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 |
|---|---|---|---|---|---|---|
IsArithFrobAt.exists_of_isInvariant | Mathlib/RingTheory/Frobenius.lean | /-- Let `G` be a finite group acting on `S`, and `R` be the fixed subring.
If `Q` is a prime of `S` with finite residue field,
then there exists a Frobenius element `σ : G` at `Q`. -/
lemma exists_of_isInvariant [Q.IsPrime] [Finite (S ⧸ Q)] : ∃ σ : G, IsArithFrobAt R σ Q | R : Type u_1
S : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
G : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulSemiringAction G S
inst✝⁴ : SMulCommClass G R S
Q : Ideal S
inst✝³ : Finite G
inst✝² : Algebra.IsInvariant R S G
inst✝¹ : Q.IsPrime
inst✝ : Finite (S ⧸ Q)
P : Ideal R := Ideal.under R Q
this✝... | dsimp [iterateFrobenius_def] | R : Type u_1
S : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
G : Type u_3
inst✝⁶ : Group G
inst✝⁵ : MulSemiringAction G S
inst✝⁴ : SMulCommClass G R S
Q : Ideal S
inst✝³ : Finite G
inst✝² : Algebra.IsInvariant R S G
inst✝¹ : Q.IsPrime
inst✝ : Finite (S ⧸ Q)
P : Ideal R := Ideal.under R Q
this✝... | 51abba1cf9ec68d8 |
Nat.totient_mul | Mathlib/Data/Nat/Totient.lean | theorem totient_mul {m n : ℕ} (h : m.Coprime n) : φ (m * n) = φ m * φ n :=
if hmn0 : m * n = 0 then by
rcases Nat.mul_eq_zero.1 hmn0 with h | h <;>
simp only [totient_zero, mul_zero, zero_mul, h]
else by
haveI : NeZero (m * n) := ⟨hmn0⟩
haveI : NeZero m := ⟨left_ne_zero_of_mul hmn0⟩
haveI : Ne... | m n : ℕ
h : m.Coprime n
hmn0 : ¬m * n = 0
⊢ φ (m * n) = φ m * φ n | haveI : NeZero (m * n) := ⟨hmn0⟩ | m n : ℕ
h : m.Coprime n
hmn0 : ¬m * n = 0
this : NeZero (m * n)
⊢ φ (m * n) = φ m * φ n | ee2d7f430e16d268 |
RootPairing.Hom.coweightHom_injective | Mathlib/LinearAlgebra/RootSystem/Hom.lean | lemma coweightHom_injective (P : RootPairing ι R M N) : Injective (coweightHom P) | case indexEquiv.H
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
f g : P.End
hfg : f.coweightMap = g.coweightMap
x : ι
y : ι := f.indexEquiv x
hy : y = f.indexEquiv x
this : f.coweight... | rw [coroot_coweightMap_apply, coroot_coweightMap_apply, Embedding.apply_eq_iff_eq, hy] at this | case indexEquiv.H
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
f g : P.End
hfg : f.coweightMap = g.coweightMap
x : ι
y : ι := f.indexEquiv x
hy : y = f.indexEquiv x
this : f.indexEqu... | 8b644fb9ed011071 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftRight.twoPowShift_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem twoPowShift_eq (aig : AIG α) (target : TwoPowShiftTarget aig w) (lhs : BitVec w)
(rhs : BitVec target.n) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, target.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < target.n), ⟦aig, target.rhs.get idx hi... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | simp | no goals | a2117c7fa85ff087 |
Matroid.IsCircuit.strong_multi_elimination_insert | Mathlib/Data/Matroid/Circuit.lean | /-- A version of `Matroid.IsCircuit.strong_multi_elimination` that is phrased using insertion. -/
lemma IsCircuit.strong_multi_elimination_insert (x : ι → α) (I : ι → Set α) (z : α)
(hxI : ∀ i, x i ∉ I i) (hC : ∀ i, M.IsCircuit (insert (x i) (I i)))
(hJx : M.IsCircuit (J ∪ range x)) (hzJ : z ∈ J) (hzI : ∀ i, z ... | α : Type u_1
M : Matroid α
ι : Type u_2
J : Set α
x : ι → α
I : ι → Set α
z : α
hxI : ∀ (i : ι), x i ∉ I i
hC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))
hJx : M.IsCircuit (J ∪ range x)
hzJ : z ∈ J
hzI : ∀ (i : ι), z ∉ I i
hι : Nonempty ι
hcl : z ∈ M.closure ((⋃ i, I i) ∪ J \ {z})
⊢ ∃ C' ⊆ J ∪ ⋃ i, I i, M.IsCircuit C... | rw [mem_closure_iff_exists_isCircuit (by simp [hzI])] at hcl | α : Type u_1
M : Matroid α
ι : Type u_2
J : Set α
x : ι → α
I : ι → Set α
z : α
hxI : ∀ (i : ι), x i ∉ I i
hC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))
hJx : M.IsCircuit (J ∪ range x)
hzJ : z ∈ J
hzI : ∀ (i : ι), z ∉ I i
hι : Nonempty ι
hcl : ∃ C ⊆ insert z ((⋃ i, I i) ∪ J \ {z}), M.IsCircuit C ∧ z ∈ C
⊢ ∃ C' ⊆ J ∪... | b2f1fee5a2272693 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastArithShiftRight.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem go_denote_eq (aig : AIG α) (distance : AIG.RefVec aig n) (curr : Nat)
(hcurr : curr ≤ n - 1) (acc : AIG.RefVec aig w)
(lhs : BitVec w) (rhs : BitVec n) (assign : α → Bool)
(hacc : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, acc.get idx hidx, assign⟧ = (BitVec.sshiftRightRec lhs rhs curr).getLsbD idx)
... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
n w : Nat
aig : AIG α
distance : aig.RefVec n
curr : Nat
hcurr : curr ≤ n - 1
acc : aig.RefVec w
lhs : BitVec w
rhs : BitVec n
assign : α → Bool
hacc :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := acc.get idx hidx }⟧ = (lhs.sshiftRightRec rhs cur... | omega | no goals | a62c971c970b9dfe |
AddMonoidAlgebra.exists_finset_adjoin_eq_top | Mathlib/RingTheory/FiniteType.lean | theorem exists_finset_adjoin_eq_top [h : FiniteType R R[M]] :
∃ G : Finset M, Algebra.adjoin R (of' R M '' G) = ⊤ | R : Type u_1
M : Type u_2
inst✝¹ : CommRing R
inst✝ : AddMonoid M
h : FiniteType R R[M]
⊢ ∃ G, adjoin R (of' R M '' ↑G) = ⊤ | obtain ⟨S, hS⟩ := h | case mk.intro
R : Type u_1
M : Type u_2
inst✝¹ : CommRing R
inst✝ : AddMonoid M
S : Finset R[M]
hS : adjoin R ↑S = ⊤
⊢ ∃ G, adjoin R (of' R M '' ↑G) = ⊤ | 73c5b741881b91fc |
Matrix.cramer_one | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | theorem cramer_one : cramer (1 : Matrix n n α) = 1 | case h.h.h
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
i j : n
⊢ (cramer 1 ∘ₗ LinearMap.single α (fun i => α) i) 1 j = (1 ∘ₗ LinearMap.single α (fun i => α) i) 1 j | convert congr_fun (cramer_row_self (1 : Matrix n n α) (Pi.single i 1) i _) j | case h.e'_3.h.e'_1
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
i j : n
⊢ 1 = det 1
case h.h.h
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
i j : n
⊢ ∀ (j : n), Pi.single i 1 j = 1 j i | 82419a2865041e60 |
exists_forall_hasDerivAt_Ioo_eq_of_contDiffAt | Mathlib/Analysis/ODE/PicardLindelof.lean | theorem exists_forall_hasDerivAt_Ioo_eq_of_contDiffAt (hv : ContDiffAt ℝ 1 v x₀) :
∃ f : ℝ → E, f t₀ = x₀ ∧
∃ ε > (0 : ℝ), ∀ t ∈ Ioo (t₀ - ε) (t₀ + ε), HasDerivAt f (v (f t)) t | case intro.intro.intro.intro.intro.intro.intro
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
v : E → E
t₀ : ℝ
x₀ : E
inst✝ : CompleteSpace E
hv : ContDiffAt ℝ 1 v x₀
ε : ℝ
hε : ε > 0
w✝² : ℝ≥0
w✝¹ w✝ : ℝ
hpl : IsPicardLindelof (fun x => v) (t₀ - ε) t₀ (t₀ + ε) x₀ w✝² w✝¹ w✝
f : ℝ → E
hf1 : f t₀ = ... | exact ⟨f, hf1, ε, hε, fun t ht =>
(hf2 t (Ioo_subset_Icc_self ht)).hasDerivAt (Icc_mem_nhds ht.1 ht.2)⟩ | no goals | 8f0ade6950e73a97 |
CategoryTheory.Sieve.functorPushforward_extend_eq | Mathlib/CategoryTheory/Sites/Sieves.lean | theorem functorPushforward_extend_eq {R : Presieve X} :
(generate R).arrows.functorPushforward F = R.functorPushforward F | case h.h
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X : C
R : Presieve X
Y : D
f : Y ⟶ F.obj X
⊢ f ∈ Presieve.functorPushforward F (generate R).arrows ↔ f ∈ Presieve.functorPushforward F R | constructor | case h.h.mp
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X : C
R : Presieve X
Y : D
f : Y ⟶ F.obj X
⊢ f ∈ Presieve.functorPushforward F (generate R).arrows → f ∈ Presieve.functorPushforward F R
case h.h.mpr
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Categ... | b61d2485d4b68090 |
Matroid.IsBasis.isBasis_of_closure_eq_closure | Mathlib/Data/Matroid/Closure.lean | lemma IsBasis.isBasis_of_closure_eq_closure (hI : M.IsBasis I X) (hY : I ⊆ Y)
(h : M.closure X = M.closure Y) (hYE : Y ⊆ M.E | α : Type u_2
M : Matroid α
X Y I : Set α
hI : M.IsBasis I X
hY : I ⊆ Y
h : M.closure X = M.closure Y
hYE : autoParam (Y ⊆ M.E) _auto✝
⊢ Y ⊆ M.closure I | rw [hI.closure_eq_closure, h] | α : Type u_2
M : Matroid α
X Y I : Set α
hI : M.IsBasis I X
hY : I ⊆ Y
h : M.closure X = M.closure Y
hYE : autoParam (Y ⊆ M.E) _auto✝
⊢ Y ⊆ M.closure Y | ced5ca6c7c094e42 |
Asymptotics.IsLittleO.comp_snd | Mathlib/Analysis/Asymptotics/Defs.lean | theorem IsLittleO.comp_snd : f =o[l] g → (f ∘ Prod.snd) =o[l' ×ˢ l] (g ∘ Prod.snd) | α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
l' : Filter β
⊢ (∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖) →
∀ ⦃c : ℝ⦄,
0 < c →
∃ pa,
(∀ᶠ (x : β) in l', pa x) ∧
∃ pb,
(∀ᶠ (y : α) in l, ... | exact fun h _ hc ↦ ⟨fun _ ↦ True, eventually_true l', _, h hc, fun _ ↦ id⟩ | no goals | 89addc72a55ec478 |
MvPFunctor.w_map_wMk | Mathlib/Data/PFunctor/Multivariate/W.lean | theorem w_map_wMk {α β : TypeVec n} (g : α ⟹ β) (a : P.A) (f' : P.drop.B a ⟹ α)
(f : P.last.B a → P.W α) : g <$$> P.wMk a f' f = P.wMk a (g ⊚ f') fun i => g <$$> f i | n : ℕ
P : MvPFunctor.{u} (n + 1)
α β : TypeVec.{u} n
g : α ⟹ β
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
this✝ : MvFunctor.map g ∘ f = fun i => ⟨(f i).fst, g ⊚ (f i).snd⟩
this : f = fun i => ⟨(f i).fst, (f i).snd⟩
⊢ g <$$> P.wMk a f' f = P.wMk a (g ⊚ f') fun i => ⟨(f i).fst, g ⊚ (f i).snd⟩ | rw [this] | n : ℕ
P : MvPFunctor.{u} (n + 1)
α β : TypeVec.{u} n
g : α ⟹ β
a : P.A
f' : P.drop.B a ⟹ α
f : P.last.B a → P.W α
this✝ : MvFunctor.map g ∘ f = fun i => ⟨(f i).fst, g ⊚ (f i).snd⟩
this : f = fun i => ⟨(f i).fst, (f i).snd⟩
⊢ (g <$$> P.wMk a f' fun i => ⟨(f i).fst, (f i).snd⟩) =
P.wMk a (g ⊚ f') fun i => ⟨((fun i =>... | 2aebefc7b173d248 |
Std.DHashMap.Internal.Raw₀.Const.toListModel_alterₘ | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem toListModel_alterₘ [BEq α] [EquivBEq α] [Hashable α] [LawfulHashable α]
{m : Raw₀ α (fun _ => β)} (h : Raw.WFImp m.1) {a : α} {f : Option β → Option β} :
Perm (toListModel (Const.alterₘ m a f).1.2) (Const.alterKey a f (toListModel m.1.2)) | α : Type u
β : Type v
inst✝³ : BEq α
inst✝² : EquivBEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : Raw.WFImp m.val
a : α
f : Option β → Option β
⊢ toListModel
(if h : m.containsₘ a = true then
let buckets' := updateBucket m.val.buckets ⋯ a fun l => AssocList.Const.alter a ... | split | case isTrue
α : Type u
β : Type v
inst✝³ : BEq α
inst✝² : EquivBEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : Raw.WFImp m.val
a : α
f : Option β → Option β
h✝ : m.containsₘ a = true
⊢ toListModel
(let buckets' := updateBucket m.val.buckets ⋯ a fun l => AssocList.Const.alter a f l;
... | 46325448e4f73ee5 |
Int.fract_eq_fract | Mathlib/Algebra/Order/Floor.lean | theorem fract_eq_fract {a b : α} : fract a = fract b ↔ ∃ z : ℤ, a - b = z :=
⟨fun h => ⟨⌊a⌋ - ⌊b⌋, by unfold fract at h; rw [Int.cast_sub, sub_eq_sub_iff_sub_eq_sub.1 h]⟩,
by
rintro ⟨z, hz⟩
refine fract_eq_iff.2 ⟨fract_nonneg _, fract_lt_one _, z + ⌊b⌋, ?_⟩
rw [eq_add_of_sub_eq hz, add_comm, Int.cast_a... | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
a b : α
h : fract a = fract b
⊢ a - b = ↑(⌊a⌋ - ⌊b⌋) | unfold fract at h | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
a b : α
h : a - ↑⌊a⌋ = b - ↑⌊b⌋
⊢ a - b = ↑(⌊a⌋ - ⌊b⌋) | 0f073cc303d23909 |
Filter.hasBasis_biInf_of_directed' | Mathlib/Order/Filter/Bases.lean | theorem hasBasis_biInf_of_directed' {ι : Type*} {ι' : ι → Sort _} {dom : Set ι}
(hdom : dom.Nonempty) {l : ι → Filter α} (s : ∀ i, ι' i → Set α) (p : ∀ i, ι' i → Prop)
(hl : ∀ i ∈ dom, (l i).HasBasis (p i) (s i)) (h : DirectedOn (l ⁻¹'o GE.ge) dom) :
(⨅ i ∈ dom, l i).HasBasis (fun ii' : Σi, ι' i => ii'.1 ∈ ... | case refine_2.intro.intro.intro
α : Type u_1
ι : Type u_6
ι' : ι → Type u_7
dom : Set ι
hdom : dom.Nonempty
l : ι → Filter α
s : (i : ι) → ι' i → Set α
p : (i : ι) → ι' i → Prop
hl : ∀ i ∈ dom, (l i).HasBasis (p i) (s i)
h : DirectedOn (l ⁻¹'o GE.ge) dom
t : Set α
i : ι
b : ι' i
hibt : s ⟨i, b⟩.fst ⟨i, b⟩.snd ⊆ t
hi : ... | exact ⟨hi, (hl i hi).mem_iff.mpr ⟨b, hb, hibt⟩⟩ | no goals | 15d436e0d28fffff |
IntermediateField.Lifts.union_isExtendible | Mathlib/FieldTheory/Extension.lean | theorem union_isExtendible [alg : Algebra.IsAlgebraic F E]
[Nonempty c] (hext : ∀ σ ∈ c, σ.IsExtendible) :
(union c hc).IsExtendible := fun S ↦ by
let Ω := adjoin F (S : Set E) →ₐ[F] K
have ⟨ω, hω⟩ : ∃ ω : Ω, ∀ π : c, ∃ θ ≥ π.1, ⟨_, ω⟩ ≤ θ ∧ θ.carrier = π.1.1 ⊔ adjoin F S | case refine_2.inr
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_... | rw [(hθ π₁).2, (hθ π₂).2] | no goals | f447a1810182010c |
EReal.mul_pos_iff | Mathlib/Data/Real/EReal.lean | lemma mul_pos_iff {a b : EReal} : 0 < a * b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 | case pos_bot
x✝ : ℝ
hx : 0 < x✝
⊢ 0 < ↑x✝ * ⊥ ↔ 0 < ↑x✝ ∧ 0 < ⊥ ∨ ↑x✝ < 0 ∧ ⊥ < 0 | simp [hx, EReal.coe_mul_bot_of_pos hx, le_of_lt] | no goals | a6466642cdafc4c3 |
FirstOrder.Language.equiv_between_cg | Mathlib/ModelTheory/PartialEquiv.lean | theorem equiv_between_cg (M_cg : Structure.CG L M) (N_cg : Structure.CG L N)
(g : L.FGEquiv M N)
(ext_dom : L.IsExtensionPair M N)
(ext_cod : L.IsExtensionPair N M) :
∃ f : M ≃[L] N, g ≤ f.toEmbedding.toPartialEquiv | case mk.intro.intro.mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun ... | have cod_top : F.cod = ⊤ := by rwa [← top_le_iff, ← Y_gen, Substructure.closure_le] | case mk.intro.intro.mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun ... | d1b972cd720b6e06 |
one_lt_natDegree_of_irrational_root | Mathlib/Data/Real/Irrational.lean | theorem one_lt_natDegree_of_irrational_root (hx : Irrational x) (p_nonzero : p ≠ 0)
(x_is_root : aeval x p = 0) : 1 < p.natDegree | x : ℝ
hx : Irrational x
a b : ℤ
p_nonzero : C a * X + C b ≠ 0
x_is_root : (aeval x) (C a * X + C b) = 0
⊢ ↑a * x = -↑b | simpa [eq_neg_iff_add_eq_zero] using x_is_root | no goals | 88891079ba21c78c |
Nat.maxPowDiv.pow_dvd | Mathlib/Data/Nat/MaxPowDiv.lean | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n | p n : ℕ
h : 1 < p ∧ 0 < n ∧ n % p = 0
⊢ n / p < n | apply Nat.div_lt_self <;> aesop | no goals | 444ebf47c1c533a4 |
Computation.of_results_bind | Mathlib/Data/Seq/Computation.lean | theorem of_results_bind {s : Computation α} {f : α → Computation β} {b k} :
Results (bind s f) b k → ∃ a m n, Results s a m ∧ Results (f a) b n ∧ k = n + m | α : Type u
β : Type v
f : α → Computation β
b : β
n : ℕ
IH : ∀ {s : Computation α}, (s.bind f).Results b n → ∃ a m n_1, s.Results a m ∧ (f a).Results b n_1 ∧ n = n_1 + m
s : Computation α
e : α
h : (f e).Results b (n + 1)
⊢ ∃ a m n_1, (pure e).Results a m ∧ (f a).Results b n_1 ∧ n + 1 = n_1 + m | exact ⟨e, _, n + 1, results_pure _, h, rfl⟩ | no goals | 3c07033b060cd97a |
exists_of_linearIndepOn_of_finite_span | Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean | theorem exists_of_linearIndepOn_of_finite_span {t : Finset V}
(hs : LinearIndepOn K id s) (hst : s ⊆ (span K ↑t : Submodule K V)) :
∃ t' : Finset V, ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = t.card | K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
s : Set V
t✝¹ : Finset V
hs : LinearIndepOn K id s
hst✝¹ : s ⊆ ↑(span K ↑t✝¹)
t✝ : Finset V
b₁ : V
t : Finset V
hb₁t : b₁ ∉ t
ih :
∀ (s' : Finset V),
↑s' ⊆ s → s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ∪ t)) → ∃ t', ↑t' ⊆ s ∪ ↑t ∧ s ⊆... | simp [eq, hb₂t', hb₁t, hb₁s'] | no goals | 084eab243cd3683b |
le_rank_iff_exists_linearIndependent | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndepOn K id s | case mp.intro.mk
K : Type u
V : Type v
inst✝⁴ : Ring K
inst✝³ : StrongRankCondition K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
c : Cardinal.{v}
this : Nontrivial K
h : c ≤ Module.rank K V
κ : Type v
t' : Basis κ K V
t : Basis (↑(range ⇑t')) K V := t'.reindexRange
⊢ ∃ s, #↑s = c ∧ LinearIndepOn K id ... | have : LinearIndepOn K id (Set.range t') := by
convert t.linearIndependent.linearIndepOn_id
ext
simp [t] | case mp.intro.mk
K : Type u
V : Type v
inst✝⁴ : Ring K
inst✝³ : StrongRankCondition K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
c : Cardinal.{v}
this✝ : Nontrivial K
h : c ≤ Module.rank K V
κ : Type v
t' : Basis κ K V
t : Basis (↑(range ⇑t')) K V := t'.reindexRange
this : LinearIndepOn K id (range ⇑t... | c96b3766841457c1 |
TensorPower.mul_one | Mathlib/LinearAlgebra/TensorPower/Basic.lean | theorem mul_one {n} (a : ⨂[R]^n M) : cast R M (add_zero _) (a ₜ* ₜ1) = a | case smul_tprod.e_a.h.e_6.h.h
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
r : R
a : Fin n → M
i : Fin n
⊢ (Fin.append a Fin.elim0 ∘ Fin.cast ⋯) i = a i | rw [Fin.append_elim0] | case smul_tprod.e_a.h.e_6.h.h
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
r : R
a : Fin n → M
i : Fin n
⊢ ((a ∘ Fin.cast ⋯) ∘ Fin.cast ⋯) i = a i | e803f9f54b1f406f |
Cardinal.lsub_lt_ord_lift_of_isRegular | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem lsub_lt_ord_lift_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c)
(hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c.ord) → Ordinal.lsub.{u, v} f < c.ord :=
lsub_lt_ord_lift (by rwa [hc.cof_eq])
| ι : Type u
f : ι → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
⊢ lift.{v, u} #ι < c.ord.cof | rwa [hc.cof_eq] | no goals | ee8b78e4aa02b22c |
Std.DHashMap.Internal.Raw.Const.get_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Raw.lean | theorem Const.get_eq [BEq α] [Hashable α] {m : Raw α (fun _ => β)} {a : α} {h : a ∈ m} :
Raw.Const.get m a h = Raw₀.Const.get
⟨m, by change dite .. = true at h; split at h <;> simp_all⟩ a
(by change dite .. = true at h; split at h <;> simp_all) :=
rfl
| α : Type u
β✝ : α → Type v
γ : Type w
δ : α → Type w
β : Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw α fun x => β
a : α
h : (if h : 0 < m.buckets.size then Raw₀.contains ⟨m, h⟩ a else false) = true
⊢ 0 < m.buckets.size | split at h <;> simp_all | no goals | aae82082bc678c71 |
IsAntichain.sperner | Mathlib/Combinatorics/SetFamily/LYM.lean | theorem _root_.IsAntichain.sperner [Fintype α] {𝒜 : Finset (Finset α)}
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) :
#𝒜 ≤ (Fintype.card α).choose (Fintype.card α / 2) | α : Type u_2
inst✝ : Fintype α
𝒜 : Finset (Finset α)
h𝒜 : IsAntichain (fun x1 x2 => x1 ⊆ x2) ↑𝒜
this : ↑(∑ x ∈ Iic (Fintype.card α), #(𝒜 # x)) / ↑((Fintype.card α).choose (Fintype.card α / 2)) ≤ 1
⊢ 0 < ↑((Fintype.card α).choose (Fintype.card α / 2)) | simp only [cast_pos] | α : Type u_2
inst✝ : Fintype α
𝒜 : Finset (Finset α)
h𝒜 : IsAntichain (fun x1 x2 => x1 ⊆ x2) ↑𝒜
this : ↑(∑ x ∈ Iic (Fintype.card α), #(𝒜 # x)) / ↑((Fintype.card α).choose (Fintype.card α / 2)) ≤ 1
⊢ 0 < (Fintype.card α).choose (Fintype.card α / 2) | 15c5e8deed1a0cf7 |
BitVec.lt_of_getLsb?_isSome | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem lt_of_getLsb?_isSome (x : BitVec w) (i : Nat) : x[i]?.isSome → i < w | case some
w : Nat
x : BitVec w
i : Nat
val✝ : Bool
h : x[i]? = some val✝
⊢ (some val✝).isSome = true → i < w | by_cases i < w <;> simp_all | no goals | c2cbd4bc6e002e18 |
NonUnitalSubring.closure_induction₂ | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | theorem closure_induction₂ {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop}
(mem_mem : ∀ (x) (y) (hx : x ∈ s) (hy : y ∈ s), p x y (subset_closure hx) (subset_closure hy))
(zero_left : ∀ x hx, p 0 x (zero_mem _) hx) (zero_right : ∀ x hx, p x 0 hx (zero_mem _))
(neg_left : ∀ x y hx hy, p x y... | case mem.neg
R : Type u
inst✝ : NonUnitalNonAssocRing R
s : Set R
p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯
neg_left : ∀ (x y : R) (hx : x ... | exact neg_left _ _ _ _ h | no goals | 412c69976f3f02c1 |
MeasureTheory.setToFun_smul_left | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToFun_smul_left (hT : DominatedFinMeasAdditive μ T C) (c : ℝ) (f : α → E) :
setToFun μ (fun s => c • T s) (hT.smul c) f = c • setToFun μ T hT f | α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : α → E
⊢ setToFun μ (fun s => c • T s)... | by_cases hf : Integrable f μ | case pos
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
c : ℝ
f : α → E
hf : Integrable f μ
⊢ ... | d237bd7e67b794d0 |
Real.exp_mul_le_cosh_add_mul_sinh | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | lemma exp_mul_le_cosh_add_mul_sinh {t : ℝ} (ht : |t| ≤ 1) (x : ℝ) :
exp (t * x) ≤ cosh x + t * sinh x | t : ℝ
ht : -1 ≤ t ∧ t ≤ 1
x : ℝ
⊢ 0 ≤ (1 - t) / 2 | linarith | no goals | 399072317cce519d |
Polynomial.Monic.eq_X_pow_iff_natDegree_le_natTrailingDegree | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | lemma eq_X_pow_iff_natDegree_le_natTrailingDegree (h₁ : p.Monic) :
p = X ^ p.natDegree ↔ p.natDegree ≤ p.natTrailingDegree | case refine_2.a.inr.inl
R : Type u
inst✝ : Semiring R
p : R[X]
h₁ : p.Monic
h : p.natDegree ≤ p.natTrailingDegree
⊢ p.coeff p.natDegree = if p.natDegree = p.natDegree then 1 else 0 | simpa only [if_pos rfl] using h₁.leadingCoeff | no goals | d8897cf4497f59d4 |
RootPairing.range_weylGroupToPerm | Mathlib/LinearAlgebra/RootSystem/WeylGroup.lean | lemma range_weylGroupToPerm :
P.weylGroupToPerm.range = Subgroup.closure (range P.reflection_perm) | case refine_2.intro.mk
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
⊢ P.weylGroupToPerm ⟨w, hw⟩ ∈ Subgroup.closure (range P.reflection_perm) | induction hw using Subgroup.closure_induction'' with
| one =>
change ((Equiv.indexHom P).restrict P.weylGroup) 1 ∈ _
simpa only [map_one] using Subgroup.one_mem _
| mem w' hw' =>
obtain ⟨i, rfl⟩ := hw'
simp only [MonoidHom.restrict_apply, Equiv.indexHom_apply, Equiv.reflection_indexEquiv]
simpa only [reflecti... | no goals | 2c5cbee2bb8c7dfd |
Matrix.vecAppend_eq_ite | Mathlib/Data/Fin/VecNotation.lean | theorem vecAppend_eq_ite {α : Type*} {o : ℕ} (ho : o = m + n) (u : Fin m → α) (v : Fin n → α) :
vecAppend ho u v = fun i : Fin o =>
if h : (i : ℕ) < m then u ⟨i, h⟩ else v ⟨(i : ℕ) - m, by omega⟩ | case h.e_e.h
m n : ℕ
α : Type u_1
o : ℕ
ho : o = m + n
u : Fin m → α
v : Fin n → α
i : Fin o
hi : ¬↑(Fin.cast ho i) < m
⊢ ⋯ ▸ v (Fin.subNat m (Fin.cast ⋯ (Fin.cast ho i)) ⋯) = v ⟨↑i - m, ⋯⟩ | simp only [eq_rec_constant] | case h.e_e.h
m n : ℕ
α : Type u_1
o : ℕ
ho : o = m + n
u : Fin m → α
v : Fin n → α
i : Fin o
hi : ¬↑(Fin.cast ho i) < m
⊢ v (Fin.subNat m (Fin.cast ⋯ (Fin.cast ho i)) ⋯) = v ⟨↑i - m, ⋯⟩ | e4a1aa9a2e45c9f0 |
Set.chainHeight_add_le_chainHeight_add | Mathlib/Order/Height.lean | theorem chainHeight_add_le_chainHeight_add (s : Set α) (t : Set β) (n m : ℕ) :
s.chainHeight + n ≤ t.chainHeight + m ↔
∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l + n ≤ length l' + m | case pos
α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
s : Set α
t : Set β
n m : ℕ
H : ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, l.length + n ≤ l'.length + m
h : ∀ (n : ℕ), ∃ l ∈ s.subchain, l.length = n
k : ℕ
⊢ ∃ l ∈ t.subchain, l.length = k | have := (le_chainHeight_TFAE t k).out 1 2 | case pos
α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
s : Set α
t : Set β
n m : ℕ
H : ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, l.length + n ≤ l'.length + m
h : ∀ (n : ℕ), ∃ l ∈ s.subchain, l.length = n
k : ℕ
this : (∃ l ∈ t.subchain, l.length = k) ↔ ∃ l ∈ t.subchain, k ≤ l.length
⊢ ∃ l ∈ t.subchain, l.length = k | c4896d58a0fe3db4 |
MeasureTheory.measure_iUnion_le | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | theorem measure_iUnion_le [Countable ι] (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) | α : Type u_1
ι : Type u_2
F : Type u_3
inst✝² : FunLike F (Set α) ℝ≥0∞
inst✝¹ : OuterMeasureClass F α
μ : F
inst✝ : Countable ι
s : ι → Set α
t : ℕ → Set α
⊢ μ (⋃ i, t i) = μ (⋃ i, disjointed t i) | rw [iUnion_disjointed] | no goals | c93bc964aa884a6a |
CochainComplex.HomComplex.Cochain.δ_leftShift | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | lemma δ_leftShift (a n' m' : ℤ) (hn' : n + a = n') (m : ℤ) (hm' : m + a = m') :
δ n' m' (γ.leftShift a n' hn') = a.negOnePow • (δ n m γ).leftShift a m' hm' | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' m' : ℤ
hn' : n + a = n'
m : ℤ
hm' : m + a = m'
hnm : n + 1 = m
hnm' : n' + 1 = m'
p q : ℤ
hpq : p + m' = q
⊢ p + a + 1 = p + 1 + a | omega | no goals | 8e3a8a0c35bd2a2a |
Algebra.adjoin_eq_span | Mathlib/RingTheory/Adjoin/Basic.lean | theorem adjoin_eq_span : Subalgebra.toSubmodule (adjoin R s) = span R (Submonoid.closure s) | case a
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ Submonoid.closure s ≤ (adjoin R s).toSubmonoid | exact Submonoid.closure_le.2 subset_adjoin | no goals | aea1f34ea7580cac |
contDiffOn_of_continuousOn_differentiableOn | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem contDiffOn_of_continuousOn_differentiableOn {n : ℕ∞}
(Hcont : ∀ m : ℕ, m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s)
(Hdiff : ∀ m : ℕ, m < n →
DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s) :
ContDiffOn 𝕜 n f s | case zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (... | intro y _ | case zero_eq
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s
Hdiff : ∀ (... | 57a7c2a9992e1de4 |
Bundle.TotalSpace.range_mk | Mathlib/Data/Bundle.lean | theorem TotalSpace.range_mk (b : B) : range ((↑) : E b → TotalSpace F E) = π F E ⁻¹' {b} | case h₂.mk
B : Type u_1
F : Type u_2
E : B → Type u_3
proj✝ : B
x : E proj✝
⊢ { proj := proj✝, snd := x } ∈ range (mk { proj := proj✝, snd := x }.proj) | exact ⟨x, rfl⟩ | no goals | d51c3874bef95ed7 |
ProbabilityTheory.IndepFun.variance_add | Mathlib/Probability/Variance.lean | theorem IndepFun.variance_add [IsProbabilityMeasure μ] {X Y : Ω → ℝ} (hX : MemLp X 2 μ)
(hY : MemLp Y 2 μ) (h : IndepFun X Y μ) : variance (X + Y) μ = variance X μ + variance Y μ :=
calc
variance (X + Y) μ = μ[fun a => X a ^ 2 + Y a ^ 2 + 2 * X a * Y a] - μ[X + Y] ^ 2 | case e_a.e_a.e_a
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X Y : Ω → ℝ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
h : IndepFun X Y μ
⊢ ∫ (x : Ω), (X * Y) x ∂μ = (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ | exact h.integral_mul_of_integrable (hX.integrable one_le_two) (hY.integrable one_le_two) | no goals | 6996a4817b022b25 |
IsCompact.exists_infEdist_eq_edist | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | theorem _root_.IsCompact.exists_infEdist_eq_edist (hs : IsCompact s) (hne : s.Nonempty) (x : α) :
∃ y ∈ s, infEdist x s = edist x y | case intro.intro
α : Type u
inst✝ : PseudoEMetricSpace α
s : Set α
hs : IsCompact s
hne : s.Nonempty
x : α
A : Continuous fun y => edist x y
y : α
ys : y ∈ s
hy : IsMinOn (fun y => edist x y) s y
⊢ ∃ y ∈ s, infEdist x s = edist x y | exact ⟨y, ys, le_antisymm (infEdist_le_edist_of_mem ys) (by rwa [le_infEdist])⟩ | no goals | 1d6aa3688cd1e329 |
MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | theorem exists_lt_lowerSemicontinuous_lintegral_ge [SigmaFinite μ] (f : α → ℝ≥0)
(fmeas : Measurable f) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0∞,
(∀ x, (f x : ℝ≥0∞) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε | case intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Mea... | calc
(f x : ℝ≥0∞) < f' x := by
simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)
_ ≤ g x := le_g x | no goals | c4a49260cca90fb4 |
Metric.glueDist_glued_points | Mathlib/Topology/MetricSpace/Gluing.lean | theorem glueDist_glued_points [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :
glueDist Φ Ψ ε (.inl (Φ p)) (.inr (Ψ p)) = ε | X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
p : Z
A : ∀ (q : Z), 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q)
⊢ ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) ≤ 0 | have : 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p) := by simp | X : Type u
Y : Type v
Z : Type w
inst✝² : MetricSpace X
inst✝¹ : MetricSpace Y
inst✝ : Nonempty Z
Φ : Z → X
Ψ : Z → Y
ε : ℝ
p : Z
A : ∀ (q : Z), 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q)
this : 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p)
⊢ ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) ≤ 0 | d31a35a51f091ed3 |
PresheafOfModules.Sheafify.map_smul | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean | lemma map_smul :
A.val.map π (smul α φ r m) = smul α φ (R.val.map π r) (A.val.map π m) | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
i... | apply A.isSeparated _ _ hS | case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjectiv... | 1eb37d76ad6054d2 |
IsBoundedBilinearMap.continuous | Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean | theorem IsBoundedBilinearMap.continuous (h : IsBoundedBilinearMap 𝕜 f) : Continuous f | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × ... | rw [h.map_sub_right, sub_self] | no goals | e893f50e1a714d43 |
List.exists_pw_disjoint_with_card | Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.lean | theorem List.exists_pw_disjoint_with_card {α : Type*} [Fintype α]
{c : List ℕ} (hc : c.sum ≤ Fintype.card α) :
∃ o : List (List α),
o.map length = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise List.Disjoint o | α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α... | let l := (ranges c).pmap klift' hc'_lt | α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α... | 463233383e1ded3a |
exists_bijective_map_powers | Mathlib/Algebra/Module/FinitePresentation.lean | /--
Let `M` be a finite `R`-module, and `N` be a finitely presented `R`-module.
If `l : M →ₗ[R] N` is a linear map whose localization at `S : Submonoid R` is bijective,
then `l` is already bijective under the localization at some `r ∈ S`.
-/
lemma exists_bijective_map_powers [Module.Finite R M] [Module.FinitePresentati... | case a.a.a.h.h
R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
S : Submonoid R
M' : Type u_1
inst✝⁷ : AddCommGroup M'
inst✝⁶ : Module R M'
f : M →ₗ[R] M'
inst✝⁵ : IsLocalizedModule S f
N' : Type u_2
inst✝⁴ : AddCommGro... | simp [lₛ, lₛ', LocalizedModule.smul'_mk, this] | no goals | 8598e5ad0b5e6c9d |
AlgebraicGeometry.StructureSheaf.const_zero | Mathlib/AlgebraicGeometry/StructureSheaf.lean | theorem const_zero (f : R) (U hu) : const R 0 f U hu = 0 :=
Subtype.eq <| funext fun x => IsLocalization.mk'_eq_iff_eq_mul.2 <| by
rw [RingHom.map_zero]
exact (mul_eq_zero_of_left rfl ((algebraMap R (Localizations R x)) _)).symm
| R : Type u
inst✝ : CommRing R
f : R
U : Opens ↑(PrimeSpectrum.Top R)
hu : ∀ x ∈ U, f ∈ x.asIdeal.primeCompl
x : ↥(unop (op U))
⊢ (algebraMap R (Localizations R ↑x)) 0 = ↑0 x * (algebraMap R (Localizations R ↑x)) ↑⟨f, ⋯⟩ | rw [RingHom.map_zero] | R : Type u
inst✝ : CommRing R
f : R
U : Opens ↑(PrimeSpectrum.Top R)
hu : ∀ x ∈ U, f ∈ x.asIdeal.primeCompl
x : ↥(unop (op U))
⊢ 0 = ↑0 x * (algebraMap R (Localizations R ↑x)) ↑⟨f, ⋯⟩ | 85585d0c44010051 |
exists_rat_pow_btwn | Mathlib/Algebra/Order/Archimedean/Basic.lean | theorem exists_rat_pow_btwn {n : ℕ} (hn : n ≠ 0) {x y : α} (h : x < y) (hy : 0 < y) :
∃ q : ℚ, 0 < q ∧ x < (q : α) ^ n ∧ (q : α) ^ n < y | α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
n : ℕ
hn : n ≠ 0
x y : α
h : x < y
hy : 0 < y
⊢ ∃ q, 0 < q ∧ x < ↑q ^ n ∧ ↑q ^ n < y | obtain ⟨q₂, hx₂, hy₂⟩ := exists_rat_btwn (max_lt h hy) | case intro.intro
α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
n : ℕ
hn : n ≠ 0
x y : α
h : x < y
hy : 0 < y
q₂ : ℚ
hx₂ : x ⊔ 0 < ↑q₂
hy₂ : ↑q₂ < y
⊢ ∃ q, 0 < q ∧ x < ↑q ^ n ∧ ↑q ^ n < y | 68c5c877ae1188c8 |
Filter.disjoint_prod | Mathlib/Order/Filter/Prod.lean | @[simp]
lemma disjoint_prod {f' : Filter α} {g' : Filter β} :
Disjoint (f ×ˢ g) (f' ×ˢ g') ↔ Disjoint f f' ∨ Disjoint g g' | α : Type u_1
β : Type u_2
f : Filter α
g : Filter β
f' : Filter α
g' : Filter β
⊢ Disjoint (f ×ˢ g) (f' ×ˢ g') ↔ Disjoint f f' ∨ Disjoint g g' | simp only [disjoint_iff, prod_inf_prod, prod_eq_bot] | no goals | e0bdc8927767fe43 |
Submonoid.eq_bot_of_subsingleton | Mathlib/Algebra/Group/Submonoid/Operations.lean | theorem eq_bot_of_subsingleton [Subsingleton S] : S = ⊥ | M : Type u_1
inst✝¹ : MulOneClass M
S : Submonoid M
inst✝ : Subsingleton ↥S
y : M
hy : y ∈ S
⊢ y = 1 | simpa using congr_arg ((↑) : S → M) <| Subsingleton.elim (⟨y, hy⟩ : S) 1 | no goals | c59074eefcc2b0bd |
Functor.mapEquiv_refl | Mathlib/Logic/Equiv/Functor.lean | theorem mapEquiv_refl : mapEquiv f (Equiv.refl α) = Equiv.refl (f α) | case H
α : Type u
f : Type u → Type v
inst✝¹ : Functor f
inst✝ : LawfulFunctor f
x : f α
⊢ ⇑(Equiv.refl α) <$> x = x | exact LawfulFunctor.id_map x | no goals | ea78297e08a39cbe |
ConvexOn.isBoundedUnder_abs | Mathlib/Analysis/Convex/Continuous.lean | lemma ConvexOn.isBoundedUnder_abs (hf : ConvexOn ℝ C f) {x₀ : E} (hC : C ∈ 𝓝 x₀) :
(𝓝 x₀).IsBoundedUnder (· ≤ ·) |f| ↔ (𝓝 x₀).IsBoundedUnder (· ≤ ·) f | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this✝ : Filter.Tendsto (fun y => 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
this : f ((1... | simp only [one_div, ← Nat.cast_smul_eq_nsmul ℝ, Nat.cast_ofNat, smul_sub, ne_eq,
OfNat.ofNat_ne_zero, not_false_eq_true, inv_smul_smul₀, add_sub_cancel, smul_eq_mul] at this | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hf : ConvexOn ℝ C f
x₀ : E
hC : C ∈ 𝓝 x₀
r : ℝ
this✝ : Filter.Tendsto (fun y => 2 • x₀ - y) (𝓝 x₀) (𝓝 x₀)
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
y : E
hx : 2 • x₀ - y ∈ C
hx' : y ∈ C
hfr : f y ≤ r
hfr' : f (2 • x₀ - y) ≤ r
this : f x₀ ... | b30d29bc4467c9b8 |
Nat.grahamConjecture_of_squarefree | Mathlib/NumberTheory/MaricaSchoenheim.lean | /-- The special case of Graham's conjecture where all numbers are squarefree. -/
lemma grahamConjecture_of_squarefree {n : ℕ} (f : ℕ → ℕ) (hf' : ∀ k < n, Squarefree (f k)) :
GrahamConjecture n f | case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
i : ℕ
hi : i < n
j : ℕ
hj : j < n
... | rw [← primeFactors_div_gcd (hf' _ hi) (hf' _ hj).ne_zero,
prod_primeFactors_of_squarefree <| hf'' _ hi _] | case calc_2
n : ℕ
f : ℕ → ℕ
hf' : ∀ k < n, Squarefree (f k)
hn : n ≠ 0
hf : StrictMonoOn f (Set.Iio n)
this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n
𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)
hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))
i : ℕ
hi : i < n
j : ℕ
hj : j < n
... | 42066deb227c3f15 |
Real.aux₃ | Mathlib/NumberTheory/DiophantineApproximation/Basic.lean | theorem aux₃ :
|(fract ξ)⁻¹ - v / (u - ⌊ξ⌋ * v)| < (((u : ℝ) - ⌊ξ⌋ * v) * (2 * (u - ⌊ξ⌋ * v) - 1))⁻¹ | case intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u' = ↑u - ↑⌊ξ⌋ * ↑v
⊢ |(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹ | replace hu'ℝ := (eq_sub_iff_add_eq.mp hu'ℝ).symm | case intro
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
hξ₀ : 0 < fract ξ
u' : ℤ := u - ⌊ξ⌋ * v
hu₀ : 0 < u'
huv : u' < v
hu' : u' = u - ⌊ξ⌋ * v
hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v
⊢ |(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹ | 9eb31167e505ece3 |
RingHom.surjectiveOnStalks_iff_of_isLocalHom | Mathlib/RingTheory/SurjectiveOnStalks.lean | lemma surjectiveOnStalks_iff_of_isLocalHom [IsLocalRing S] [IsLocalHom f] :
f.SurjectiveOnStalks ↔ Function.Surjective f | R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
f : R →+* S
inst✝¹ : IsLocalRing S
inst✝ : IsLocalHom f
H : f.SurjectiveOnStalks
x : S
⊢ ∃ a, f a = x | obtain ⟨y, r, c, hc, hr, e⟩ :=
(surjective_localRingHom_iff _).mp (H (IsLocalRing.maximalIdeal _) inferInstance) x | case intro.intro.intro.intro.intro
R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
f : R →+* S
inst✝¹ : IsLocalRing S
inst✝ : IsLocalHom f
H : f.SurjectiveOnStalks
x : S
y r : R
c : S
hc : c ∉ IsLocalRing.maximalIdeal S
hr : f r ∉ IsLocalRing.maximalIdeal S
e : c * f r * x = c * f y
⊢ ∃ a, f a = x | c34928fcee33d128 |
List.getElem_concat_length | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem getElem_concat_length : ∀ (l : List α) (a : α) (i) (_ : i = l.length) (w), (l ++ [a])[i]'w = a
| [], a, _, h, _ => by subst h; simp
| _ :: l, a, _, h, _ => by simp [getElem_concat_length, h]
| α : Type u_1
a : α
x✝¹ : Nat
h : x✝¹ = [].length
x✝ : x✝¹ < ([] ++ [a]).length
⊢ ([] ++ [a])[x✝¹] = a | subst h | α : Type u_1
a : α
x✝ : [].length < ([] ++ [a]).length
⊢ ([] ++ [a])[[].length] = a | 0cb8e725f5561d53 |
AlgebraicGeometry.Scheme.AffineZariskiSite.generate_presieveOfSections | Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.lean | lemma generate_presieveOfSections
{U V : X.AffineZariskiSite} {s : Set Γ(X, U.toOpens)} {f : V ⟶ U} :
Sieve.generate (presieveOfSections U s) f ↔ ∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens | case mk.mp.intro.mk.intro.up.up.intro.intro.intro.intro.intro.intro
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
f₂ : ↑Γ(X, U.toOpens)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : ↑Γ(X, toOpens ⟨X.basicOpen f₂, hW⟩)
hV : IsAffineOpen (X.basicOpen f₁)
f₃ : ↑Γ(X, ↑U)
hf₃ : X.basicOpen f₃ = X.basic... | exact X.basicOpen_le _ | no goals | f488c05e7e519ddc |
Polynomial.cyclotomic_pos | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | theorem cyclotomic_pos {n : ℕ} (hn : 2 < n) {R} [LinearOrderedCommRing R] (x : R) :
0 < eval x (cyclotomic n R) | case h.inl.hb
R : Type u_1
inst✝ : LinearOrderedCommRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
h : Odd n ∨ 0 < x + 1
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i ... | rcases h with hk | hx | case h.inl.hb.inl
R : Type u_1
inst✝ : LinearOrderedCommRing R
x : R
n : ℕ
ih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)
hn : 2 < n
hn' : 0 < n
hn'' : 1 < n
this : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i
i : ℕ
hi : i ≠ 1 ∧ i ∣ n ∧ i < n
hk : Odd n
⊢ 0... | 0959ed2b08a19c09 |
Polynomial.natDegree_eq_zero_of_derivative_eq_zero | Mathlib/Algebra/Polynomial/Derivative.lean | theorem natDegree_eq_zero_of_derivative_eq_zero [NoZeroSMulDivisors ℕ R] {f : R[X]}
(h : derivative f = 0) : f.natDegree = 0 | case inr
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < f.natDegree
⊢ False | let m := f.natDegree - 1 | case inr
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
f : R[X]
h : derivative f = 0
hf : f ≠ 0
f_nat_degree_pos : 0 < f.natDegree
m : ℕ := f.natDegree - 1
⊢ False | 1a3451df08199868 |
mul_neg_geom_series | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem mul_neg_geom_series (x : R) (h : ‖x‖ < 1) : (1 - x) * ∑' i : ℕ, x ^ i = 1 | R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ (1 - x) * ∑' (i : ℕ), x ^ i = 1 | have := (summable_geometric_of_norm_lt_one h).hasSum.mul_left (1 - x) | R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
this : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * ∑' (b : ℕ), x ^ b)
⊢ (1 - x) * ∑' (i : ℕ), x ^ i = 1 | 9c96f6b2cfbada68 |
Matrix.PosDef.eigenvalues_pos | Mathlib/LinearAlgebra/Matrix/PosDef.lean | /-- The eigenvalues of a positive definite matrix are positive -/
lemma eigenvalues_pos [DecidableEq n] {A : Matrix n n 𝕜}
(hA : Matrix.PosDef A) (i : n) : 0 < hA.1.eigenvalues i | n : Type u_2
𝕜 : Type u_4
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.PosDef
i : n
⊢ 0 < ⋯.eigenvalues i | simp only [hA.1.eigenvalues_eq] | n : Type u_2
𝕜 : Type u_4
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.PosDef
i : n
⊢ 0 <
RCLike.re
(star ((WithLp.equiv 2 (n → 𝕜)) (⋯.eigenvectorBasis i)) ⬝ᵥ A *ᵥ (WithLp.equiv 2 (n → 𝕜)) (⋯.eigenvectorBasis i)) | bd4f4c2e85760b3f |
BddAbove.bddAbove_image2_of_bddBelow | Mathlib/Order/Bounds/Image.lean | theorem BddAbove.bddAbove_image2_of_bddBelow :
BddAbove s → BddBelow t → BddAbove (Set.image2 f s t) | α : Type u
β : Type v
γ : Type w
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
f : α → β → γ
s : Set α
t : Set β
h₀ : ∀ (b : β), Monotone (swap f b)
h₁ : ∀ (a : α), Antitone (f a)
⊢ BddAbove s → BddBelow t → BddAbove (image2 f s t) | rintro ⟨a, ha⟩ ⟨b, hb⟩ | case intro.intro
α : Type u
β : Type v
γ : Type w
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : Preorder γ
f : α → β → γ
s : Set α
t : Set β
h₀ : ∀ (b : β), Monotone (swap f b)
h₁ : ∀ (a : α), Antitone (f a)
a : α
ha : a ∈ upperBounds s
b : β
hb : b ∈ lowerBounds t
⊢ BddAbove (image2 f s t) | f7a0a4c3fe283aef |
AlgebraicTopology.DoldKan.comp_P_eq_self_iff | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | theorem comp_P_eq_self_iff {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} :
φ ≫ (P q).f (n + 1) = φ ↔ HigherFacesVanish q φ | case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
⊢ φ ≫ (P q).f (n + 1) = φ → HigherFacesVanish q φ | intro hφ | case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
Y : C
n q : ℕ
φ : Y ⟶ X _⦋n + 1⦌
hφ : φ ≫ (P q).f (n + 1) = φ
⊢ HigherFacesVanish q φ | ce078dd85e69ec2d |
Polynomial.coeff_det_X_add_C_card | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | theorem coeff_det_X_add_C_card (A B : Matrix n n α) :
coeff (det ((X : α[X]) • A.map C + B.map C)) (Fintype.card n) = det A | n : Type u_1
α : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
⊢ (X • A.map ⇑C + B.map ⇑C).det.coeff (Fintype.card n) = A.det | rw [det_apply, det_apply, finset_sum_coeff] | n : Type u_1
α : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
⊢ ∑ b : Equiv.Perm n, (sign b • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (b i) i).coeff (Fintype.card n) =
∑ σ : Equiv.Perm n, sign σ • ∏ i : n, A (σ i) i | 62c349ac9b0d0cc9 |
Computation.liftRel_def | Mathlib/Data/Seq/Computation.lean | theorem liftRel_def {R : α → β → Prop} {ca cb} :
LiftRel R ca cb ↔ (Terminates ca ↔ Terminates cb) ∧ ∀ {a b}, a ∈ ca → b ∈ cb → R a b :=
⟨fun h =>
⟨terminates_of_liftRel h, fun {a b} ma mb => by
let ⟨b', mb', ab⟩ := h.left ma
rwa [mem_unique mb mb']⟩,
fun ⟨l, r⟩ =>
⟨fun {_} ma =>
let... | α : Type u
β : Type v
R : α → β → Prop
ca : Computation α
cb : Computation β
h : LiftRel R ca cb
a : α
b : β
ma : a ∈ ca
mb : b ∈ cb
⊢ R a b | let ⟨b', mb', ab⟩ := h.left ma | α : Type u
β : Type v
R : α → β → Prop
ca : Computation α
cb : Computation β
h : LiftRel R ca cb
a : α
b : β
ma : a ∈ ca
mb : b ∈ cb
b' : β
mb' : b' ∈ cb
ab : R a b'
⊢ R a b | 0a68114e015fd455 |
MeasureTheory.aemeasurable_withDensity_ennreal_iff' | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem aemeasurable_withDensity_ennreal_iff' {f : α → ℝ≥0}
(hf : AEMeasurable f μ) {g : α → ℝ≥0∞} :
AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
AEMeasurable (fun x => (f x : ℝ≥0∞) * g x) μ | case intro.intro.mp.intro.intro
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᶠ[ae μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : g =ᶠ[ae (μ.withDensity fun x => ↑(f x))] g'
⊢ AEMeasurable (fun x => ↑(f x) * g x) μ | have A : MeasurableSet {x | f' x ≠ 0} := hf'_m (measurableSet_singleton _).compl | case intro.intro.mp.intro.intro
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᶠ[ae μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : g =ᶠ[ae (μ.withDensity fun x => ↑(f x))] g'
A : MeasurableSet {x | f' x ≠ 0}
⊢ AEMeasurabl... | 8b1f28ad3ee0d17a |
AlgebraicGeometry.Scheme.Hom.ext' | Mathlib/AlgebraicGeometry/Scheme.lean | /-- An alternative ext lemma for scheme morphisms. -/
protected lemma ext' {f g : X ⟶ Y} (h : f.toLRSHom = g.toLRSHom) : f = g | case mk.mk
X Y : Scheme
toHom_1✝¹ toHom_1✝ : X.Hom Y.toLocallyRingedSpace
h : { toHom_1 := toHom_1✝¹ }.toLRSHom = { toHom_1 := toHom_1✝ }.toLRSHom
⊢ { toHom_1 := toHom_1✝¹ } = { toHom_1 := toHom_1✝ } | congr 1 | no goals | 86f141582f23fac1 |
WittVector.exists_eq_pow_p_mul | Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean | theorem exists_eq_pow_p_mul (a : 𝕎 k) (ha : a ≠ 0) :
∃ (m : ℕ) (b : 𝕎 k), b.coeff 0 ≠ 0 ∧ a = (p : 𝕎 k) ^ m * b | case h.e'_3.h.e'_2
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
a : 𝕎 k
ha : a ≠ 0
m : ℕ
b : 𝕎 k
hc : b.coeff 0 ^ p ^ m ≠ 0
hcm : a = (⇑verschiebung ∘ ⇑frobenius)^[m] b
this : (⇑verschiebung ∘ ⇑frobenius)^[m] b = (⇑verschiebung)^[m] ((⇑frobenius)^[m] b)
⊢ (... | ext1 x | case h.e'_3.h.e'_2.h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : PerfectRing k p
a : 𝕎 k
ha : a ≠ 0
m : ℕ
b : 𝕎 k
hc : b.coeff 0 ^ p ^ m ≠ 0
hcm : a = (⇑verschiebung ∘ ⇑frobenius)^[m] b
this : (⇑verschiebung ∘ ⇑frobenius)^[m] b = (⇑verschiebung)^[m] ((⇑frobenius)^[m] b)
x... | 6d33c7c8f2fbbc5d |
SimpleGraph.Walk.toSubgraph_adj_iff | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | theorem toSubgraph_adj_iff {u v u' v'} (w : G.Walk u v) :
w.toSubgraph.Adj u' v' ↔ ∃ i, s(w.getVert i, w.getVert (i + 1)) =
s(u', v') ∧ i < w.length | case mpr
V : Type u
G : SimpleGraph V
u v u' v' : V
w : G.Walk u v
⊢ (∃ i, s(w.getVert i, w.getVert (i + 1)) = s(u', v') ∧ i < w.length) → w.toSubgraph.Adj u' v' | rintro ⟨i, hi⟩ | case mpr.intro
V : Type u
G : SimpleGraph V
u v u' v' : V
w : G.Walk u v
i : ℕ
hi : s(w.getVert i, w.getVert (i + 1)) = s(u', v') ∧ i < w.length
⊢ w.toSubgraph.Adj u' v' | dd0760f68cffb54c |
CategoryTheory.ComposableArrows.Mk₁.map_comp | Mathlib/CategoryTheory/ComposableArrows.lean | lemma map_comp {i j k : Fin 2} (hij : i ≤ j) (hjk : j ≤ k) :
map f i k (hij.trans hjk) = map f i j hij ≫ map f j k hjk | case inr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X₀ X₁ : C
f : X₀ ⟶ X₁
i j : Fin 2
hij : i ≤ j
hjk : j ≤ j
⊢ map f i j ⋯ = map f i j hij ≫ map f j j hjk | rw [map_id, comp_id] | no goals | efea0c5e4fad734c |
MeasureTheory.SimpleFunc.iSup_eapprox_apply | Mathlib/MeasureTheory/Function/SimpleFunc.lean | lemma iSup_eapprox_apply (hf : Measurable f) (a : α) : ⨆ n, (eapprox f n : α →ₛ ℝ≥0∞) a = f a | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ¬f a > ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k | intro h | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
h : f a > ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k
⊢ False | fe6521a286df8e09 |
smul_singleton_mem_nhds_of_sigmaCompact | Mathlib/Topology/Algebra/Group/OpenMapping.lean | theorem smul_singleton_mem_nhds_of_sigmaCompact
{U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x | G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_... | obtain ⟨h, rfl⟩ : ∃ h, h • x = y := exists_smul_eq G x y | case intro
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V... | 09fff4470c6f8515 |
List.findSome?_guard | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem findSome?_guard (l : List α) : findSome? (Option.guard fun x => p x) l = find? p l | case cons
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : findSome? (Option.guard fun x => p x = true) xs = find? p xs
⊢ (match Option.guard (fun x => p x = true) x with
| some b => some b
| none => findSome? (Option.guard fun x => p x = true) xs) =
match p x with
| true => some x
| false => find? ... | split <;> rename_i h | case cons.h_1
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : findSome? (Option.guard fun x => p x = true) xs = find? p xs
x✝ : Option α
b✝ : α
h : Option.guard (fun x => p x = true) x = some b✝
⊢ some b✝ =
match p x with
| true => some x
| false => find? p xs
case cons.h_2
α : Type u_1
p : α → Bool
x : α... | 16b4b47d114c59c3 |
dvd_of_one_le_padicValNat | Mathlib/NumberTheory/Padics/PadicVal/Basic.lean | theorem dvd_of_one_le_padicValNat {n : ℕ} (hp : 1 ≤ padicValNat p n) : p ∣ n | p n : ℕ
hp : 1 ≤ padicValNat p n
⊢ p ∣ n | by_contra h | p n : ℕ
hp : 1 ≤ padicValNat p n
h : ¬p ∣ n
⊢ False | 38ea41d491bc4ff7 |
PadicInt.appr_spec | Mathlib/NumberTheory/Padics/RingHoms.lean | theorem appr_spec (n : ℕ) : ∀ x : ℤ_[p], x - appr x n ∈ Ideal.span {(p : ℤ_[p]) ^ n} | case neg.intro.h
p : ℕ
hp_prime : Fact (Nat.Prime p)
n : ℕ
ih : ∀ (x : ℤ_[p]), ↑p ^ n ∣ x - ↑(x.appr n)
x : ℤ_[p]
h : ¬x - ↑(x.appr n) = 0
c : ℤ_[p]
hc : x - ↑(x.appr n) = ↑p ^ n * c
hc' : c ≠ 0
⊢ ↑p ∣ c - (toZMod ↑(unitCoeff h) * 0 ^ (↑c.valuation).natAbs).cast | obtain hc0 | hc0 := eq_or_ne c.valuation 0 | case neg.intro.h.inl
p : ℕ
hp_prime : Fact (Nat.Prime p)
n : ℕ
ih : ∀ (x : ℤ_[p]), ↑p ^ n ∣ x - ↑(x.appr n)
x : ℤ_[p]
h : ¬x - ↑(x.appr n) = 0
c : ℤ_[p]
hc : x - ↑(x.appr n) = ↑p ^ n * c
hc' : c ≠ 0
hc0 : c.valuation = 0
⊢ ↑p ∣ c - (toZMod ↑(unitCoeff h) * 0 ^ (↑c.valuation).natAbs).cast
case neg.intro.h.inr
p : ℕ
hp_... | 6d18cd4591ade642 |
MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone_aux | Mathlib/MeasureTheory/Integral/Lebesgue.lean | /-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. Auxiliary version assuming moreover that the
functions in the sequence are ae measurable. -... | case h
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i ... | exact ha.choose_spec | no goals | 99ba951a0bb94bbf |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.nodup_insertRupUnits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem nodup_insertRupUnits {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(units : CNF.Clause (PosFin n)) :
∀ i : Fin (f.insertRupUnits units).1.rupUnits.size, ∀ j : Fin (f.insertRupUnits units).1.rupUnits.size,
i ≠ j → (f.insertRupUnits units).1.rupUnits[i] ≠ (f.insertRupUnits... | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
i j : Fin (f.insertRupUnits units).fst.rupUnits.size
i_ne_j : i ≠ j
li : PosFin n
bi : Bool
hi : (f.insertRupUnits units).fst.rupUnits[i] = (li, bi)
lj : PosFin n
bj : Bool
hj : (f.insertRupUnits units).fst.rupUnits[j] = (lj,... | have j_ne_k1 : j ≠ k1 := by rw [← i_eq_k1]; exact i_ne_j.symm | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
i j : Fin (f.insertRupUnits units).fst.rupUnits.size
i_ne_j : i ≠ j
li : PosFin n
bi : Bool
hi : (f.insertRupUnits units).fst.rupUnits[i] = (li, bi)
lj : PosFin n
bj : Bool
hj : (f.insertRupUnits units).fst.rupUnits[j] = (lj,... | 3e2853ef929060be |
List.inter_nil' | Mathlib/Data/List/Lattice.lean | theorem inter_nil' (l : List α) : l ∩ [] = [] | case nil
α : Type u_1
inst✝ : DecidableEq α
⊢ [] ∩ [] = [] | rfl | no goals | b125656355b71b4b |
IsDenseInducing.extend_Z_bilin_key | Mathlib/Topology/Algebra/UniformGroup/Basic.lean | theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' | case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopological... | obtain ⟨x₁, x₁_in⟩ : U₁.Nonempty := (de.comap_nhds_neBot _).nonempty_of_mem U₁_nhd | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopol... | 965b7a7f03c398d2 |
LinearMap.IsSymmetric.orthogonalFamily_iInf_eigenspaces | Mathlib/Analysis/InnerProductSpace/JointEigenspace.lean | theorem orthogonalFamily_iInf_eigenspaces (hT : ∀ i, (T i).IsSymmetric) :
OrthogonalFamily 𝕜 (fun γ : n → 𝕜 ↦ (⨅ j, eigenspace (T j) (γ j) : Submodule 𝕜 E))
fun γ : n → 𝕜 ↦ (⨅ j, eigenspace (T j) (γ j)).subtypeₗᵢ | case intro
𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
hT : ∀ (i : n), IsSymmetric (T i)
f g : n → 𝕜
hfg : f ≠ g
Ef : ↥(⨅ j, (T j).eigenspace (f j))
Eg : ↥(⨅ j, (T j).eigenspace (g j))
a : n
ha : f a ≠ g a
H :
(fun i ... | simp only [Submodule.coe_subtypeₗᵢ, Submodule.coe_subtype, Subtype.forall] at H | case intro
𝕜 : Type u_1
E : Type u_2
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : n → Module.End 𝕜 E
hT : ∀ (i : n), IsSymmetric (T i)
f g : n → 𝕜
hfg : f ≠ g
Ef : ↥(⨅ j, (T j).eigenspace (f j))
Eg : ↥(⨅ j, (T j).eigenspace (g j))
a : n
ha : f a ≠ g a
H : ∀ a_1 ∈ (... | aa50dd28c049731f |
Algebra.Generators.cotangentSpaceBasis_repr_tmul | Mathlib/RingTheory/Kaehler/CotangentComplex.lean | @[simp]
lemma cotangentSpaceBasis_repr_tmul (r x i) :
P.cotangentSpaceBasis.repr (r ⊗ₜ[P.Ring] KaehlerDifferential.D R P.Ring x : _) i =
r * aeval P.val (pderiv i x) | R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Generators R S
r : S
x : P.Ring
i : P.vars
⊢ (P.cotangentSpaceBasis.repr (r ⊗ₜ[P.Ring] (D R P.Ring) x)) i = r * (aeval P.val) ((pderiv i) x) | simp only [cotangentSpaceBasis, Basis.baseChange_repr_tmul, mvPolynomialBasis_repr_apply,
Algebra.smul_def, mul_comm r, algebraMap_apply, toExtension] | no goals | 00189045c42bca4c |
AlgebraicGeometry.IsAffineOpen.isoSpec_hom_base_apply | Mathlib/AlgebraicGeometry/AffineScheme.lean | lemma isoSpec_hom_base_apply (x : U) :
hU.isoSpec.hom.base x = (Spec.map (X.presheaf.germ U x x.2)).base (closedPoint _) | X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥U
⊢ (TopCat.Hom.hom (Spec.map (X.presheaf.map (eqToHom ⋯).op)).base)
((ConcreteCategory.hom (Spec.map ((↑U).presheaf.germ ⊤ x trivial)).base) (closedPoint ↑((↑U).presheaf.stalk x))) =
(ConcreteCategory.hom (Spec.map (X.presheaf.germ U ↑x ⋯)).base) (closedPoint ↑... | rw [← Scheme.comp_base_apply, ← Spec.map_comp,
(Iso.eq_comp_inv _).mpr (Scheme.Opens.germ_stalkIso_hom U (V := ⊤) x trivial),
X.presheaf.germ_res_assoc, Spec.map_comp, Scheme.comp_base_apply] | X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥U
⊢ (ConcreteCategory.hom (Spec.map (X.presheaf.germ U ↑x ⋯)).base)
((ConcreteCategory.hom (Spec.map (U.stalkIso x).inv).base) (closedPoint ↑((↑U).presheaf.stalk x))) =
(ConcreteCategory.hom (Spec.map (X.presheaf.germ U ↑x ⋯)).base) (closedPoint ↑(X.presheaf.sta... | 32ec53d746203035 |
Finset.small_alternating_pow_of_small_tripling | Mathlib/Combinatorics/Additive/SmallTripling.lean | /-- If `A` has small tripling, say with constant `K`, then `A` has small alternating powers, in the
sense that `|A^±1 * ... * A^±1|` is at most `|A|` times a constant exponential in the number of
terms in the product.
When `A` is symmetric (`A⁻¹ = A`), the base of the exponential can be lowered from `K ^ 3` to `K`,
wh... | case inr.intro.inr.intro.inl.inl
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
m : ℕ
hm : 3 ≤ m
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
ε : Fin m → ℤ
hε : ∀ (i : Fin m), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin m), ε i ≠ 0
hA₀ : A.Nonempty
hK₁ : 1 ≤ K
δ : Fin 3 → ℤ
this✝ : K ≤ K ^ 3
this : K ^ 2 ≤ K ^ 3
hδ₀... | nlinarith [small_neg_pos_pos_mul hA] | no goals | 647fdb3ae1591fd5 |
Real.hasStrictDerivAt_arsinh | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | theorem hasStrictDerivAt_arsinh (x : ℝ) : HasStrictDerivAt arsinh (√(1 + x ^ 2))⁻¹ x | case h.e'_9.h.e'_3
x : ℝ
⊢ √(1 + x ^ 2) = cosh (↑sinhHomeomorph.toPartialHomeomorph.symm x) | exact (cosh_arsinh _).symm | no goals | bd4c2c2cf5d5bb31 |
CategoryTheory.Monoidal.Reflective.isIso_tfae | Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean | theorem isIso_tfae : List.TFAE
[ ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
, ∀ (c : C) (d : D), IsIso ((pre (adj.unit.app d)).app (R.obj c))
, ∀ (d d' : D), IsIso (L.map ((adj.unit.app d) ▷ d'))
, ∀ (d d' : D), IsIso (L.map ((adj.unit.app d) ⊗ (adj.unit.app d')))] | C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D)... | simp only [comp_obj, op_tensorObj, coyoneda_obj_obj, unop_tensorObj, id_obj,
yoneda_obj_obj, tensorLeft_obj, EquivLike.comp_bijective, EquivLike.bijective_comp] | C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
(∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D)... | 8ccd82e9c49cc7ac |
pow_dvd_of_mul_eq_pow | Mathlib/Algebra/GCDMonoid/Basic.lean | theorem pow_dvd_of_mul_eq_pow [GCDMonoid α] {a b c d₁ d₂ : α} (ha : a ≠ 0) (hab : IsUnit (gcd a b))
{k : ℕ} (h : a * b = c ^ k) (hc : c = d₁ * d₂) (hd₁ : d₁ ∣ a) : d₁ ^ k ≠ 0 ∧ d₁ ^ k ∣ a | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
⊢ d₁ ^ k ∣ a | apply (dvd_gcd_mul_of_dvd_mul h2).trans | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c d₁ d₂ : α
ha : a ≠ 0
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
hc : c = d₁ * d₂
hd₁ : d₁ ∣ a
h1 : IsUnit (gcd (d₁ ^ k) b)
h2 : d₁ ^ k ∣ b * a
⊢ gcd (d₁ ^ k) b * a ∣ a | d13628a1c1101622 |
LinearMap.range_dualMap_dual_eq_span_singleton | Mathlib/LinearAlgebra/Dual.lean | @[simp]
lemma LinearMap.range_dualMap_dual_eq_span_singleton (f : Dual R M₁) :
range f.dualMap = R ∙ f | case h.refine_2.h
R : Type u
inst✝² : CommSemiring R
M₁ : Type v
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R M₁
f m : Dual R M₁
x✝¹ : ∃ a, a • f = m
r : R
hr : r • f = m
x✝ : M₁
⊢ ((dualMap f) (r • id)) x✝ = m x✝ | simp [dualMap_apply', ← hr] | no goals | 098f3391c79eb8b9 |
SpectrumRestricts.nnreal_le_iff | Mathlib/Analysis/Normed/Algebra/Spectrum.lean | lemma nnreal_le_iff [Algebra ℝ A] {a : A}
(ha : SpectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
(∀ x ∈ spectrum ℝ≥0 a, r ≤ x) ↔ ∀ x ∈ spectrum ℝ a, r ≤ x | A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
ha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
r : ℝ≥0
⊢ (∀ x ∈ spectrum ℝ≥0 a, r ≤ x) ↔ ∀ x ∈ spectrum ℝ a, ↑r ≤ x | simp [← ha.algebraMap_image] | no goals | e8e81c88b7501712 |
IsApproximateSubgroup.pow_inter_pow | Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean | @[to_additive]
lemma pow_inter_pow (hA : IsApproximateSubgroup K A) (hB : IsApproximateSubgroup L B) (hm : 2 ≤ m)
(hn : 2 ≤ n) :
IsApproximateSubgroup (K ^ (2 * m - 1) * L ^ (2 * n - 1)) (A ^ m ∩ B ^ n) where
one_mem := ⟨Set.one_mem_pow hA.one_mem, Set.one_mem_pow hB.one_mem⟩
inv_eq_self | G : Type u_1
inst✝ : Group G
A B : Set G
K L : ℝ
m n : ℕ
hA : IsApproximateSubgroup K A
hB : IsApproximateSubgroup L B
hm : 2 ≤ m
hn : 2 ≤ n
⊢ 2 ≤ 2 * m | omega | no goals | 7d3de84205efacd7 |
norm_derivWithin_eq_norm_fderivWithin | Mathlib/Analysis/Calculus/Deriv/Basic.lean | theorem norm_derivWithin_eq_norm_fderivWithin : ‖derivWithin f s x‖ = ‖fderivWithin 𝕜 f s x‖ | 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
⊢ ‖derivWithin f s x‖ = ‖fderivWithin 𝕜 f s x‖ | simp [← derivWithin_fderivWithin] | no goals | e096c62fe5a38dc0 |
SimpleGraph.dart_edge_fiber_card | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | theorem dart_edge_fiber_card [DecidableEq V] (e : Sym2 V) (h : e ∈ G.edgeSet) :
#{d : G.Dart | d.edge = e} = 2 | case h.e'_3
V : Type u
G : SimpleGraph V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
e : Sym2 V
v w : V
h : Quot.mk (Sym2.Rel V) (v, w) ∈ G.edgeSet
d : G.Dart := { toProd := (v, w), adj := h }
⊢ 2 = #{d, d.symm} | rw [card_insert_of_not_mem, card_singleton] | case h.e'_3
V : Type u
G : SimpleGraph V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
e : Sym2 V
v w : V
h : Quot.mk (Sym2.Rel V) (v, w) ∈ G.edgeSet
d : G.Dart := { toProd := (v, w), adj := h }
⊢ d ∉ {d.symm} | 7f8fe026966e3f75 |
NFA.mem_stepSet | Mathlib/Computability/NFA.lean | theorem mem_stepSet (s : σ) (S : Set σ) (a : α) : s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a | α : Type u
σ : Type v
M : NFA α σ
s : σ
S : Set σ
a : α
⊢ s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a | simp [stepSet] | no goals | 615f0529465e8936 |
List.erase_orderedInsert | Mathlib/Data/List/Sort.lean | theorem erase_orderedInsert [DecidableEq α] [IsRefl α r] (x : α) (xs : List α) :
(xs.orderedInsert r x).erase x = xs | case h
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : DecidableEq α
inst✝ : IsRefl α r
x : α
xs : List α
h : x ∈ takeWhile (fun b => decide ¬r x b) xs
⊢ False | replace h := mem_takeWhile_imp h | case h
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : DecidableEq α
inst✝ : IsRefl α r
x : α
xs : List α
h : (decide ¬r x x) = true
⊢ False | 216662178eb0482c |
CategoryTheory.rightDistributor_ext₂_left | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | theorem rightDistributor_ext₂_left {J : Type} [Fintype J]
{f : J → C} {X Y Z : C} {g h : ((⨁ f) ⊗ X) ⊗ Y ⟶ Z}
(w : ∀ j, ((biproduct.ι f j ▷ X) ▷ Y) ≫ g = ((biproduct.ι f j ▷ X) ▷ Y) ≫ h) :
g = h | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
f : J → C
X Y Z : C
g h : ((⨁ f) ⊗ X) ⊗ Y ⟶ Z
w : ∀ (j : J), biproduct.ι f j ▷ X ▷ Y ≫ g = biproduct.ι f j ▷ X ▷ Y ≫ h
⊢ g = h | apply (cancel_epi (α_ _ _ _).inv).mp | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
f : J → C
X Y Z : C
g h : ((⨁ f) ⊗ X) ⊗ Y ⟶ Z
w : ∀ (j : J), biproduct.ι f j ▷ X ▷ Y ≫ g = biproduct.ι f j ▷ X ▷ Y ≫ h
⊢ (α_ (⨁ f) X Y).... | 1115a03b5b5ae969 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_go_eq_divRec_q | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_go_eq_divRec_q (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case hq
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.g... | rw [denote_blastDivSubtractShift_q (rbv := rbv) (qbv := qbv) (lhs := lhs) (rhs := rhs)] | case hq
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.g... | edb5936f3962e4a9 |
orderOf_eq_one_iff | Mathlib/GroupTheory/OrderOfElement.lean | theorem orderOf_eq_one_iff : orderOf x = 1 ↔ x = 1 | G : Type u_1
inst✝ : Monoid G
x : G
⊢ orderOf x = 1 ↔ x = 1 | rw [orderOf, minimalPeriod_eq_one_iff_isFixedPt, IsFixedPt, mul_one] | no goals | d72bd8ea564f94bc |
Equiv.Perm.Basis.ofPermHomFun_apply_of_cycleOf_mem | Mathlib/GroupTheory/Perm/Centralizer.lean | theorem ofPermHomFun_apply_of_cycleOf_mem {x : α} {c : g.cycleFactorsFinset}
(hx : x ∈ c.val.support) {m : ℤ} (hm : (g ^ m) (a c) = x) :
ofPermHomFun a τ x = (g ^ m) (a ((τ : Perm g.cycleFactorsFinset) c)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
x : α
c : { x // x ∈ g.cycleFactorsFinset }
hx : x ∈ (↑c).support
m : ℤ
hm : (g ^ m) (a c) = x
hx' : ↑c = g.cycleOf x
⊢ a.ofPermHomFun τ x = (g ^ m) (a (↑τ c)) | have hx'' : g.cycleOf x ∈ g.cycleFactorsFinset := hx' ▸ c.prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
x : α
c : { x // x ∈ g.cycleFactorsFinset }
hx : x ∈ (↑c).support
m : ℤ
hm : (g ^ m) (a c) = x
hx' : ↑c = g.cycleOf x
hx'' : g.cycleOf x ∈ g.cycleFactorsFinset
⊢ a.ofPermHomFun τ x = (g ^ m) (a (↑τ c)) | 06eae76e090deab3 |
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