
inst✝ : Ring A
relations : Relations A
r : relations.R
⊢ relations.map (Finsupp.single r 1) = relations.relation r
|
simp [map]
|
no goals
|
6c53bb1ffaf7695f
|
LinearMap.isOrthogonal_of_forall_apply_same
|
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
lemma isOrthogonal_of_forall_apply_same {F : Type*} [FunLike F M M] [LinearMapClass F R M M]
(f : F) (h : IsLeftRegular (2 : R)) (hB : B.IsSymm) (hf : ∀ x, B (f x) (f x) = B x x) :
B.IsOrthogonal f
|
R : Type u_20
M : Type u_21
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : LinearMap.BilinForm R M
F : Type u_22
inst✝¹ : FunLike F M M
inst✝ : LinearMapClass F R M M
f : F
h : IsLeftRegular 2
hB : IsSymm B
hf : ∀ (x : M), (B (f x)) (f x) = (B x) x
x y : M
⊢ (B (f x)) (f y) = (B x) y
|
suffices 2 * B (f x) (f y) = 2 * B x y from h this
|
R : Type u_20
M : Type u_21
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
B : LinearMap.BilinForm R M
F : Type u_22
inst✝¹ : FunLike F M M
inst✝ : LinearMapClass F R M M
f : F
h : IsLeftRegular 2
hB : IsSymm B
hf : ∀ (x : M), (B (f x)) (f x) = (B x) x
x y : M
⊢ 2 * (B (f x)) (f y) = 2 * (B x) y
|
87f8ecad436eb885
|
minpoly.two_le_natDegree_iff
|
Mathlib/FieldTheory/Minpoly/Basic.lean
|
theorem two_le_natDegree_iff (int : IsIntegral A x) :
2 ≤ (minpoly A x).natDegree ↔ x ∉ (algebraMap A B).range
|
A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : Ring B
inst✝¹ : Algebra A B
x : B
inst✝ : Nontrivial B
int : IsIntegral A x
h : (minpoly A x).natDegree < 2
⊢ (minpoly A x).natDegree = 1
|
linarith only [minpoly.natDegree_pos int, h]
|
no goals
|
217915a11973e4b8
|
DirichletCharacter.even_or_odd
|
Mathlib/NumberTheory/DirichletCharacter/Basic.lean
|
lemma even_or_odd [NoZeroDivisors S] : ψ.Even ∨ ψ.Odd
|
S : Type u_2
inst✝¹ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
inst✝ : NoZeroDivisors S
this : ψ (-1) ^ 2 = 1
⊢ ψ.Even ∨ ψ.Odd
|
convert sq_eq_one_iff.mp this
|
no goals
|
80d6617dc6d1d7a7
|
ContinuousLinearMap.IsInvertible.inverse_comp_of_left
|
Mathlib/Topology/Algebra/Module/Equiv.lean
|
lemma IsInvertible.inverse_comp_of_left {g : M₂ →L[R] M₃} {f : M →L[R] M₂}
(hg : g.IsInvertible) : (g ∘L f).inverse = f.inverse ∘L g.inverse
|
R : Type u_3
M : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁹ : TopologicalSpace M
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : TopologicalSpace M₃
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M₂
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
g : M₂ →L[R] M₃
f : M →L[R] M₂
hg : g.IsInvertible
⊢ (g.comp f).inverse = f.inverse.comp g.inverse
|
rcases hg with ⟨N, rfl⟩
|
case intro
R : Type u_3
M : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁹ : TopologicalSpace M
inst✝⁸ : TopologicalSpace M₂
inst✝⁷ : TopologicalSpace M₃
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M₂
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
f : M →L[R] M₂
N : M₂ ≃L[R] M₃
⊢ ((↑N).comp f).inverse = f.inverse.comp (↑N).inverse
|
2a57f4a02d19e1d6
|
AlgHom.IsArithFrobAt.restrict_injective
|
Mathlib/RingTheory/Frobenius.lean
|
lemma restrict_injective [Q.IsPrime] :
Function.Injective H.restrict
|
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : Q.IsPrime
x : S ⧸ Q
hx : H.restrict x = 0
⊢ x = 0
|
simpa [restrict_apply, H.card_pos.ne'] using hx
|
no goals
|
f0a4d893fa08d0c7
|
MeasureTheory.OuterMeasure.isCaratheodory_compl
|
Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
|
theorem isCaratheodory_compl : IsCaratheodory m s₁ → IsCaratheodory m s₁ᶜ
|
α : Type u
m : OuterMeasure α
s₁ : Set α
⊢ m.IsCaratheodory s₁ → m.IsCaratheodory s₁ᶜ
|
simp [IsCaratheodory, diff_eq, add_comm]
|
no goals
|
a291054f540212f9
|
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