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 |
|---|---|---|---|---|---|---|
BoxIntegral.hasIntegral_GP_pderiv | Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean | theorem hasIntegral_GP_pderiv (f : (Fin (n + 1) → ℝ) → E)
(f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E) (s : Set (Fin (n + 1) → ℝ))
(hs : s.Countable) (Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x)
(Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) (i : Fin (n + 1)) :
HasI... | case refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc ... | have : ContinuousWithinAt (fun δ : ℝ => (2 * δ) ^ (n + 1) * ‖f' x (Pi.single i 1)‖)
(Ioi 0) 0 := ((continuousWithinAt_id.const_mul _).pow _).mul_const _ | case refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc ... | a5c18ce91cb82696 |
Nat.totient_gcd_mul_totient_mul | Mathlib/Data/Nat/Totient.lean | theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b | case a
a b : ℕ
shuffle : ∀ (a1 a2 b1 b2 c1 c2 : ℕ), b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2)
⊢ ∏ p ∈ (a * b).primeFactors, p ∣ a * b | apply prod_primeFactors_dvd | no goals | 35f1994c784b46d9 |
bind₁_rename_expand_wittPolynomial | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | theorem bind₁_rename_expand_wittPolynomial (Φ : MvPolynomial idx ℤ) (n : ℕ)
(IH :
∀ m : ℕ,
m < n + 1 →
map (Int.castRingHom ℚ) (wittStructureInt p Φ m) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) m) :
bind₁ (fun b => rename (fun i => (b, i)) (expand p (W_ ℤ n))) Φ =
... | case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
(bind₁ fun i => (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ fun ... | rw [key] | case a
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH :
∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
key :
(bind₁ fun i => (expand p) ((rename (Prod.mk i)) (W_ ℚ n))) ((map (Int.castRingHom ℚ)) Φ) =
(bind₁ fun ... | 227cfd398809aa75 |
CategoryTheory.MonoidalCategory.tensor_dite | Mathlib/CategoryTheory/Monoidal/Category.lean | theorem tensor_dite {P : Prop} [Decidable P] {W X Y Z : C} (f : W ⟶ X) (g : P → (Y ⟶ Z))
(g' : ¬P → (Y ⟶ Z)) : (f ⊗ if h : P then g h else g' h) =
if h : P then f ⊗ g h else f ⊗ g' h | C : Type u
𝒞 : Category.{v, u} C
inst✝¹ : MonoidalCategory C
P : Prop
inst✝ : Decidable P
W X Y Z : C
f : W ⟶ X
g : P → (Y ⟶ Z)
g' : ¬P → (Y ⟶ Z)
⊢ (f ⊗ if h : P then g h else g' h) = if h : P then f ⊗ g h else f ⊗ g' h | split_ifs <;> rfl | no goals | 5641c0be9efc3186 |
Batteries.RBNode.Path.zoom_zoomed₂ | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | theorem zoom_zoomed₂ (e : zoom cut t path = (t', path'))
(hp : path.Zoomed cut) : path'.Zoomed cut :=
match t, e with
| nil, rfl => hp
| node .., e => by
revert e; unfold zoom; split
· next h => exact fun e => zoom_zoomed₂ e ⟨h, hp⟩
· next h => exact fun e => zoom_zoomed₂ e ⟨h, hp⟩
· intro e; ... | α✝ : Type u_1
cut : α✝ → Ordering
t : RBNode α✝
path : Path α✝
t'✝ : RBNode α✝
path'✝ : Path α✝
e✝ : zoom cut t path = (t'✝, path'✝)
hp : Zoomed cut path
path' : Path α✝
t' : RBNode α✝
c✝ : RBColor
l✝ : RBNode α✝
v✝ : α✝
r✝ : RBNode α✝
e : zoom cut (node c✝ l✝ v✝ r✝) path = (t', path')
⊢ Zoomed cut path' | revert e | α✝ : Type u_1
cut : α✝ → Ordering
t : RBNode α✝
path : Path α✝
t'✝ : RBNode α✝
path'✝ : Path α✝
e : zoom cut t path = (t'✝, path'✝)
hp : Zoomed cut path
path' : Path α✝
t' : RBNode α✝
c✝ : RBColor
l✝ : RBNode α✝
v✝ : α✝
r✝ : RBNode α✝
⊢ zoom cut (node c✝ l✝ v✝ r✝) path = (t', path') → Zoomed cut path' | f1d1917e985dc60d |
GenContFract.IntFractPair.stream_succ | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | theorem stream_succ (h : Int.fract v ≠ 0) (n : ℕ) :
IntFractPair.stream v (n + 1) = IntFractPair.stream (Int.fract v)⁻¹ n | case zero
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
h : Int.fract v ≠ 0
H : (IntFractPair.of v).fr = Int.fract v
⊢ IntFractPair.stream v (0 + 1) = IntFractPair.stream (Int.fract v)⁻¹ 0 | rw [stream_zero, stream_succ_of_some (stream_zero v) (ne_of_eq_of_ne H h), H] | no goals | 555c114cd3b55bfe |
Behrend.exists_large_sphere | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | theorem exists_large_sphere (n d : ℕ) :
∃ k, ((d ^ n :) / (n * d ^ 2 :) : ℝ) ≤ #(sphere n d k) | case intro.intro
n d k : ℕ
hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(#(sphere n d k))
⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(#(sphere n d k)) | obtain rfl | hn := n.eq_zero_or_pos | case intro.intro.inl
d k : ℕ
hk : ↑(d ^ 0) / (↑(0 * (d - 1) ^ 2) + 1) ≤ ↑(#(sphere 0 d k))
⊢ ↑(d ^ 0) / ↑(0 * d ^ 2) ≤ ↑(#(sphere 0 d k))
case intro.intro.inr
n d k : ℕ
hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(#(sphere n d k))
hn : n > 0
⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(#(sphere n d k)) | 2a838ce406a4d3ec |
Module.exists_basis_of_basis_baseChange | Mathlib/RingTheory/LocalRing/Module.lean | /-- If `M` is of finite presentation over a local ring `(R, 𝔪, k)` such that
`𝔪 ⊗ M → M` is injective, then every family of elements that is a `k`-basis of
`k ⊗ M` is an `R`-basis of `M`. -/
lemma exists_basis_of_basis_baseChange [Module.FinitePresentation R M]
{ι : Type u} (v : ι → M) (hli : LinearIndependent k ... | case refine_2
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H : ... | apply hi'.injective | case refine_2.a
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsLocalRing R
inst✝ : FinitePresentation R M
ι : Type u
v : ι → M
hli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)
hsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1) ∘ v)) = ⊤
H ... | 59c6db943feaa269 |
max_eq_left | Mathlib/Order/Defs/LinearOrder.lean | lemma max_eq_left (h : b ≤ a) : max a b = a | case h
α : Type u_1
inst✝ : LinearOrder α
a b : α
h : b ≤ a
⊢ a = max a b | apply eq_max (le_refl _) h | case h
α : Type u_1
inst✝ : LinearOrder α
a b : α
h : b ≤ a
⊢ ∀ {d : α}, a ≤ d → b ≤ d → a ≤ d | 20ab61547662627b |
isSplittingField_AdjoinRoot_X_pow_sub_C | Mathlib/FieldTheory/KummerExtension.lean | lemma isSplittingField_AdjoinRoot_X_pow_sub_C :
haveI := Fact.mk H
letI : Algebra K K[n√a] := inferInstance
IsSplittingField K K[n√a] (X ^ n - C a) | case adjoin_rootSet'
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a))
this : Algebra K K[n√a] := inferInstance
⊢ Algebra.adjoin K ((X ^ n - C a).rootSet K[n√a]) = ⊤ | rw [eq_top_iff, ← AdjoinRoot.adjoinRoot_eq_top] | case adjoin_rootSet'
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a))
this : Algebra K K[n√a] := inferInstance
⊢ Algebra.adjoin K {root (X ^ n - C a)} ≤ Algebra.adjoin K ((X ^ n - C a).rootSet K[n√a]) | 9baaa324cdd6264f |
Matrix.rank_le_card_width | Mathlib/Data/Matrix/Rank.lean | theorem rank_le_card_width [StrongRankCondition R] (A : Matrix m n R) :
A.rank ≤ Fintype.card n | m : Type u_2
n : Type u_3
R : Type u_5
inst✝² : Fintype n
inst✝¹ : CommRing R
inst✝ : StrongRankCondition R
A : Matrix m n R
this : Module.Finite R (n → R)
⊢ A.rank ≤ Fintype.card n | haveI : Module.Free R (n → R) := Module.Free.pi _ _ | m : Type u_2
n : Type u_3
R : Type u_5
inst✝² : Fintype n
inst✝¹ : CommRing R
inst✝ : StrongRankCondition R
A : Matrix m n R
this✝ : Module.Finite R (n → R)
this : Free R (n → R)
⊢ A.rank ≤ Fintype.card n | 43bd14c4cf1a1e79 |
SimpleGraph.exists_isTuranMaximal | Mathlib/Combinatorics/SimpleGraph/Turan.lean | lemma exists_isTuranMaximal (hr : 0 < r):
∃ H : SimpleGraph V, ∃ _ : DecidableRel H.Adj, H.IsTuranMaximal r | V : Type u_1
inst✝ : Fintype V
r : ℕ
hr : 0 < r
⊢ ∃ H x, H.IsTuranMaximal r | let c := {H : SimpleGraph V | H.CliqueFree (r + 1)} | V : Type u_1
inst✝ : Fintype V
r : ℕ
hr : 0 < r
c : Set (SimpleGraph V) := {H | H.CliqueFree (r + 1)}
⊢ ∃ H x, H.IsTuranMaximal r | 441aa4bf814a2786 |
Sylow.conj_eq_normalizer_conj_of_mem_centralizer | Mathlib/GroupTheory/Sylow.lean | theorem conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
(P : Sylow p G) (x g : G) (hx : x ∈ centralizer P)
(hy : g⁻¹ * x * g ∈ centralizer P) :
∃ n ∈ P.normalizer, g⁻¹ * x * g = n⁻¹ * x * n | p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ ∃ n ∈ (↑P).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | have h1 : P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le] | p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ ∃ n ∈ (↑P).normalizer, g⁻¹ * x * g = n⁻¹ * x * n | 9634d8bb29e7eed2 |
ENat.iInf_toNat | Mathlib/Data/ENat/Lattice.lean | lemma iInf_toNat : (⨅ i, (f i : ℕ∞)).toNat = ⨅ i, f i | case inr
ι : Sort u_1
f : ι → ℕ
h✝ : Nonempty ι
⊢ (⨅ i, ↑(f i)).toNat = ⨅ i, f i | norm_cast | no goals | de0017ef08eab043 |
StieltjesFunction.ae_hasDerivAt | Mathlib/Analysis/Calculus/Monotone.lean | theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x | f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Ten... | have : Tendsto (fun y : ℝ => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + ↑1 * (x - x) ^ 2)) :=
tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul ↑1) | f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Ten... | 5dba92ed00d1be0b |
Order.length_le_height | Mathlib/Order/KrullDimension.lean | lemma length_le_height {p : LTSeries α} {x : α} (hlast : p.last ≤ x) :
p.length ≤ height x | case hab
α : Type u_1
inst✝ : Preorder α
p : LTSeries α
x : α
hlast : RelSeries.last p ≤ x
hlen0 : p.length ≠ 0
⊢ (RelSeries.eraseLast p).last < ?b | apply p.step ⟨p.length - 1, by omega⟩ | no goals | de1d1b5b1d97e68b |
PartialHomeomorph.symm_map_nhds_eq | Mathlib/Topology/PartialHomeomorph.lean | theorem symm_map_nhds_eq {x} (hx : x ∈ e.source) : map e.symm (𝓝 (e x)) = 𝓝 x :=
(e.symm.map_nhds_eq <| e.map_source hx).trans <| by rw [e.left_inv hx]
| X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
x : X
hx : x ∈ e.source
⊢ 𝓝 (↑e.symm (↑e x)) = 𝓝 x | rw [e.left_inv hx] | no goals | 2c34750dc1eea27e |
Module.Baer.ExtensionOfMaxAdjoin.extensionToFun_wd | Mathlib/Algebra/Module/Injective.lean | theorem ExtensionOfMaxAdjoin.extensionToFun_wd (h : Module.Baer R Q) {y : N}
(x : supExtensionOfMaxSingleton i f y) (a : (extensionOfMax i f).domain)
(r : R) (eq1 : ↑x = ↑a + r • y) :
ExtensionOfMaxAdjoin.extensionToFun i f h x =
(extensionOfMax i f).toLinearPMap a + ExtensionOfMaxAdjoin.extendIdealTo... | case mk
R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
x : ↥(supExtensionOfMaxSingleton... | congr | case mk.e_a.e_a
R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
x : ↥(supExtensionOfMaxS... | be5bc40bd455ceb2 |
Multiset.toFinset_union | Mathlib/Data/Finset/Basic.lean | theorem toFinset_union (s t : Multiset α) : (s ∪ t).toFinset = s.toFinset ∪ t.toFinset | case h
α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
a✝ : α
⊢ a✝ ∈ (s ∪ t).toFinset ↔ a✝ ∈ s.toFinset ∪ t.toFinset | simp | no goals | 446c07ef725aa370 |
MulAction.IsTrivialBlock.smul_iff | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem IsTrivialBlock.smul_iff {B : Set α} (g : M) :
IsTrivialBlock (g • B) ↔ IsTrivialBlock B | case mp
M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
g : M
⊢ IsTrivialBlock (g • B) → IsTrivialBlock B | intro H | case mp
M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
g : M
H : IsTrivialBlock (g • B)
⊢ IsTrivialBlock B | e100c6db6ef07927 |
List.Sublist.sym | Mathlib/Data/List/Sym.lean | theorem Sublist.sym (n : ℕ) {xs ys : List α} (h : xs <+ ys) : xs.sym n <+ ys.sym n :=
match n, h with
| 0, _ => by simp [List.sym]
| n + 1, .slnil => by simp only [refl]
| n + 1, .cons a h => by
rw [List.sym, ← nil_append (List.sym (n + 1) xs)]
apply Sublist.append (nil_sublist _)
exact h.sym (n + 1... | α : Type u_1
n✝ : ℕ
xs ys : List α
h✝ : xs <+ ys
n : ℕ
l₁✝ l₂✝ : List α
a : α
h : l₁✝ <+ l₂✝
⊢ List.sym (n + 1) (a :: l₁✝) <+ List.sym (n + 1) (a :: l₂✝) | rw [List.sym, List.sym] | α : Type u_1
n✝ : ℕ
xs ys : List α
h✝ : xs <+ ys
n : ℕ
l₁✝ l₂✝ : List α
a : α
h : l₁✝ <+ l₂✝
⊢ map (fun p => a ::ₛ p) (List.sym n (a :: l₁✝)) ++ List.sym (n + 1) l₁✝ <+
map (fun p => a ::ₛ p) (List.sym n (a :: l₂✝)) ++ List.sym (n + 1) l₂✝ | d3891099e3ec45ad |
SimpleGraph.isUniform_singleton | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | @[simp] lemma isUniform_singleton : G.IsUniform ε {a} {b} ↔ 0 < ε | α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
a b : α
⊢ G.IsUniform ε {a} {b} ↔ 0 < ε | refine ⟨IsUniform.pos, fun hε s' hs' t' ht' hs ht ↦ ?_⟩ | α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
a b : α
hε : 0 < ε
s' : Finset α
hs' : s' ⊆ {a}
t' : Finset α
ht' : t' ⊆ {b}
hs : ↑(#{a}) * ε ≤ ↑(#s')
ht : ↑(#{b}) * ε ≤ ↑(#t')
⊢ |↑(G.edgeDensity s' t') - ↑(G.edgeDensity {a} {b})| < ε | e22101c2938ea16c |
PerfectPairing.exists_basis_basis_of_span_eq_top_of_mem_algebraMap | Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean | /-- If a perfect pairing over a field `L` takes values in a subfield `K` along two `K`-subspaces
whose `L` span is full, then these subspaces induce a `K`-structure in the sense of
[*Algebra I*, Bourbaki : Chapter II, §8.1 Definition 1][bourbaki1989]. -/
lemma exists_basis_basis_of_span_eq_top_of_mem_algebraMap
(M'... | K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairing L M N
M' : Submodule K M
N' : Submodul... | exact hv₃.of_comp | no goals | 28e714660261030d |
List.prefix_of_prefix_length_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem prefix_of_prefix_length_le :
∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
| [], _, _, _, _, _ => nil_prefix
| _ :: _, b :: _, _, ⟨_, rfl⟩, ⟨_, e⟩, ll => by
injection e with _ e'; subst b
rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩ (le_of_succ_le_succ ll... | case intro
α : Type u_1
head✝ : α
tail✝ w✝¹ w✝ r₃ : List α
e' : (tail✝ ++ r₃).append w✝ = tail✝.append w✝¹
ll : (head✝ :: tail✝).length ≤ (head✝ :: (tail✝ ++ r₃)).length
⊢ head✝ :: tail✝ <+: head✝ :: (tail✝ ++ r₃) | exact ⟨r₃, rfl⟩ | no goals | 555eaf8606db38b1 |
ChainComplex.next_nat_zero | Mathlib/Algebra/Homology/HomologicalComplex.lean | theorem next_nat_zero : (ComplexShape.down ℕ).next 0 = 0 | ⊢ (ComplexShape.down ℕ).next 0 = 0 | classical
refine dif_neg ?_
push_neg
intro
apply Nat.noConfusion | no goals | 5712c554fbf39f11 |
contDiff_update | Mathlib/Analysis/Calculus/ContDiff/Operations.lean | theorem contDiff_update [DecidableEq ι] (k : WithTop ℕ∞) (x : ∀ i, F' i) (i : ι) :
ContDiff 𝕜 k (update x i) | case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : WithTop ℕ∞
x : (i : ι) → F' i
i j : ι
h : ¬j = i
⊢ ContDiff 𝕜 k fun x_1 => x j | exact contDiff_const | no goals | 3545f34c1606627c |
Array.find?_some | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean | theorem find?_some {xs : Array α} (h : find? p xs = some a) : p a | α : Type u_1
p : α → Bool
a : α
xs : Array α
h : find? p xs = some a
⊢ p a = true | cases xs | case mk
α : Type u_1
p : α → Bool
a : α
toList✝ : List α
h : find? p { toList := toList✝ } = some a
⊢ p a = true | 4eb25ecfa7c17746 |
AddCircle.norm_eq | Mathlib/Analysis/Normed/Group/AddCircle.lean | theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p| | x : ℝ
h₁ : BddBelow (abs '' {m | ↑m = ↑x})
h₂ : (abs '' {m | ↑m = ↑x}).Nonempty
⊢ sInf (norm '' {m | ↑m = ↑x}) = fract x ⊓ (1 - fract x) | apply le_antisymm | case a
x : ℝ
h₁ : BddBelow (abs '' {m | ↑m = ↑x})
h₂ : (abs '' {m | ↑m = ↑x}).Nonempty
⊢ sInf (norm '' {m | ↑m = ↑x}) ≤ fract x ⊓ (1 - fract x)
case a
x : ℝ
h₁ : BddBelow (abs '' {m | ↑m = ↑x})
h₂ : (abs '' {m | ↑m = ↑x}).Nonempty
⊢ fract x ⊓ (1 - fract x) ≤ sInf (norm '' {m | ↑m = ↑x}) | e14e944eccb5c30b |
LinearMap.nondegenerate_restrict_of_disjoint_orthogonal | Mathlib/LinearAlgebra/SesquilinearForm.lean | theorem nondegenerate_restrict_of_disjoint_orthogonal {B : M →ₗ[R] M →ₗ[R] M₁} (hB : B.IsRefl)
{W : Submodule R M} (hW : Disjoint W (W.orthogonalBilin B)) :
(B.domRestrict₁₂ W W).Nondegenerate | case mk
R : Type u_1
M : Type u_5
M₁ : Type u_6
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
hB : B.IsRefl
W : Submodule R M
hW : Disjoint W (W.orthogonalBilin B)
x : M
hx : x ∈ W
y : M
hy : y ∈ W
b₁ : (B x) y = 0
⊢ B.IsOrtho y x | rw [hB.ortho_comm] | case mk
R : Type u_1
M : Type u_5
M₁ : Type u_6
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
hB : B.IsRefl
W : Submodule R M
hW : Disjoint W (W.orthogonalBilin B)
x : M
hx : x ∈ W
y : M
hy : y ∈ W
b₁ : (B x) y = 0
⊢ B.IsOrtho x y | 46214cbf79cc3be4 |
HasFPowerSeriesWithinOnBall.tendstoUniformlyOn | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesWithinOnBall.tendstoUniformlyOn {r' : ℝ≥0}
(hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
((x + ·)⁻¹' (insert x s) ∩ Metric.ball (0 : E) r') | case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesWithi... | refine Metric.tendstoUniformlyOn_iff.2 fun ε εpos => ?_ | case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesWithi... | a7eb39e6d75bf303 |
AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_inv_ι_app | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | theorem colimitPresheafObjIsoComponentwiseLimit_inv_ι_app (F : J ⥤ PresheafedSpace.{_, _, v} C)
(U : Opens (Limits.colimit F).carrier) (j : J) :
(colimitPresheafObjIsoComponentwiseLimit F U).inv ≫ (colimit.ι F j).c.app (op U) =
limit.π _ (op j) | J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.ob... | erw [← (F.obj j).presheaf.map_id] | J : Type u'
inst✝⁴ : Category.{v', u'} J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimitsOfShape J TopCat
inst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst✝ : HasLimitsOfShape Jᵒᵖ C
F : J ⥤ PresheafedSpace C
U : Opens ↑↑(Limits.colimit F)
j : J
f : (componentwiseDiagram F U).obj (op j) ⟶ (F.ob... | af27d4a27a45b31b |
Matroid.existsMaximalSubsetProperty_of_bdd | Mathlib/Data/Matroid/IndepAxioms.lean | theorem _root_.Matroid.existsMaximalSubsetProperty_of_bdd {P : Set α → Prop}
(hP : ∃ (n : ℕ), ∀ Y, P Y → Y.encard ≤ n) (X : Set α) : ExistsMaximalSubsetProperty P X | case h.intro.intro.intro.intro.intro.intro
α : Type u_1
P : Set α → Prop
X : Set α
n : ℕ
I : Set α
hI : P I
hIX : I ⊆ X
hP : ∀ (Y : Set α), P Y → ∃ n₀, Y.encard = ↑n₀ ∧ n₀ ≤ n
Y : Set α
hY : P Y
n₀ : ℕ
heq : Y.encard = ↑n₀
hle : n₀ ≤ n
⊢ Y.ncard ≤ n | rwa [ncard_def, heq, ENat.toNat_coe] | no goals | 25330014d05a8b6c |
Complex.GammaSeq_mul | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem GammaSeq_mul (z : ℂ) {n : ℕ} (hn : n ≠ 0) :
GammaSeq z n * GammaSeq (1 - z) n =
n / (n + ↑1 - z) * (↑1 / (z * ∏ j ∈ Finset.range n, (↑1 - z ^ 2 / ((j : ℂ) + 1) ^ 2))) | case hb
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this✝ : ↑n ^ z * ↑n ^ (1 - z) = ↑n
this : ∀ (j : ℕ), (z + ↑(j + 1)) * (1 - z + ↑j) = ↑((j + 1) ^ 2) * (1 - z ^ 2 / (↑j + 1) ^ 2)
⊢ ↑n ! ^ 2 ≠ 0 | exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr <| Nat.factorial_ne_zero n) | no goals | 83d8f1e363e5e8ed |
AlgebraicGeometry.PresheafedSpace.GlueData.ι_image_preimage_eq | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | theorem ι_image_preimage_eq (i j : D.J) (U : Opens (D.U i).carrier) :
(Opens.map (𝖣.ι j).base).obj ((D.ι_isOpenEmbedding i).isOpenMap.functor.obj U) =
(opensFunctor (D.f j i)).obj
((Opens.map (𝖣.t j i).base).obj ((Opens.map (𝖣.f i j).base).obj U)) | case h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)
⊢ ⇑(ConcreteCategory.hom ((D.mapGlueData (forget C)).ι j)) ⁻¹'
(⇑(ConcreteCategory.hom ((D.mapGlueData (forget C)).ι i)) '' ↑U) =
⇑(ConcreteCategory.hom (D.f j i).base) ''
(⇑(ConcreteCategory.ho... | refine Eq.trans (D.toTopGlueData.preimage_image_eq_image' _ _ _) ?_ | case h
C : Type u
inst✝¹ : Category.{v, u} C
D : GlueData C
inst✝ : HasLimits C
i j : D.J
U : Opens ↑↑(D.U i)
⊢ ⇑(ConcreteCategory.hom (D.toTopGlueData.t i j ≫ D.toTopGlueData.f j i)) ''
(⇑(ConcreteCategory.hom (D.toTopGlueData.f i j)) ⁻¹' ↑U) =
⇑(ConcreteCategory.hom (D.f j i).base) ''
(⇑(ConcreteCateg... | 9e77f46fd9263de3 |
Vector.foldl_filterMap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem foldl_filterMap (f : α → Option β) (g : γ → β → γ) (l : Vector α n) (init : γ) :
(l.filterMap f).foldl g init = l.foldl (fun x y => match f y with | some b => g x b | none => x) init | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → Option β
g : γ → β → γ
init : γ
l : Array α
⊢ Array.foldl g init (Array.filterMap f { toArray := l, size_toArray := ⋯ }.toArray) =
foldl
(fun x y =>
match f y with
| some b => g x b
| none => x)
init { toArray := l, size_toAr... | simp [Array.foldl_filterMap'] | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → Option β
g : γ → β → γ
init : γ
l : Array α
⊢ Array.foldl
(fun x y =>
match f y with
| some b => g x b
| none => x)
init l =
Array.foldl
(fun x y =>
match f y with
| some b => g x b
| none => x)
... | b301c2e379232c7a |
CategoryTheory.Limits.isPushout_coequalizer_coprod | Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equalizer.lean | /-- The coequalizer of `f g : X ⟶ Y` is the pushout of the diagonal map `X ⨿ X ⟶ X`
along the map `(f, g) : X ⨿ X ⟶ Y`. -/
lemma isPushout_coequalizer_coprod [HasCoequalizer f g] [HasBinaryCoproduct X X] :
IsPushout (coprod.desc f g) (coprod.desc (𝟙 _) (𝟙 _))
(coequalizer.π f g) (f ≫ coequalizer.π f g) | case refine_2
C : Type u
inst✝² : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝¹ : HasCoequalizer f g
inst✝ : HasBinaryCoproduct X X
⊢ ∀ (s : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))),
coequalizer.π f g ≫ coequalizer.desc s.inl ⋯ = s.inl | exact fun s ↦ by simp | no goals | 52b23e1366b8871f |
QuasispectrumRestricts.cfc | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean | theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalContinuousFunctionalCalculus R p where
predicate_zero := h0
compactSpace_quasispectrum a | R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹⁴ : NonUnita... | exact ((h a).mp ha).2.nonUnitalStarAlgHom_id <| cfcₙHom_id ((h a).mp ha).1 | no goals | 3af5d542f22c6b37 |
SimpleGraph.left_nonuniformWitnesses_subset | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | theorem left_nonuniformWitnesses_subset (h : ¬G.IsUniform ε s t) :
(G.nonuniformWitnesses ε s t).1 ⊆ s | α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
h : ¬G.IsUniform ε s t
⊢ (⋯.choose, ⋯.choose).1 ⊆ s | exact (not_isUniform_iff.1 h).choose_spec.1 | no goals | 0b2e973a7cb97f6b |
Matrix.submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det | Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean | theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : ℕ}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : ∑ j, M j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.submatrix (Fin.succAbove j₁) id).det =
Int.negOnePow (j₁ - j₂) • (M.submatrix (Fin.succAbove j₂) id).det | case succ.e_M.a
R : Type u_1
inst✝ : CommRing R
n : ℕ
M : Matrix (Fin (n + 1)) (Fin n) R
hv : ∑ j : Fin (n + 1), M j = 0
j₁ j₂ : Fin (n + 1)
i : Fin n
h_ind : (M.submatrix i.castSucc.succAbove id).det = (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det
a b : Fin n
⊢ (M.submatrix i.succ.succAbove id).upd... | simp_rw [neg_one_smul, updateRow_apply, Finset.sum_neg_distrib, Pi.neg_apply,
Finset.sum_apply, submatrix_apply, id_eq] | case succ.e_M.a
R : Type u_1
inst✝ : CommRing R
n : ℕ
M : Matrix (Fin (n + 1)) (Fin n) R
hv : ∑ j : Fin (n + 1), M j = 0
j₁ j₂ : Fin (n + 1)
i : Fin n
h_ind : (M.submatrix i.castSucc.succAbove id).det = (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det
a b : Fin n
⊢ (if a = i then -∑ x : Fin n, M (i.suc... | f724e374e85a14e0 |
orthogonalProjection_tendsto_self | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem orthogonalProjection_tendsto_self {ι : Type*} [Preorder ι]
(U : ι → Submodule 𝕜 E) [∀ t, HasOrthogonalProjection (U t)] (hU : Monotone U) (x : E)
(hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure) :
Filter.Tendsto (fun t => (orthogonalProjection (U t) x : E)) atTop (𝓝 x) | case h.e'_5.h.e'_3
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : Preorder ι
U : ι → Submodule 𝕜 E
inst✝ : ∀ (t : ι), HasOrthogonalProjection (U t)
hU : Monotone U
x : E
hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure
this : HasOrthogonalProjec... | rw [eq_comm, orthogonalProjection_eq_self_iff, top_unique hU'] | case h.e'_5.h.e'_3
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : Preorder ι
U : ι → Submodule 𝕜 E
inst✝ : ∀ (t : ι), HasOrthogonalProjection (U t)
hU : Monotone U
x : E
hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure
this : HasOrthogonalProjec... | c0527473232db349 |
ENNReal.rpow_arith_mean_le_arith_mean_rpow | Mathlib/Analysis/MeanInequalitiesPow.lean | theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
(hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p | case refine_1
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ⊤ ↔ w i * z i ^ p = ⊤
h : ∑ i ∈ s, w i * z i = 0 ∧ p < 0 ∨ (∃ a ∈ s, w a * z a = ⊤) ∧ 0 < p
⊢ ∃ a ∈ s, w a * z a ^ p = ⊤ | simp only [and_false, hp_not_neg, false_or] at h | case refine_1
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ⊤ ↔ w i * z i ^ p = ⊤
h : (∃ a ∈ s, w a * z a = ⊤) ∧ 0 < p
⊢ ∃ a ∈ s, w a * z a ^ p = ⊤ | 706e672499704425 |
Odd.pow_add_pow_eq_zero | Mathlib/Algebra/Ring/Parity.lean | lemma Odd.pow_add_pow_eq_zero [IsCancelAdd α] (hn : Odd n) (hab : a + b = 0) :
a ^ n + b ^ n = 0 | α : Type u_2
inst✝¹ : Semiring α
a b : α
inst✝ : IsCancelAdd α
hab : a + b = 0
k : ℕ
ih : a ^ (2 * k + 1) + b ^ (2 * k + 1) = 0
⊢ 0 = b ^ 2 + a * b | rw [sq, ← add_mul, add_comm, hab, zero_mul] | no goals | 4377842bdf198980 |
EuclideanGeometry.angle_eq_zero_iff_ne_and_wbtw | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | theorem angle_eq_zero_iff_ne_and_wbtw {p₁ p₂ p₃ : P} :
∠ p₁ p₂ p₃ = 0 ↔ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ | case mpr.inr.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
hp₃p₂ : p₃ ≠ p₂
h : Wbtw ℝ p₂ p₃ p₁
⊢ ∠ p₁ p₂ p₃ = 0 | exact h.angle₃₁₂_eq_zero_of_ne hp₃p₂ | no goals | 457541c99dbfa83a |
CategoryTheory.ObjectProperty.shift_shift | Mathlib/CategoryTheory/ObjectProperty/Shift.lean | lemma shift_shift (a b c : A) (h : a + b = c) [P.IsClosedUnderIsomorphisms] :
(P.shift b).shift a = P.shift c | case h.a
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
P : ObjectProperty C
A : Type u_2
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
a b c : A
h : a + b = c
inst✝ : P.IsClosedUnderIsomorphisms
X : C
⊢ (P.shift b).shift a X ↔ P.shift c X | exact P.prop_iff_of_iso ((shiftFunctorAdd' C a b c h).symm.app X) | no goals | c7d623eee69386e3 |
Cardinal.not_injective_limitation_set | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | theorem not_injective_limitation_set : ¬ InjOn g (Iio (ord <| succ #α)) | α : Type u
g : Ordinal.{u} → α
h_inj : InjOn g (Iio (succ #α).ord)
h : lift.{u, u + 1} #↑(Iio (succ #α).ord) ≤ lift.{u + 1, u} #α
⊢ False | have mk_initialSeg_subtype :
#(Iio (ord <| succ #α)) = lift.{u + 1} (succ #α) := by
simpa only [coe_setOf, card_typein, card_ord] using mk_Iio_ordinal (ord <| succ #α) | α : Type u
g : Ordinal.{u} → α
h_inj : InjOn g (Iio (succ #α).ord)
h : lift.{u, u + 1} #↑(Iio (succ #α).ord) ≤ lift.{u + 1, u} #α
mk_initialSeg_subtype : #↑(Iio (succ #α).ord) = lift.{u + 1, u} (succ #α)
⊢ False | 3c3f184f52cf5024 |
IsNoetherian.iff_rank_lt_aleph0 | Mathlib/FieldTheory/Finiteness.lean | theorem iff_rank_lt_aleph0 : IsNoetherian K V ↔ Module.rank K V < ℵ₀ | case mpr
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
b : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V
hbfinite : (Basis.ofVectorSpaceIndex K V).Finite
⊢ IsNoetherian K ↥⊤ | refine isNoetherian_of_fg_of_noetherian _ ⟨Set.Finite.toFinset hbfinite, ?_⟩ | case mpr
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
b : Basis (↑(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V
hbfinite : (Basis.ofVectorSpaceIndex K V).Finite
⊢ span K ↑hbfinite.toFinset = ⊤ | c231033ee29253fb |
Vector.mapIdx_eq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/MapIdx.lean | theorem mapIdx_eq_iff {l : Vector α n} :
mapIdx f l = l' ↔ ∀ (i : Nat) (h : i < n), f i l[i] = l'[i] | case mk.mk
α : Type u_1
α✝ : Type u_2
f : Nat → α → α✝
l : Array α
l' : Array α✝
h : l'.size = l.size
⊢ mapIdx f { toArray := l, size_toArray := ⋯ } = { toArray := l', size_toArray := h } ↔
∀ (i : Nat) (h_1 : i < l.size), f i { toArray := l, size_toArray := ⋯ }[i] = { toArray := l', size_toArray := h }[i] | simp only [mapIdx_mk, eq_mk, Array.mapIdx_eq_iff, getElem_mk] | case mk.mk
α : Type u_1
α✝ : Type u_2
f : Nat → α → α✝
l : Array α
l' : Array α✝
h : l'.size = l.size
⊢ (∀ (i : Nat), l'[i]? = Option.map (f i) l[i]?) ↔ ∀ (i : Nat) (h_1 : i < l.size), f i l[i] = l'[i] | 76b17555eac9fccf |
Filter.frequently_bot | Mathlib/Order/Filter/Basic.lean | theorem frequently_bot {p : α → Prop} : ¬∃ᶠ x in ⊥, p x | α : Type u
p : α → Prop
⊢ ¬∃ᶠ (x : α) in ⊥, p x | simp | no goals | 04f61edf5e519ac3 |
StieltjesFunction.ae_hasDerivAt | Mathlib/Analysis/Calculus/Monotone.lean | theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x | f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Ten... | apply tendsto_apply_add_mul_sq_div_sub (nhdsLT_le_nhdsNE x) L2 | f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Ten... | 5dba92ed00d1be0b |
ArithmeticFunction.pdiv_zeta | Mathlib/NumberTheory/ArithmeticFunction.lean | theorem pdiv_zeta [DivisionSemiring R] (f : ArithmeticFunction R) :
pdiv f zeta = f | R : Type u_1
inst✝ : DivisionSemiring R
f : ArithmeticFunction R
⊢ f.pdiv ↑ζ = f | ext n | case h
R : Type u_1
inst✝ : DivisionSemiring R
f : ArithmeticFunction R
n : ℕ
⊢ (f.pdiv ↑ζ) n = f n | 604af81fd9505878 |
AkraBazziRecurrence.eventually_atTop_sumTransform_le | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma eventually_atTop_sumTransform_le :
∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, sumTransform (p a b) g (r i n) n ≤ c * g n | case h.h.hab
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ ... | rw [Set.mem_Icc] | case h.h.hab
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ ... | 9769bbce1ca34368 |
List.prod_map_ite_eq | Mathlib/Algebra/BigOperators/Group/List/Basic.lean | @[to_additive]
lemma prod_map_ite_eq {A : Type*} [DecidableEq A] (l : List A) (f g : A → G) (a : A) :
(l.map fun x => if x = a then f x else g x).prod
= (f a / g a) ^ (l.count a) * (l.map g).prod | case pos
G : Type u_7
inst✝¹ : CommGroup G
A : Type u_8
inst✝ : DecidableEq A
f g : A → G
a x : A
xs : List A
hx : x = a
⊢ (if x = a then f x else g x) * ((f a / g a) ^ count a xs * (map g xs).prod) =
(f a / g a) ^ (count a xs + if (x == a) = true then 1 else 0) * (g x * (map g xs).prod) | simp only [hx, ite_true, div_pow, pow_add, pow_one, div_eq_mul_inv, mul_assoc, mul_comm,
mul_left_comm, mul_inv_cancel_left, beq_self_eq_true] | no goals | cf149ec2fcf674c8 |
IsCyclic.ext | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | @[to_additive]
lemma IsCyclic.ext [Finite G] [IsCyclic G] {d : ℕ} {a b : ZMod d}
(hGcard : Nat.card G = d) (h : ∀ t : G, t ^ a.val = t ^ b.val) : a = b | case intro
G : Type u_2
inst✝² : Group G
inst✝¹ : Finite G
inst✝ : IsCyclic G
this : NeZero (Nat.card G)
g : G
hg : ∀ (x : G), x ∈ zpowers g
a b : ZMod (Nat.card G)
h : ↑a.val = ↑b.val
⊢ a = b | simpa [ZMod.natCast_val, ZMod.cast_id'] using h | no goals | 5b2b921e905984aa |
IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow | Mathlib/NumberTheory/Cyclotomic/Rat.lean | theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K | case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ k)
x : K
h : IsIntegral ℤ x
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K :... | have H := discr_mul_isIntegral_mem_adjoin ℚ hint h | case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
hcycl : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ k)
x : K
h : IsIntegral ℤ x
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensional ℚ K :... | 076c1160c95c322f |
BitVec.udiv_eq_divRec | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem udiv_eq_divRec (hd : 0#w < d) :
let out := divRec w {n, d} (DivModState.init w)
n / d = out.q | w : Nat
d n : BitVec w
hd : 0#w < d
this✝ : DivModState.Lawful { n := n, d := d } (DivModState.init w)
this : DivModState.Lawful { n := n, d := d } (divRec (DivModState.init w).wn { n := n, d := d } (DivModState.init w))
⊢ let out := divRec w { n := n, d := d } (DivModState.init w);
n / d = out.q | apply DivModState.udiv_eq_of_lawful this (wn_divRec ..) | no goals | 949b362303afc98b |
MeasureTheory.Measure.InnerRegular.innerRegularWRT_isClosed_isOpen | Mathlib/MeasureTheory/Measure/Regular.lean | lemma innerRegularWRT_isClosed_isOpen [R1Space α] [OpensMeasurableSpace α] [h : InnerRegular μ] :
InnerRegularWRT μ IsClosed IsOpen | α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : R1Space α
inst✝ : OpensMeasurableSpace α
h : μ.InnerRegular
U : Set α
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ U
⊢ ∃ K ⊆ U, IsClosed K ∧ r < μ K | rcases h.innerRegular hU.measurableSet r hr with ⟨K, KU, K_comp, hK⟩ | case intro.intro.intro
α : Type u_1
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : R1Space α
inst✝ : OpensMeasurableSpace α
h : μ.InnerRegular
U : Set α
hU : IsOpen U
r : ℝ≥0∞
hr : r < μ U
K : Set α
KU : K ⊆ U
K_comp : IsCompact K
hK : r < μ K
⊢ ∃ K ⊆ U, IsClosed K ∧ r < μ K | 14e1087cc08b8ad9 |
Set.image_congr | Mathlib/Data/Set/Image.lean | theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s | case h
α : Type u_1
β : Type u_2
f g : α → β
s : Set α
h : ∀ a ∈ s, f a = g a
x : β
⊢ x ∈ f '' s ↔ x ∈ g '' s | exact exists_congr fun a ↦ and_congr_right fun ha ↦ by rw [h a ha] | no goals | d2822e4c2f99628e |
AkraBazziRecurrence.GrowsPolynomially.rpow | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | protected lemma GrowsPolynomially.rpow (p : ℝ) (hf : GrowsPolynomially f)
(hf_nonneg : ∀ᶠ x in atTop, 0 ≤ f x) : GrowsPolynomially fun x => (f x) ^ p | case intro.intro.intro.intro.inr
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp :... | match hf.eventually_atTop_zero_or_pos_or_neg with
| .inl hzero => refine ⟨1, by norm_num, 1, by norm_num, ?_⟩
filter_upwards [hzero, hfnew] with x hx hx'
intro u hu
simp only [hx, ne_eq, zero_rpow (ne_of_lt hp), mul_zero, le_refl, not_true, lt_self_iff_false,
Set.Icc_self, Set.mem_singleton_iff]
simp only [... | no goals | cc486cdea8b0b615 |
Std.DHashMap.Internal.Raw₀.Const.get?_of_isEmpty | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem get?_of_isEmpty [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {a : α} :
m.1.isEmpty = true → get? m a = none | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw₀ α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
a : α
⊢ m.val.isEmpty = true → get? m a = none | simp_to_model | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw₀ α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
a : α
⊢ (toListModel m.val.buckets).isEmpty = true → getValue? a (toListModel m.val.buckets) = none | 935d58f9e8fd1d8f |
PerfectionMap.mk' | Mathlib/RingTheory/Perfection.lean | theorem mk' {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : Perfection.lift p P R f = g) :
PerfectionMap p f :=
{ injective := fun x y hxy =>
g.injective <|
(RingHom.ext_iff.1 hfg x).symm.trans <|
Eq.symm <| (RingHom.ext_iff.1 hfg y).symm.trans <| Perfection.ext fun n => (hxy n).symm
... | p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
R : Type u₁
inst✝⁴ : CommSemiring R
inst✝³ : CharP R p
P : Type u₃
inst✝² : CommSemiring P
inst✝¹ : CharP P p
inst✝ : PerfectRing P p
f : P →+* R
g : P ≃+* Ring.Perfection R p
hfg : (Perfection.lift p P R) f = ↑g
y : ℕ → R
hy : ∀ (n : ℕ), y (n + 1) ^ p = y n
x : P
hx : g x = ⟨y, hy⟩
n ... | simp [hfg, hx] | no goals | 627a0bc38a3717b3 |
HallMarriageTheorem.hall_hard_inductive_step_B | Mathlib/Combinatorics/Hall/Finite.lean | theorem hall_hard_inductive_step_B {n : ℕ} (hn : Fintype.card ι = n + 1)
(ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(ih :
∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →
∃ f : ι' → α, Function.Injective f ∧... | ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Inje... | have h := hsf'' ⟨x'', hx''⟩ | ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Inje... | de2e47a85ff6c58c |
AddChar.mulShift_one | Mathlib/Algebra/Group/AddChar.lean | @[simp]
lemma mulShift_one (ψ : AddChar R M) : mulShift ψ 1 = ψ | R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
⊢ ψ.mulShift 1 = ψ | ext | case h
R : Type u_1
M : Type u_2
inst✝¹ : Ring R
inst✝ : CommMonoid M
ψ : AddChar R M
x✝ : R
⊢ (ψ.mulShift 1) x✝ = ψ x✝ | 954e8c76d64d5408 |
OrderIso.image_setOf_maximal | Mathlib/Order/Minimal.lean | theorem image_setOf_maximal (f : α ≃o β) (P : α → Prop) :
f '' {x | Maximal P x} = {x | Maximal (fun x ↦ P (f.symm x)) x} | α : Type u_1
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α ≃o β
P : α → Prop
⊢ ∀ ⦃x y : α⦄, ?m.90682 x → ?m.90682 y → (f x ≤ f y ↔ x ≤ y) | simp [f.le_iff_le] | no goals | c31eacbcadcc348e |
List.destutter'_sublist | Mathlib/Data/List/Destutter.lean | theorem destutter'_sublist (a) : l.destutter' R a <+ a :: l | case pos
α : Type u_1
l✝ : List α
R : α → α → Prop
inst✝ : DecidableRel R
b : α
l : List α
hl : ∀ (a : α), destutter' R a l <+ a :: l
a : α
h✝ : R a b
⊢ a :: destutter' R b l <+ a :: b :: l | exact Sublist.cons₂ a (hl b) | no goals | b6577bd9a48bccd4 |
Bool.bne_self_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Bool.lean | theorem bne_self_right : ∀(a b : Bool), ((a != b) != b) = a | ⊢ ∀ (a b : Bool), ((a != b) != b) = a | decide | no goals | 86bf0553458675bf |
AkraBazziRecurrence.GrowsPolynomially.add_isLittleO | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x | f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.... | calc ‖g u‖ ≤ 1/2 * ‖f u‖ := hfg' _ hu_lb
_ = 1/2 * f u := by congr; simp only [norm_eq_abs, abs_eq_self, hfu_nonneg] | no goals | dd533a6f58fe3819 |
Set.maximal_iff_forall_insert | Mathlib/Order/Minimal.lean | theorem Set.maximal_iff_forall_insert (hP : ∀ ⦃s t⦄, P t → s ⊆ t → P s) :
Maximal P s ↔ P s ∧ ∀ x ∉ s, ¬ P (insert x s) | α : Type u_1
P : Set α → Prop
s : Set α
hP : ∀ ⦃s t : Set α⦄, P t → s ⊆ t → P s
⊢ Maximal P s ↔ P s ∧ ∀ (x : α), P (insert x s) → x ∈ s | exact ⟨fun h ↦ ⟨h.1, fun x ↦ h.mem_of_prop_insert⟩,
fun h ↦ ⟨h.1, fun t ht hst x hxt ↦ h.2 x (hP ht <| insert_subset hxt hst)⟩⟩ | no goals | 43583144eee9fcb5 |
mem_pNilradical | Mathlib/FieldTheory/IsPerfectClosure.lean | theorem mem_pNilradical {R : Type*} [CommSemiring R] {p : ℕ} {x : R} :
x ∈ pNilradical R p ↔ ∃ n : ℕ, x ^ p ^ n = 0 | R : Type u_1
inst✝ : CommSemiring R
p : ℕ
x : R
hp : ¬1 < p
h : x = 0
⊢ x ^ p ^ 0 = 0 | rw [pow_zero, pow_one, h] | no goals | 138a84fbc4a7b5d6 |
AlgebraicGeometry.Spec.preimage_comp | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | @[simp, reassoc]
lemma Spec.preimage_comp {R S T : CommRingCat} (f : Spec R ⟶ Spec S) (g : Spec S ⟶ Spec T) :
Spec.preimage (f ≫ g) = Spec.preimage g ≫ Spec.preimage f :=
Spec.map_injective (by simp)
| R S T : CommRingCat
f : Spec R ⟶ Spec S
g : Spec S ⟶ Spec T
⊢ map (preimage (f ≫ g)) = map (preimage g ≫ preimage f) | simp | no goals | 8ec4a3591ae1e261 |
CategoryTheory.SimplicialThickening.SimplicialCategory.comp_id | Mathlib/AlgebraicTopology/SimplicialNerve.lean | @[simp]
lemma comp_id (i j : SimplicialThickening J) :
(ρ_ (Hom i j)).inv ≫ Hom i j ◁ id j ≫ comp i j j = 𝟙 (Hom i j) | J : Type u_1
inst✝ : LinearOrder J
i j : SimplicialThickening J
n✝ : SimplexCategoryᵒᵖ
a✝ : (Hom i j ⊗ 𝟙_ SSet).obj n✝
x✝ : Fin ((Opposite.unop n✝).len + 1)
⊢ ((Hom i j ◁ id j ≫ comp i j j).app n✝ a✝).obj x✝ = (((ρ_ (Hom i j)).hom ≫ 𝟙 (Hom i j)).app n✝ a✝).obj x✝ | simp | no goals | 5ee044c1d9fc7e6c |
Subgroup.eq_bot_or_eq_top_of_prime_card | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | theorem Subgroup.eq_bot_or_eq_top_of_prime_card
(H : Subgroup G) [hp : Fact (Nat.card G).Prime] : H = ⊥ ∨ H = ⊤ | G : Type u_2
inst✝ : Group G
H : Subgroup G
hp : Fact (Nat.Prime (Nat.card G))
this✝ : Finite G
this : Nat.card ↥H ∣ Nat.card G
⊢ H = ⊥ ∨ H = ⊤ | rwa [Nat.dvd_prime hp.1, ← eq_bot_iff_card, card_eq_iff_eq_top] at this | no goals | eb97e2d809a7468d |
gaussSum_pow_eq_prod_jacobiSum_aux | Mathlib/NumberTheory/JacobiSum/Basic.lean | lemma gaussSum_pow_eq_prod_jacobiSum_aux (χ : MulChar F R) (ψ : AddChar F R) {n : ℕ}
(hn₁ : 0 < n) (hn₂ : n < orderOf χ) :
gaussSum χ ψ ^ n = gaussSum (χ ^ n) ψ * ∏ j ∈ Ico 1 n, jacobiSum χ (χ ^ j) | F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
χ : MulChar F R
ψ : AddChar F R
n✝ n : ℕ
hn : Nat.succ 0 ≤ n
hn₂ : n + 1 < orderOf χ
ih : gaussSum χ ψ ^ n = gaussSum (χ ^ n) ψ * ∏ j ∈ Ico 1 n, jacobiSum χ (χ ^ j)
hχn : χ * χ ^ n ≠ 1
⊢ gaussSum (χ ^ n) ψ * gaussSum χ ... | rw [mul_comm, ← jacobiSum_mul_nontrivial hχn, mul_comm, ← pow_succ'] | no goals | 77a2321db694f8d2 |
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastUmod | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean | theorem denote_blastUmod (aig : AIG α) (lhs rhs : BitVec w) (assign : α → Bool)
(input : BinaryRefVec aig w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.rhs.get idx hidx, assign⟧ = rhs.getLsbD i... | case hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := ... | rw [AIG.LawfulVecOperator.denote_mem_prefix (f := blastConst)] | case hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := ... | 3032a72cd8852392 |
Set.Countable.isPathConnected_compl_of_one_lt_rank | Mathlib/Analysis/NormedSpace/Connected.lean | theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ | case H
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x ... | rw [sub_eq_add_neg _ x] | case H
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x ... | a2405a5e761a2c4d |
hasDerivAt_ofReal_cpow_const' | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | theorem hasDerivAt_ofReal_cpow_const' {x : ℝ} (hx : x ≠ 0) {r : ℂ} (hr : r ≠ -1) :
HasDerivAt (fun y : ℝ => (y : ℂ) ^ (r + 1) / (r + 1)) (x ^ r) x | case inr
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => ↑(-y) ^ (r + 1)) (-(r + 1) * ↑(-x) ^ r) x | suffices HasDerivAt (fun y : ℝ => (y : ℂ) ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x) by
convert @HasDerivAt.scomp ℝ _ ℂ _ _ x ℝ _ _ _ _ _ _ _ _ this (hasDerivAt_neg x) using 1
rw [real_smul, ofReal_neg 1, ofReal_one]; ring | case inr
x : ℝ
hx✝ : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
hx : 0 > x
this : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)
⊢ HasDerivAt (fun y => ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x) | e70c1aa0a6bd69b4 |
Subgroup.exists_finset_card_le_mul | Mathlib/GroupTheory/Schreier.lean | theorem exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = ⊤) :
∃ T : Finset H, T.card ≤ H.index * S.card ∧ closure (T : Set H) = ⊤ | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : H.FiniteIndex
S : Finset G
hS : closure ↑S = ⊤
this✝¹ : Fintype (G ⧸ H) := H.fintypeQuotientOfFiniteIndex
this✝ : DecidableEq G
R₀ : Set G
hR1 : 1 ∈ R₀
this : Fintype ↑R₀
R : Finset G := R₀.toFinset
hR : IsComplement ↑H ↑R
⊢ 1 ∈ R | rwa [Set.mem_toFinset] | no goals | 3de9489fe069ff02 |
CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app | Mathlib/CategoryTheory/Triangulated/Opposite/Basic.lean | lemma shiftFunctorAdd'_op_hom_app (X : Cᵒᵖ) (a₁ a₂ a₃ : ℤ) (h : a₁ + a₂ = a₃)
(b₁ b₂ b₃ : ℤ) (h₁ : a₁ + b₁ = 0) (h₂ : a₂ + b₂ = 0) (h₃ : a₃ + b₃ = 0) :
(shiftFunctorAdd' Cᵒᵖ a₁ a₂ a₃ h).hom.app X =
(shiftFunctorOpIso C _ _ h₃).hom.app X ≫
((shiftFunctorAdd' C b₁ b₂ b₃ (by omega)).inv.app X.unop).o... | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
a₁ a₂ a₃ : ℤ
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : ℤ
h₁ : a₁ + b₁ = 0
h₂ : a₂ + b₂ = 0
h₃ : a₃ + b₃ = 0
⊢ b₂ = (AddMonoidHom.mk' (fun n => -n) CategoryTheory.Pretriangulated.Opposite.OppositeShiftAux.proof_1) a₂ | dsimp | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
a₁ a₂ a₃ : ℤ
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : ℤ
h₁ : a₁ + b₁ = 0
h₂ : a₂ + b₂ = 0
h₃ : a₃ + b₃ = 0
⊢ b₂ = -a₂ | b022e71c424a408a |
mul_lt_iff_lt_one_left | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | theorem mul_lt_iff_lt_one_left [MulPosStrictMono α] [MulPosReflectLT α] (b0 : 0 < b) :
a * b < b ↔ a < 1 :=
Iff.trans (by rw [one_mul]) (mul_lt_mul_right b0)
| α : Type u_3
a b : α
inst✝⁴ : MulOneClass α
inst✝³ : Zero α
inst✝² : Preorder α
inst✝¹ : MulPosStrictMono α
inst✝ : MulPosReflectLT α
b0 : 0 < b
⊢ a * b < b ↔ a * b < 1 * b | rw [one_mul] | no goals | 580c810185df4a0b |
OrthogonalFamily.summable_iff_norm_sq_summable | Mathlib/Analysis/InnerProductSpace/Subspace.lean | theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
(Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 | case h.e'_3
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpa... | have : s₁ ⊓ s₂ ⊆ s₁ := Finset.inter_subset_left | case h.e'_3
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpa... | ef6c5d8e8db28ff0 |
PNat.gcd_props | Mathlib/Data/PNat/Xgcd.lean | theorem gcd_props :
let d := gcdD a b
let w := gcdW a b
let x := gcdX a b
let y := gcdY a b
let z := gcdZ a b
let a' := gcdA' a b
let b' := gcdB' a b
w * z = succPNat (x * y) ∧
a = a' * d ∧
b = b' * d ∧
z * a' = succPNat (x * b') ∧
w * b' = succPNat (y... | case right
a b : ℕ+
d : ℕ+ := a.gcdD b
w : ℕ+ := a.gcdW b
x : ℕ := a.gcdX b
y : ℕ := a.gcdY b
z : ℕ+ := a.gcdZ b
a' : ℕ+ := a.gcdA' b
b' : ℕ+ := a.gcdB' b
u : XgcdType := XgcdType.start a b
ur : XgcdType := u.reduce
x✝ : d = ur.a
hb : d = ur.b
ha' : ↑a' = ↑w + x
hb' : ↑b' = y + ↑z
hdet : w * z = (x * y).succPNat
⊢ a = ... | have hdet' : (w * z : ℕ) = x * y + 1 := by rw [← mul_coe, hdet, succPNat_coe] | case right
a b : ℕ+
d : ℕ+ := a.gcdD b
w : ℕ+ := a.gcdW b
x : ℕ := a.gcdX b
y : ℕ := a.gcdY b
z : ℕ+ := a.gcdZ b
a' : ℕ+ := a.gcdA' b
b' : ℕ+ := a.gcdB' b
u : XgcdType := XgcdType.start a b
ur : XgcdType := u.reduce
x✝ : d = ur.a
hb : d = ur.b
ha' : ↑a' = ↑w + x
hb' : ↑b' = y + ↑z
hdet : w * z = (x * y).succPNat
hdet' ... | 61616b6c576ab0fb |
CategoryTheory.PreGaloisCategory.mkSelfProdFib_map_π | Mathlib/CategoryTheory/Galois/Decomposition.lean | @[simp]
private lemma mkSelfProdFib_map_π (t : F.obj X) : F.map (Pi.π _ t) (mkSelfProdFib F X) = t | C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
t : (F.obj X).carrier
⊢ F.map (Pi.π (fun x => X) t) (CategoryTheory.PreGaloisCategory.mkSelfProdFib F X) = t | rw [← congrFun (piComparison_comp_π F _ t), FintypeCat.comp_apply,
← PreservesProduct.iso_hom] | C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
t : (F.obj X).carrier
⊢ Pi.π (fun b => F.obj X) t
((PreservesProduct.iso F fun x => X).hom (CategoryTheory.PreGaloisCategory.mkSelfProdFib F X)) =
t | 91c50544bed2cfbc |
Real.BohrMollerup.logGammaSeq_add_one | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem logGammaSeq_add_one (x : ℝ) (n : ℕ) :
logGammaSeq (x + 1) n = logGammaSeq x (n + 1) + log x - (x + 1) * (log (n + 1) - log n) | case a.h.e'_1
x : ℝ
n m : ℕ
a✝ : m ∈ Finset.range (n + 1)
⊢ x + 1 + ↑m = x + (↑m + 1) | abel | no goals | f1787e73c63507b4 |
Matrix.SpecialLinearGroup.fin_two_induction | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | theorem fin_two_induction (P : SL(2, R) → Prop)
(h : ∀ (a b c d : R) (hdet : a * d - b * c = 1), P ⟨!![a, b; c, d], by rwa [det_fin_two_of]⟩)
(g : SL(2, R)) : P g | case h.e'_1.h.e'_3
R : Type v
inst✝ : CommRing R
P : SL(2, R) → Prop
h : ∀ (a b c d : R) (hdet : a * d - b * c = 1), P ⟨!![a, b; c, d], ⋯⟩
m : Matrix (Fin 2) (Fin 2) R
hm : m.det = 1
⊢ m = !![m 0 0, m 0 1; m 1 0, m 1 1] | ext i j | case h.e'_1.h.e'_3.a
R : Type v
inst✝ : CommRing R
P : SL(2, R) → Prop
h : ∀ (a b c d : R) (hdet : a * d - b * c = 1), P ⟨!![a, b; c, d], ⋯⟩
m : Matrix (Fin 2) (Fin 2) R
hm : m.det = 1
i j : Fin 2
⊢ m i j = !![m 0 0, m 0 1; m 1 0, m 1 1] i j | 8af824a529a8df74 |
seminormFromConst_apply_of_isMul | Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean | theorem seminormFromConst_apply_of_isMul {x : R} (hx : ∀ y : R, f (x * y) = f x * f y) :
seminormFromConst' hf1 hc hpm x = f x :=
have hlim : Tendsto (seminormFromConst_seq c f x) atTop (𝓝 (f x)) | case pos
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
hx : ∀ (y : R), f (x * y) = f x * f y
n : ℕ
hn : n = 0
⊢ seminormFromConst_seq c f x n = f x | simp only [seminormFromConst_seq, hn, pow_zero, mul_one, div_one] | no goals | e9e8771da6df1462 |
List.set_eq_take_append_cons_drop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem set_eq_take_append_cons_drop (l : List α) (n : Nat) (a : α) :
l.set n a = if n < l.length then l.take n ++ a :: l.drop (n + 1) else l | case neg.intro
α : Type u_1
l : List α
n : Nat
a : α
h : n < l.length
k : Nat
h' : ¬n + k + 1 < n
h'' : ¬n + k + 1 = n
h''' : n < n + k + 1
p : n + k + 1 - n = k + 1
⊢ l[n + k + 1]? = (a :: drop (n + 1) l)[n + k + 1 - n]? | rw [p] | case neg.intro
α : Type u_1
l : List α
n : Nat
a : α
h : n < l.length
k : Nat
h' : ¬n + k + 1 < n
h'' : ¬n + k + 1 = n
h''' : n < n + k + 1
p : n + k + 1 - n = k + 1
⊢ l[n + k + 1]? = (a :: drop (n + 1) l)[k + 1]? | 9fb4c99baced165c |
sup_himp_distrib | Mathlib/Order/Heyting/Basic.lean | theorem sup_himp_distrib (a b c : α) : a ⊔ b ⇨ c = (a ⇨ c) ⊓ (b ⇨ c) :=
eq_of_forall_le_iff fun d => by
rw [le_inf_iff, le_himp_comm, sup_le_iff]
simp_rw [le_himp_comm]
| α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c d : α
⊢ a ≤ d ⇨ c ∧ b ≤ d ⇨ c ↔ d ≤ a ⇨ c ∧ d ≤ b ⇨ c | simp_rw [le_himp_comm] | no goals | a4aa4ba7e594ec49 |
Finsupp.indicator_injective | Mathlib/Data/Finsupp/Indicator.lean | theorem indicator_injective : Injective fun f : ∀ i ∈ s, α => indicator s f | ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
⊢ Injective fun f => indicator s f | intro a b h | ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
a b : (i : ι) → i ∈ s → α
h : (fun f => indicator s f) a = (fun f => indicator s f) b
⊢ a = b | 5cf50d46838d4703 |
Associates.dvd_of_mem_factors | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem dvd_of_mem_factors {a p : Associates α} (hm : p ∈ factors a) :
p ∣ a | case inr.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
a0 : α
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ a.factors.prod | rw [← ha', factors_mk a0 nza] at hm ⊢ | case inr.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
ha0 : a ≠ 0
a0 : α
hm : p ∈ ↑(factors' a0)
nza : a0 ≠ 0
ha' : Associates.mk a0 = a
⊢ p ∣ FactorSet.prod ↑(factors' a0) | 5299e19f6694fb2c |
AlgebraicGeometry.eq_top_of_sigmaSpec_subset_of_isCompact | Mathlib/AlgebraicGeometry/PointsPi.lean | lemma eq_top_of_sigmaSpec_subset_of_isCompact
(U : (Spec (.of (Π i, R i))).Opens) (V : Set (Spec (.of (Π i, R i))))
(hV : ↑(sigmaSpec R).opensRange ⊆ V)
(hV' : IsCompact (X := Spec (.of (Π i, R i))) V)
(hVU : V ⊆ U) : U = ⊤ | ι : Type u
R : ι → CommRingCat
U : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens
V : Set ↑↑(Spec (CommRingCat.of ((i : ι) → ↑(R i)))).toPresheafedSpace
hV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V
hV' : IsCompact V
hVU : V ⊆ ↑U
s : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))
hs : U.carrierᶜ = zeroLocus s
t : Set ↑(C... | rfl | no goals | 016c4327c8b84e91 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRat | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem sat_of_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (p : PosFin n → Bool)
(pf : p ⊨ f) :
(insertRatUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inr.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : Pos... | simp only [addAssignment, ← b_eq_false, addNegAssignment, ite_false, reduceCtorEq] at h2 | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inr.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : Pos... | c0b439908684a182 |
WithTop.untopD_zero_mul | Mathlib/Algebra/Order/Ring/WithTop.lean | @[simp]
lemma untopD_zero_mul (a b : WithTop α) : (a * b).untopD 0 = a.untopD 0 * b.untopD 0 | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
ha : ¬a = 0
hb : b = 0
⊢ untopD 0 (a * b) = untopD 0 a * untopD 0 b | rw [hb, mul_zero, ← coe_zero, untopD_coe, mul_zero] | no goals | ebaa5001124c917e |
Subgroup.normalCore_eq_ker | Mathlib/GroupTheory/GroupAction/Quotient.lean | theorem normalCore_eq_ker : H.normalCore = (MulAction.toPermHom G (G ⧸ H)).ker | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ H.normalCore = (MulAction.toPermHom G (G ⧸ H)).ker | apply le_antisymm | case a
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ H.normalCore ≤ (MulAction.toPermHom G (G ⧸ H)).ker
case a
G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ (MulAction.toPermHom G (G ⧸ H)).ker ≤ H.normalCore | c667aa832c2ff481 |
BitVec.Nat.sub_mul_eq_mod_of_lt_of_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem Nat.sub_mul_eq_mod_of_lt_of_le (hlo : w * n ≤ i) (hhi : i < w * (n + 1)) :
i - w * n = i % w | w n i : Nat
hlo : w * n ≤ i
hhi : i < w * (n + 1)
⊢ i < (n + 1) * w | rw [Nat.mul_comm] | w n i : Nat
hlo : w * n ≤ i
hhi : i < w * (n + 1)
⊢ i < w * (n + 1) | b3934d511b5b407b |
BitVec.allOnes_and | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem allOnes_and {x : BitVec w} : allOnes w &&& x = x | case pred
w : Nat
x : BitVec w
i : Nat
h : i < w
⊢ (allOnes w &&& x).getLsbD i = x.getLsbD i | simp [h] | no goals | 2c9cfa09a5591f05 |
Cardinal.cantorFunction_injective | Mathlib/Data/Real/Cardinality.lean | theorem cantorFunction_injective (h1 : 0 < c) (h2 : c < 1 / 2) :
Function.Injective (cantorFunction c) | c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
hfg : cantorFunction c f = cantorFunction c g
h : ¬f = g
⊢ False | revert hfg | c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
h : ¬f = g
⊢ cantorFunction c f = cantorFunction c g → False | 1fee4878056d7b26 |
wellQuasiOrderedLE_iff | Mathlib/Order/WellQuasiOrder.lean | theorem wellQuasiOrderedLE_iff :
WellQuasiOrderedLE α ↔ WellFoundedLT α ∧ ∀ s : Set α, IsAntichain (· ≤ ·) s → s.Finite | case intro.inl
α : Type u_1
inst✝ : Preorder α
x✝ : WellFoundedLT α ∧ ∀ (s : Set α), IsAntichain (fun x1 x2 => x1 ≤ x2) s → s.Finite
hwf : WellFoundedLT α
hc : ∀ (s : Set α), IsAntichain (fun x1 x2 => x1 ≤ x2) s → s.Finite
f : ℕ → α
g : ℕ ↪o ℕ
h1 : ∀ (m n : ℕ), m < n → f (g m) > f (g n)
⊢ False | apply RelEmbedding.not_wellFounded_of_decreasing_seq _ hwf.wf | α : Type u_1
inst✝ : Preorder α
x✝ : WellFoundedLT α ∧ ∀ (s : Set α), IsAntichain (fun x1 x2 => x1 ≤ x2) s → s.Finite
hwf : WellFoundedLT α
hc : ∀ (s : Set α), IsAntichain (fun x1 x2 => x1 ≤ x2) s → s.Finite
f : ℕ → α
g : ℕ ↪o ℕ
h1 : ∀ (m n : ℕ), m < n → f (g m) > f (g n)
⊢ (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x... | 8b7235cbbf8f1942 |
PEquiv.mem_ofSet_iff | Mathlib/Data/PEquiv.lean | theorem mem_ofSet_iff {s : Set α} [DecidablePred (· ∈ s)] {a b : α} :
a ∈ ofSet s b ↔ a = b ∧ a ∈ s | α : Type u
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
a b : α
⊢ a ∈ (ofSet s) b ↔ a = b ∧ a ∈ s | dsimp [ofSet] | α : Type u
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
a b : α
⊢ (a ∈ if b ∈ s then some b else none) ↔ a = b ∧ a ∈ s | 11951a2d8659894a |
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastRotateLeft | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/RotateLeft.lean | theorem denote_blastRotateLeft (aig : AIG α) (target : ShiftTarget aig w)
(assign : α → Bool) :
∀ (idx : Nat) (hidx : idx < w),
⟦
(blastRotateLeft aig target).aig,
(blastRotateLeft aig target).vec.get idx hidx,
assign
⟧
=
if hidx2 : idx < target.di... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
target : aig.ShiftTarget w
assign : α → Bool
idx✝ : Nat
hidx✝ : idx✝ < w
⊢ ⟦assign,
{
aig :=
(match target with
| { vec := input, distance := distance } =>
{ aig := aig, vec := blastRotateLeft.go inp... | dsimp only | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
target : aig.ShiftTarget w
assign : α → Bool
idx✝ : Nat
hidx✝ : idx✝ < w
⊢ ⟦assign, { aig := aig, ref := (blastRotateLeft.go target.vec target.distance 0 ⋯ RefVec.empty).get idx✝ hidx✝ }⟧ =
if hidx2 : idx✝ < target.distance % w then
⟦assig... | feee6d93cbc92b68 |
EuclideanGeometry.Sphere.sbtw_secondInter | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | theorem Sphere.sbtw_secondInter {s : Sphere P} {p p' : P} (hp : p ∈ s)
(hp' : dist p' s.center < s.radius) : Sbtw ℝ p p' (s.secondInter p (p' -ᵥ p)) | case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : s.radius < s.radius
h : p' = s.secondInter p (p' -ᵥ p)
⊢ False | exact lt_irrefl _ hp' | no goals | f4c80e9c705e3709 |
Equiv.Perm.signBijAux_injOn | Mathlib/GroupTheory/Perm/Sign.lean | theorem signBijAux_injOn {n : ℕ} {f : Perm (Fin n)} :
(finPairsLT n : Set (Σ _, Fin n)).InjOn (signBijAux f) | case neg
n : ℕ
f : Perm (Fin n)
a₁ a₂ b₁ b₂ : Fin n
ha : b₁ < b₂
hb : b₂ < b₁
this : b₂ ≤ b₁
h✝¹ : f b₁ < f b₂
h✝ : f b₁ ≤ f b₂
h : a₁ = b₂ ∧ a₂ = b₁
⊢ b₂ = b₁ ∧ b₁ = b₂ | exact absurd this (not_le.mpr ha) | no goals | 1f949d06d26d705c |
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