Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
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import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Algebra.Group.UniqueProds
#align_import algebra.monoid_algebra.no_zero_divisors from "leanprover-community/mathlib"@"3e067975886cf5801e597925328c335609511b1a"
open Finsupp
variable {R A : Type*} [Semiring R]
namespace MonoidAlgebra
| Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean | 68 | 79 | theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
(h : UniqueMul f.support g.support a0 b0) :
(f * g) (a0 * b0) = f a0 * g b0 := by |
classical
simp_rw [mul_apply, sum, ← Finset.sum_product']
refine (Finset.sum_eq_single (a0, b0) ?_ ?_).trans (if_pos rfl) <;> simp_rw [Finset.mem_product]
· refine fun ab hab hne => if_neg (fun he => hne <| Prod.ext ?_ ?_)
exacts [(h hab.1 hab.2 he).1, (h hab.1 hab.2 he).2]
· refine fun hnmem => ite_eq_r... |
import Mathlib.Analysis.Calculus.Deriv.Add
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v
open Filter Set
open scoped Topology Classical
section Module
variable {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E ... | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 86 | 96 | theorem mem_posTangentConeAt_of_segment_subset {s : Set E} {x y : E} (h : segment ℝ x y ⊆ s) :
y - x ∈ posTangentConeAt s x := by |
let c := fun n : ℕ => (2 : ℝ) ^ n
let d := fun n : ℕ => (c n)⁻¹ • (y - x)
refine ⟨c, d, Filter.univ_mem' fun n => h ?_, tendsto_pow_atTop_atTop_of_one_lt one_lt_two, ?_⟩
· show x + d n ∈ segment ℝ x y
rw [segment_eq_image']
refine ⟨(c n)⁻¹, ⟨?_, ?_⟩, rfl⟩
exacts [inv_nonneg.2 (pow_nonneg zero_le_tw... |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 145 | 154 | theorem mapAccumr₂_mapAccumr₂_right_right (f₁ : β → γ → σ₁ → σ₁ × φ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ ys (mapAccumr₂ f₂ xs ys s₂).snd s₁)
= let m := mapAccumr₂ (fun x y (s₁, s₂) =>
let r₂ := f₂ x y s₂
let r₁ := f₁ y r₂.snd s₁
((r₁.fst, r₂.fst), r₁.snd)
... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.SupIndep
import Mathlib.Order.Atoms
#align_import order.partition.finpartition from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset Function
variable {α : Type*}
@[ext]
structure Finpartition [Lattice α]... | Mathlib/Order/Partition/Finpartition.lean | 178 | 182 | theorem ne_bot {b : α} (hb : b ∈ P.parts) : b ≠ ⊥ := by |
intro h
refine P.not_bot_mem (?_)
rw [h] at hb
exact hb
|
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/mathlib"@"1c4e18434eeb5546b212e830b2b39de6a83c473c"
-- Porting note: the lemmas `mk_complex` and `mk_univ_complex` should be in the namespace `Cardinal`
-- like their real counter... | Mathlib/Data/Complex/Cardinality.lean | 25 | 26 | theorem mk_complex : #ℂ = 𝔠 := by |
rw [mk_congr Complex.equivRealProd, mk_prod, lift_id, mk_real, continuum_mul_self]
|
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 209 | 211 | theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c • p) = c • ofFractionRing p := by |
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
|
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 89 | 101 | theorem locallyConnectedSpace_iff_connectedComponentIn_open :
LocallyConnectedSpace α ↔
∀ F : Set α, IsOpen F → ∀ x ∈ F, IsOpen (connectedComponentIn F x) := by |
constructor
· intro h
exact fun F hF x _ => hF.connectedComponentIn
· intro h
rw [locallyConnectedSpace_iff_open_connected_subsets]
refine fun x U hU =>
⟨connectedComponentIn (interior U) x,
(connectedComponentIn_subset _ _).trans interior_subset, h _ isOpen_interior x ?_,
... |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.InsertNth
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import data.vector.basic from "leanprover-community/mathlib"... | Mathlib/Data/Vector/Basic.lean | 112 | 115 | theorem tail_map {β : Type*} (v : Vector α (n + 1)) (f : α → β) :
(v.map f).tail = v.tail.map f := by |
obtain ⟨a, v', h⟩ := Vector.exists_eq_cons v
rw [h, map_cons, tail_cons, tail_cons]
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 107 | 110 | theorem relabel_id (t : L.Term α) : t.relabel id = t := by |
induction' t with _ _ _ _ ih
· rfl
· simp [ih]
|
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 83 | 84 | theorem Invariant.comp_const (hκ : Invariant κ μ) : κ ∘ₖ const α μ = const α μ := by |
rw [← const_bind_eq_comp_const κ μ, hκ.def]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Nat
import Mathlib.Data.Set.Basic
#align_import data.set.equitable from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α β : Type*}
namespace Set
def Equ... | Mathlib/Data/Set/Equitable.lean | 42 | 54 | theorem equitableOn_iff_exists_le_le_add_one {s : Set α} {f : α → ℕ} :
s.EquitableOn f ↔ ∃ b, ∀ a ∈ s, b ≤ f a ∧ f a ≤ b + 1 := by |
refine ⟨?_, fun ⟨b, hb⟩ x y hx hy => (hb x hx).2.trans (add_le_add_right (hb y hy).1 _)⟩
obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
· simp
intro hs
by_cases h : ∀ y ∈ s, f x ≤ f y
· exact ⟨f x, fun y hy => ⟨h _ hy, hs hy hx⟩⟩
push_neg at h
obtain ⟨w, hw, hwx⟩ := h
refine ⟨f w, fun y hy => ⟨Nat.le... |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.matrix.determinant from "leanprover-community/mathlib"@"c30... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 68 | 69 | theorem det_apply' (M : Matrix n n R) : M.det = ∑ σ : Perm n, ε σ * ∏ i, M (σ i) i := by |
simp [det_apply, Units.smul_def]
|
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 449 | 458 | theorem derivSeries_apply_diag (n : ℕ) (x : E) :
derivSeries p n (fun _ ↦ x) x = (n + 1) • p (n + 1) fun _ ↦ x := by |
simp only [derivSeries, compFormalMultilinearSeries_apply, changeOriginSeries,
compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe, LinearIsometryEquiv.coe_coe,
Function.comp_apply, ContinuousMultilinearMap.sum_apply, map_sum, coe_sum', Finset.sum_apply,
continuousMultilinearCurryFin1_apply, ... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 454 | 459 | theorem HasDerivWithinAt.clm_comp (hc : HasDerivWithinAt c c' s x)
(hd : HasDerivWithinAt d d' s x) :
HasDerivWithinAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x := by |
have := (hc.hasFDerivWithinAt.clm_comp hd.hasFDerivWithinAt).hasDerivWithinAt
rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
|
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 102 | 105 | theorem get_extract_aux {a : ByteArray} {start stop} (h : i < (a.extract start stop).size) :
start + i < a.size := by |
apply Nat.add_lt_of_lt_sub'; apply Nat.lt_of_lt_of_le h
rw [size_extract, ← Nat.sub_min_sub_right]; exact Nat.min_le_right ..
|
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 41 | 42 | theorem coeFn_le (f g : Lp E p μ) : f ≤ᵐ[μ] g ↔ f ≤ g := by |
rw [← Subtype.coe_le_coe, ← AEEqFun.coeFn_le]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 513 | 538 | theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : ℝ}
(hx : DifferentiableWithinAt ℝ f (Ici x) x) :
∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (derivWithin f (Ici x) x) r ε := by |
have := hx.hasDerivWithinAt
simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
refine ⟨m - x, by linarith [show x < m from xm], fun r hr => ?_⟩
have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
refine... |
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
namespace Sigma
variable {ι : Type*} {E : ι → Type... | Mathlib/Topology/MetricSpace/Gluing.lean | 358 | 361 | theorem fst_eq_of_dist_lt_one (x y : Σi, E i) (h : dist x y < 1) : x.1 = y.1 := by |
cases x; cases y
contrapose! h
apply one_le_dist_of_ne h
|
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 146 | 149 | theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set X) :
closure s = f ⁻¹' closure (f '' s) := by |
ext x
rw [Set.mem_preimage, ← closure_induced, hf.induced]
|
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 109 | 112 | theorem IsTensorProduct.map_eq (hf : IsTensorProduct f) (hg : IsTensorProduct g) (i₁ : M₁ →ₗ[R] N₁)
(i₂ : M₂ →ₗ[R] N₂) (x₁ : M₁) (x₂ : M₂) : hf.map hg i₁ i₂ (f x₁ x₂) = g (i₁ x₁) (i₂ x₂) := by |
delta IsTensorProduct.map
simp
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.ENNReal
#align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filte... | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | 73 | 75 | theorem summable_iff_not_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by |
rw [← not_iff_not, Classical.not_not, not_summable_iff_tendsto_nat_atTop_of_nonneg hf]
|
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 56 | 60 | theorem pow_le_pow_right' {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m :=
let ⟨k, hk⟩ := Nat.le.dest h
calc
a ^ n ≤ a ^ n * a ^ k := le_mul_of_one_le_right' (one_le_pow_of_one_le' ha _)
_ = a ^ m := by | rw [← hk, pow_add]
|
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 59 | 62 | theorem supportedEquivMvPolynomial_symm_C (s : Set σ) (x : R) :
(supportedEquivMvPolynomial s).symm (C x) = algebraMap R (supported R s) x := by |
ext1
simp [supportedEquivMvPolynomial, MvPolynomial.algebraMap_eq]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
theorem powerset_insert (s : Set α) (a : α)... | Mathlib/Data/Set/Image.lean | 693 | 695 | theorem image_univ {f : α → β} : f '' univ = range f := by |
ext
simp [image, range]
|
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 91 | 98 | theorem map_coeIdeal (I : Ideal R) : (I : FractionalIdeal S P).map g = I := by |
ext x
simp only [mem_coeIdeal]
constructor
· rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩
exact ⟨y, hy, (g.commutes y).symm⟩
· rintro ⟨y, hy, rfl⟩
exact ⟨_, ⟨y, hy, rfl⟩, g.commutes y⟩
|
import Mathlib.Data.PNat.Basic
#align_import data.pnat.find from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
namespace PNat
variable {p q : ℕ+ → Prop} [DecidablePred p] [DecidablePred q] (h : ∃ n, p n)
instance decidablePredExistsNat : DecidablePred fun n' : ℕ => ∃ (n : ℕ+) (_ : n'... | Mathlib/Data/PNat/Find.lean | 118 | 124 | theorem find_comp_succ (h : ∃ n, p n) (h₂ : ∃ n, p (n + 1)) (h1 : ¬p 1) :
PNat.find h = PNat.find h₂ + 1 := by |
refine (find_eq_iff _).2 ⟨PNat.find_spec h₂, fun n => PNat.recOn n ?_ ?_⟩
· simp [h1]
intro m _ hm
simp only [add_lt_add_iff_right, lt_find_iff] at hm
exact hm _ le_rfl
|
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.cofix from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 64 | 66 | theorem corecF_eq {α : TypeVec n} {β : Type u} (g : β → F (α.append1 β)) (x : β) :
M.dest q.P (corecF g x) = appendFun id (corecF g) <$$> repr (g x) := by |
rw [corecF, M.dest_corec]
|
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 117 | 119 | theorem vadd_vsub_assoc (g : G) (p₁ p₂ : P) : g +ᵥ p₁ -ᵥ p₂ = g + (p₁ -ᵥ p₂) := by |
apply vadd_right_cancel p₂
rw [vsub_vadd, add_vadd, vsub_vadd]
|
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 198 | 199 | theorem map [MonoidHomClass F M N] (f : F) {x : M} (h : IsUnit x) : IsUnit (f x) := by |
rcases h with ⟨y, rfl⟩; exact (Units.map (f : M →* N) y).isUnit
|
import Mathlib.Topology.GDelta
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.Instances.EReal
import Mathlib.Analysis.Normed.Group.Basic
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
noncomputable ... | Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 63 | 69 | theorem borel_eq_top_of_countable [TopologicalSpace α] [T1Space α] [Countable α] : borel α = ⊤ := by |
refine top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ ?_
apply MeasurableSet.biUnion s.to_countable
intro x _
apply MeasurableSet.of_compl
apply GenerateMeasurable.basic
exact isClosed_singleton.isOpen_compl
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 92 | 104 | theorem cons_update : cons x (update p i y) = update (cons x p) i.succ y := by |
ext j
by_cases h : j = 0
· rw [h]
simp [Ne.symm (succ_ne_zero i)]
· let j' := pred j h
have : j'.succ = j := succ_pred j h
rw [← this, cons_succ]
by_cases h' : j' = i
· rw [h']
simp
· have : j'.succ ≠ i.succ := by rwa [Ne, succ_inj]
rw [update_noteq h', update_noteq this, co... |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 114 | 114 | theorem abs_circleMap_zero (R : ℝ) (θ : ℝ) : abs (circleMap 0 R θ) = |R| := by | simp [circleMap]
|
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 84 | 86 | theorem nontrivial_of_einfsep_lt_top (hs : s.einfsep < ∞) : s.Nontrivial := by |
rcases einfsep_lt_top.1 hs with ⟨_, hx, _, hy, hxy, _⟩
exact ⟨_, hx, _, hy, hxy⟩
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
variable {R : Type*} [CommRing R] (M : Submonoid R) ... | Mathlib/RingTheory/Localization/Submodule.lean | 94 | 96 | theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S := by |
rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h ⊢
exact OrderEmbedding.wellFounded (IsLocalization.orderEmbedding M S).dual h
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 65 | 67 | theorem exp_inj_of_neg_pi_lt_of_le_pi {x y : ℂ} (hx₁ : -π < x.im) (hx₂ : x.im ≤ π) (hy₁ : -π < y.im)
(hy₂ : y.im ≤ π) (hxy : exp x = exp y) : x = y := by |
rw [← log_exp hx₁ hx₂, ← log_exp hy₁ hy₂, hxy]
|
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 94 | 108 | theorem updateRow_eq_transvection [Finite n] (c : R) :
updateRow (1 : Matrix n n R) i ((1 : Matrix n n R) i + c • (1 : Matrix n n R) j) =
transvection i j c := by |
cases nonempty_fintype n
ext a b
by_cases ha : i = a
· by_cases hb : j = b
· simp only [updateRow_self, transvection, ha, hb, Pi.add_apply, StdBasisMatrix.apply_same,
one_apply_eq, Pi.smul_apply, mul_one, Algebra.id.smul_eq_mul, add_apply]
· simp only [updateRow_self, transvection, ha, hb, StdB... |
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 92 | 114 | theorem mem_closure_isSwap {S : Set (Perm α)} (hS : ∀ f ∈ S, f.IsSwap) {f : Perm α} :
f ∈ closure S ↔ (fixedBy α f)ᶜ.Finite ∧ ∀ x, f x ∈ orbit (closure S) x := by |
refine ⟨fun hf ↦ ⟨?_, fun x ↦ mem_orbit_iff.mpr ⟨⟨f, hf⟩, rfl⟩⟩, ?_⟩
· exact finite_compl_fixedBy_closure_iff.mpr (fun f hf ↦ (hS f hf).finite_compl_fixedBy) _ hf
rintro ⟨fin, hf⟩
set supp := (fixedBy α f)ᶜ with supp_eq
suffices h : (fixedBy α f)ᶜ ⊆ supp → f ∈ closure S from h supp_eq.symm.subset
clear_val... |
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Fin Topology
-- Porting note: added explicit universes to fix compile
universe u u' v w x
... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 119 | 124 | theorem congr (p : FormalMultilinearSeries 𝕜 E F) {m n : ℕ} {v : Fin m → E} {w : Fin n → E}
(h1 : m = n) (h2 : ∀ (i : ℕ) (him : i < m) (hin : i < n), v ⟨i, him⟩ = w ⟨i, hin⟩) :
p m v = p n w := by |
subst n
congr with ⟨i, hi⟩
exact h2 i hi hi
|
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.preadditive.of_biproducts from "leanprover-community/mathlib"@"061ea99a5610cfc72c286aa930d3c1f47f74f3d0"
noncomputable section
universe v u
op... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 54 | 68 | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by |
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := by
intro f
ext
· aesop_cat
· simp [biprod.lift_snd... |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Function
namespace SemiconjBy
@[simp... | Mathlib/Algebra/Ring/Semiconj.lean | 39 | 41 | theorem add_left [Distrib R] {a b x y : R} (ha : SemiconjBy a x y) (hb : SemiconjBy b x y) :
SemiconjBy (a + b) x y := by |
simp only [SemiconjBy, left_distrib, right_distrib, ha.eq, hb.eq]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanprover-community/mathlib"@"a6ece35404f60597c651689c1b46ead86de5ac1b"
open Function
variable {R A : Type*}
def IsSelfAdjoint [Star R] (x : R) : Prop :=
... | Mathlib/Algebra/Star/SelfAdjoint.lean | 141 | 142 | theorem sub {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x - y) := by |
simp only [isSelfAdjoint_iff, star_sub, hx.star_eq, hy.star_eq]
|
import Mathlib.Topology.ContinuousFunction.ZeroAtInfty
open Topology Filter
variable {E F 𝓕 : Type*}
variable [SeminormedAddGroup E] [SeminormedAddCommGroup F]
variable [FunLike 𝓕 E F] [ZeroAtInftyContinuousMapClass 𝓕 E F]
theorem ZeroAtInftyContinuousMapClass.norm_le (f : 𝓕) (ε : ℝ) (hε : 0 < ε) :
∃ (r ... | Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean | 38 | 49 | theorem zero_at_infty_of_norm_le (f : E → F)
(h : ∀ (ε : ℝ) (_hε : 0 < ε), ∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε) :
Tendsto f (cocompact E) (𝓝 0) := by |
rw [tendsto_zero_iff_norm_tendsto_zero]
intro s hs
rw [mem_map, Metric.mem_cocompact_iff_closedBall_compl_subset 0]
rw [Metric.mem_nhds_iff] at hs
rcases hs with ⟨ε, hε, hs⟩
rcases h ε hε with ⟨r, hr⟩
use r
intro
aesop
|
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 126 | 127 | theorem mul_inverse_cancel_left (x y : M₀) (h : IsUnit x) : x * (inverse x * y) = y := by |
rw [← mul_assoc, mul_inverse_cancel x h, one_mul]
|
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 75 | 92 | theorem isIntegrallyClosed_dvd {s : S} (hs : IsIntegral R s) {p : R[X]}
(hp : Polynomial.aeval s p = 0) : minpoly R s ∣ p := by |
let K := FractionRing R
let L := FractionRing S
let _ : Algebra K L := FractionRing.liftAlgebra R L
have := FractionRing.isScalarTower_liftAlgebra R L
have : minpoly K (algebraMap S L s) ∣ map (algebraMap R K) (p %ₘ minpoly R s) := by
rw [map_modByMonic _ (minpoly.monic hs), modByMonic_eq_sub_mul_div]
... |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 91 | 93 | theorem convexHull_subset_space (hs : s ∈ K.faces) : convexHull 𝕜 ↑s ⊆ K.space := by |
convert subset_biUnion_of_mem hs
rfl
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 86 | 91 | theorem weightedVSubOfPoint_congr {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P}
(hp : ∀ i ∈ s, p₁ i = p₂ i) (b : P) :
s.weightedVSubOfPoint p₁ b w₁ = s.weightedVSubOfPoint p₂ b w₂ := by |
simp_rw [weightedVSubOfPoint_apply]
refine sum_congr rfl fun i hi => ?_
rw [hw i hi, hp i hi]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 101 | 101 | theorem card_box : (box n d).card = d ^ n := by | simp [box]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 118 | 120 | theorem HasStrictDerivAt.scomp (hg : HasStrictDerivAt g₁ g₁' (h x)) (hh : HasStrictDerivAt h h' x) :
HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x := by |
simpa using ((hg.restrictScalars 𝕜).comp x hh).hasStrictDerivAt
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 79 | 104 | theorem norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi {x y : V}
(h : angle x (x - y) = angle y (y - x)) (hpi : angle x y ≠ π) : ‖x‖ = ‖y‖ := by |
replace h := Real.arccos_injOn (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x (x - y)))
(abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one y (y - x))) h
by_cases hxy : x = y
· rw [hxy]
· rw [← norm_neg (y - x), neg_sub, mul_comm, mul_comm ‖y‖, div_eq_mul_inv, div_eq_mul_inv,
mul_inv_rev, mul_... |
import Mathlib.Data.Set.Basic
#align_import order.well_founded from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104e7bbb471381592"
variable {α β γ : Type*}
namespace WellFounded
variable {r r' : α → α → Prop}
#align well_founded_relation.r WellFoundedRelation.rel
protected theorem isAsymm (h : Well... | Mathlib/Order/WellFounded.lean | 82 | 89 | theorem wellFounded_iff_has_min {r : α → α → Prop} :
WellFounded r ↔ ∀ s : Set α, s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬r x m := by |
refine ⟨fun h => h.has_min, fun h => ⟨fun x => ?_⟩⟩
by_contra hx
obtain ⟨m, hm, hm'⟩ := h {x | ¬Acc r x} ⟨x, hx⟩
refine hm ⟨_, fun y hy => ?_⟩
by_contra hy'
exact hm' y hy' hy
|
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Preadditive.LeftExact
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.Algebra.Homology.Exact
import Mathli... | Mathlib/CategoryTheory/Abelian/Exact.lean | 115 | 120 | theorem exact_epi_comp_iff {W : C} (h : W ⟶ X) [Epi h] : Exact (h ≫ f) g ↔ Exact f g := by |
refine ⟨fun hfg => ?_, fun h => exact_epi_comp h⟩
let hc := isCokernelOfComp _ _ (colimit.isColimit (parallelPair (h ≫ f) 0))
(by rw [← cancel_epi h, ← Category.assoc, CokernelCofork.condition, comp_zero]) rfl
refine (exact_iff' _ _ (limit.isLimit _) hc).2 ⟨?_, ((exact_iff _ _).1 hfg).2⟩
exact zero_of_epi_... |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 135 | 141 | theorem mapsTo_toEnd_weightSpace_add_of_mem_rootSpace (α χ : H → R)
{x : L} (hx : x ∈ rootSpace H α) :
MapsTo (toEnd R L M x) (weightSpace M χ) (weightSpace M (α + χ)) := by |
intro m hm
let x' : rootSpace H α := ⟨x, hx⟩
let m' : weightSpace M χ := ⟨m, hm⟩
exact (rootSpaceWeightSpaceProduct R L H M α χ (α + χ) rfl (x' ⊗ₜ m')).property
|
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 61 | 61 | theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by | use 1, i; simp
|
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 37 | 88 | theorem smul_singleton_mem_nhds_of_sigmaCompact
{U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by |
/- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
space. By Baire, one of them... |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureTheory.Constructions.Polish
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Topology.Algebra.Order.Floor
#align_import measure_theory.integral.periodic from "leanprover-c... | Mathlib/MeasureTheory/Integral/Periodic.lean | 267 | 274 | theorem intervalIntegral_add_eq (hf : Periodic f T) (t s : ℝ) :
∫ x in t..t + T, f x = ∫ x in s..s + T, f x := by |
rcases lt_trichotomy (0 : ℝ) T with (hT | rfl | hT)
· exact hf.intervalIntegral_add_eq_of_pos hT t s
· simp
· rw [← neg_inj, ← integral_symm, ← integral_symm]
simpa only [← sub_eq_add_neg, add_sub_cancel_right] using
hf.neg.intervalIntegral_add_eq_of_pos (neg_pos.2 hT) (t + T) (s + T)
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.Normed.Group.Completion
#align_import analysis.normed.group.hom_completion from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
noncomputable section
open Set NormedAddGroupHom UniformSpace
section Completion
variable {G... | Mathlib/Analysis/Normed/Group/HomCompletion.lean | 107 | 113 | theorem NormedAddGroupHom.completion_comp (f : NormedAddGroupHom G H) (g : NormedAddGroupHom H K) :
g.completion.comp f.completion = (g.comp f).completion := by |
ext x
rw [NormedAddGroupHom.coe_comp, NormedAddGroupHom.completion_def,
NormedAddGroupHom.completion_coe_to_fun, NormedAddGroupHom.completion_coe_to_fun,
Completion.map_comp g.uniformContinuous f.uniformContinuous]
rfl
|
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c75b07bceaaac130ce23044d1386"
universe u v
namespace SimpleGraph
variable {V : Type u} (G : SimpleGraph V)
structure Partition where
parts : Set (Set V)
... | Mathlib/Combinatorics/SimpleGraph/Partition.lean | 93 | 95 | theorem mem_partOfVertex (v : V) : v ∈ P.partOfVertex v := by |
obtain ⟨⟨_, h⟩, _⟩ := (P.isPartition.2 v).choose_spec
exact h
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 112 | 114 | theorem map_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) : f n = f 0 + n • b := by |
simpa using map_add_nat' f 0 n
|
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 84 | 86 | theorem remainder_summable {m : ℝ} (hm : 1 < m) (k : ℕ) :
Summable fun i : ℕ => 1 / m ^ (i + (k + 1))! := by |
convert (summable_nat_add_iff (k + 1)).2 (LiouvilleNumber.summable hm)
|
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.properties from "leanprover-community/mathlib"@"88... | Mathlib/AlgebraicGeometry/Properties.lean | 84 | 93 | theorem isReducedOfOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[IsReduced Y] : IsReduced X := by |
constructor
intro U
have : U = (Opens.map f.1.base).obj (H.base_open.isOpenMap.functor.obj U) := by
ext1; exact (Set.preimage_image_eq _ H.base_open.inj).symm
rw [this]
exact isReduced_of_injective (inv <| f.1.c.app (op <| H.base_open.isOpenMap.functor.obj U))
(asIso <| f.1.c.app (op <| H.base_open.i... |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
assert_not_exists Finset
assert_not_exists Module
assert_not_exists Submonoid
assert_not_exists FloorRing
structure Real w... | Mathlib/Data/Real/Basic.lean | 130 | 132 | theorem ofCauchy_sub (a b) : (⟨a - b⟩ : ℝ) = ⟨a⟩ - ⟨b⟩ := by |
rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]
rfl
|
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 265 | 274 | theorem mem_toList_iff {y : α} : y ∈ toList p x ↔ SameCycle p x y ∧ x ∈ p.support := by |
simp only [toList, mem_range, mem_map]
constructor
· rintro ⟨n, hx, rfl⟩
refine ⟨⟨n, rfl⟩, ?_⟩
contrapose! hx
rw [← support_cycleOf_eq_nil_iff] at hx
simp [hx]
· rintro ⟨h, hx⟩
simpa using h.exists_pow_eq_of_mem_support hx
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 96 | 111 | theorem isPreconnected_of_forall {s : Set α} (x : α)
(H : ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) : IsPreconnected s := by |
rintro u v hu hv hs ⟨z, zs, zu⟩ ⟨y, ys, yv⟩
have xs : x ∈ s := by
rcases H y ys with ⟨t, ts, xt, -, -⟩
exact ts xt
-- Porting note (#11215): TODO: use `wlog xu : x ∈ u := hs xs using u v y z, v u z y`
cases hs xs with
| inl xu =>
rcases H y ys with ⟨t, ts, xt, yt, ht⟩
have := ht u v hu hv (ts... |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 54 | 54 | theorem mem_support_pure_iff : a' ∈ (pure a).support ↔ a' = a := by | simp
|
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 66 | 67 | theorem approx_eta {n : ℕ} (x : CofixA F (n + 1)) : x = CofixA.intro (head' x) (children' x) := by |
cases x; rfl
|
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : ℕ → ℕ
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 73 | 76 | theorem factorial_dvd_factorial {m n} (h : m ≤ n) : m ! ∣ n ! := by |
induction' h with n _ ih
· exact Nat.dvd_refl _
· exact Nat.dvd_trans ih (Nat.dvd_mul_left _ _)
|
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 47 | 51 | theorem continuous_algebraMap_iff_smul [TopologicalSemiring A] :
Continuous (algebraMap R A) ↔ Continuous fun p : R × A => p.1 • p.2 := by |
refine ⟨fun h => ?_, fun h => have : ContinuousSMul R A := ⟨h⟩; continuous_algebraMap _ _⟩
simp only [Algebra.smul_def]
exact (h.comp continuous_fst).mul continuous_snd
|
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 307 | 309 | theorem HasStrictFDerivAt.smul_const (hc : HasStrictFDerivAt c c' x) (f : F) :
HasStrictFDerivAt (fun y => c y • f) (c'.smulRight f) x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasStrictFDerivAt_const f x)
|
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Hom
open Set Pointwise
theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) := by
lift m to Mˣ using hm
rw [Functio... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 87 | 91 | theorem preimage_smul_setₛₗ_of_units (hc : IsUnit c) (hc' : IsUnit (σ c)) :
h ⁻¹' (σ c • t) = c • h ⁻¹' t := by |
apply preimage_smul_setₛₗ'
· exact (MulAction.smul_bijective_of_is_unit hc).surjective
· exact (MulAction.smul_bijective_of_is_unit hc').injective
|
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 98 | 100 | theorem singletonFinsubgraph_le_adj_right {u v : V} {e : G.Adj u v} :
singletonFinsubgraph v ≤ finsubgraphOfAdj e := by |
simp [singletonFinsubgraph, finsubgraphOfAdj]
|
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 149 | 150 | theorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by |
rw [← toComplex_zero, toComplex_inj]
|
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 147 | 150 | theorem hasLineDerivAt_iff_isLittleO_nhds_zero :
HasLineDerivAt 𝕜 f f' x v ↔
(fun t : 𝕜 => f (x + t • v) - f x - t • f') =o[𝓝 0] fun t => t := by |
simp only [HasLineDerivAt, hasDerivAt_iff_isLittleO_nhds_zero, zero_add, zero_smul, add_zero]
|
import Mathlib.Data.Fin.VecNotation
#align_import data.fin.tuple.monotone from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Fin Matrix Function
variable {α : Type*}
| Mathlib/Data/Fin/Tuple/Monotone.lean | 21 | 24 | theorem liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :
((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f := by |
simp only [liftFun_iff_succ r, forall_fin_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,
castSucc_zero]
|
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.mv_polynomial from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
set_option linter.uppercaseLean3 false
varia... | Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean | 75 | 80 | theorem det_mvPolynomialX_ne_zero [DecidableEq m] [Fintype m] [CommRing R] [Nontrivial R] :
det (mvPolynomialX m m R) ≠ 0 := by |
intro h_det
have := congr_arg Matrix.det (mvPolynomialX_mapMatrix_eval (1 : Matrix m m R))
rw [det_one, ← RingHom.map_det, h_det, RingHom.map_zero] at this
exact zero_ne_one this
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 131 | 131 | theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 := by | simp [rpow_def, *]
|
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.matrix.dot_product from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
variable {m n p R : Type*}
namespace Matrix
section Semiring
variable [Semiring... | Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 49 | 51 | theorem dotProduct_stdBasis_one [DecidableEq n] (v : n → R) (i : n) :
dotProduct v (LinearMap.stdBasis R (fun _ => R) i 1) = v i := by |
rw [dotProduct_stdBasis_eq_mul, mul_one]
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 249 | 250 | theorem boole_mul {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 := by | simp
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 261 | 263 | theorem HasGradientAtFilter.congr_of_eventuallyEq (h : HasGradientAtFilter f f' x L)
(hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x) : HasGradientAtFilter f₁ f' x L := by |
rwa [hL.hasGradientAtFilter_iff hx rfl]
|
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 263 | 277 | theorem finsuppTensorFinsupp_apply (f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ) :
finsuppTensorFinsupp R S M N ι κ (f ⊗ₜ g) (i, k) = f i ⊗ₜ g k := by |
apply Finsupp.induction_linear f
· simp
· intro f₁ f₂ hf₁ hf₂
simp [add_tmul, hf₁, hf₂]
intro i' m
apply Finsupp.induction_linear g
· simp
· intro g₁ g₂ hg₁ hg₂
simp [tmul_add, hg₁, hg₂]
intro k' n
classical
simp_rw [finsuppTensorFinsupp_single, Finsupp.single_apply, Prod.mk.inj_iff, ite_an... |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 94 | 98 | theorem idealOfSet_closed [T2Space R] (s : Set X) :
IsClosed (idealOfSet R s : Set C(X, R)) := by |
simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall]
exact isClosed_iInter fun x => isClosed_iInter fun _ =>
isClosed_eq (continuous_eval_const x) continuous_const
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Tactic.TFAE
#align_import ring_theory.ideal.local_ring from "leanprover-community/mathlib"@"ec1c7d810034d4202b0dd239112d1792be9f6fdc"
un... | Mathlib/RingTheory/Ideal/LocalRing.lean | 93 | 96 | theorem isUnit_or_isUnit_of_isUnit_add {a b : R} (h : IsUnit (a + b)) : IsUnit a ∨ IsUnit b := by |
rcases h with ⟨u, hu⟩
rw [← Units.inv_mul_eq_one, mul_add] at hu
apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right
|
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 119 | 122 | theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re := by |
apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
intro z
apply LinearIsometry.im_apply_eq_im h
|
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 161 | 169 | theorem StrongMono.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) [h : StrongMono f] : StrongMono g :=
{ mono := by |
rw [Arrow.iso_w' e]
haveI := mono_comp f e.hom.right
apply mono_comp
rlp := fun {X Y} z => by
intro
apply HasLiftingProperty.of_arrow_iso_right z e }
|
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6"
namespace MvPolynomial
variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S]
noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno... | Mathlib/Algebra/MvPolynomial/Expand.lean | 82 | 84 | theorem rename_expand (f : σ → τ) (p : ℕ) (φ : MvPolynomial σ R) :
rename f (expand p φ) = expand p (rename f φ) := by |
simp [expand, bind₁_rename, rename_bind₁, Function.comp]
|
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 38 | 43 | theorem discr_prime_pow_ne_two' [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hk : p ^ (k + 1) ≠ 2) :
discr ℚ (hζ.subOnePowerBasis ℚ).basis =
(-1) ^ ((p ^ (k + 1) : ℕ).totient / 2) * p ^ ((p : ℕ) ^ k * ((p - 1) * (k + 1) - 1)) := by |
rw [← discr_prime_pow_ne_two hζ (cyclotomic.irreducible_rat (p ^ (k + 1)).pos) hk]
exact hζ.discr_zeta_eq_discr_zeta_sub_one.symm
|
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
open MeasureTheory Topology Metric Filter Set ENNReal NNReal
open scoped Topology ENNReal NNReal BoundedContinuousFunction
section auxiliary
namespace MeasureTheory
variable {Ω : Type*} [TopologicalSpace Ω] [Mea... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 56 | 65 | theorem tendsto_lintegral_nn_filter_of_le_const {ι : Type*} {L : Filter ι} [L.IsCountablyGenerated]
(μ : Measure Ω) [IsFiniteMeasure μ] {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
(fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c) {f : Ω → ℝ≥0}
(fs_lim : ∀ᵐ ω : Ω ∂μ, Tendsto (fun i ↦ fs i ω) L (𝓝 (f ω))) :
Tendsto (... |
refine tendsto_lintegral_filter_of_dominated_convergence (fun _ ↦ c)
(eventually_of_forall fun i ↦ (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_
(@lintegral_const_lt_top _ _ μ _ _ (@ENNReal.coe_ne_top c)).ne ?_
· simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const
· si... |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 75 | 77 | theorem eq_reverse_iff [h : HasInvolutiveReverse V] {a b : V} (f : a ⟶ b)
(g : b ⟶ a) : f = reverse g ↔ reverse f = g := by |
rw [← reverse_inj, reverse_reverse]
|
import Mathlib.Data.Finset.Grade
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finset.interval from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
variable {α β : Type*}
namespace Finset
section Decidable
variable [DecidableEq α] (s t : Finset α)
instance instLocally... | Mathlib/Data/Finset/Interval.lean | 120 | 121 | theorem card_Ioo_finset (h : s ⊆ t) : (Ioo s t).card = 2 ^ (t.card - s.card) - 2 := by |
rw [card_Ioo_eq_card_Icc_sub_two, card_Icc_finset h]
|
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 194 | 197 | theorem eq_iSup_inf_genEigenspace [FiniteDimensional K V]
(h : ∀ x ∈ p, f x ∈ p) (h' : ⨆ μ, ⨆ k, f.genEigenspace μ k = ⊤) :
p = ⨆ μ, ⨆ k, p ⊓ f.genEigenspace μ k := by |
rw [← inf_iSup_genEigenspace h, h', inf_top_eq]
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Sigma
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Vector
import Mathlib.Algebra.BigOperators.Option
#align_import data.fintype.big_operators from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde5525935... | Mathlib/Data/Fintype/BigOperators.lean | 37 | 37 | theorem prod_bool [CommMonoid α] (f : Bool → α) : ∏ b, f b = f true * f false := by | simp
|
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Topology.Category.CompHaus.Limits
namespace Profinite
universe u w
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
section Pullbacks
variable ... | Mathlib/Topology/Category/Profinite/Limits.lean | 128 | 131 | theorem pullback_snd_eq :
Profinite.pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
|
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
open Ideal Polynomial PrimeSpectrum Set
namespace AlgebraicGeometry
names... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 38 | 40 | theorem isOpen_imageOfDf : IsOpen (imageOfDf f) := by |
rw [imageOfDf, setOf_exists fun i (x : PrimeSpectrum R) => coeff f i ∉ x.asIdeal]
exact isOpen_iUnion fun i => isOpen_basicOpen
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanprover-community/mathlib"@"a6ece35404f60597c651689c1b46ead86de5ac1b"
open Function
variable {R A : Type*}
def IsSelfAdjoint [Star R] (x : R) : Prop :=
... | Mathlib/Algebra/Star/SelfAdjoint.lean | 224 | 225 | theorem mul {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x * y) := by |
simp only [isSelfAdjoint_iff, star_mul', hx.star_eq, hy.star_eq]
|
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (𝕜 A : Type*) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A]
variable [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 167 | 180 | theorem antilipschitzWith_addEquiv :
AntilipschitzWith 2 (addEquiv 𝕜 A) := by |
refine AddMonoidHomClass.antilipschitz_of_bound (addEquiv 𝕜 A) fun x => ?_
rw [norm_eq_sup, Prod.norm_def, NNReal.coe_two]
refine max_le ?_ ?_
· rw [mul_max_of_nonneg _ _ (zero_le_two : (0 : ℝ) ≤ 2)]
exact le_max_of_le_left ((le_add_of_nonneg_left (norm_nonneg _)).trans_eq (two_mul _).symm)
· nontrivial... |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 482 | 491 | theorem schur_complement_eq₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A]
(hA : A.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star (x + (A⁻¹ * B) *ᵥ y)) ᵥ* A ⬝ᵥ (x + (A⁻¹ * B) *ᵥ... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 97 | 107 | theorem finite_mulSupport {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => v.maxPowDividing I).Finite :=
haveI h_subset : {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} ⊆
{v : HeightOneSpectrum R |
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) ≠ 0} := by |
intro v hv h_zero
have hv' : v.maxPowDividing I = 1 := by
rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
pow_zero _]
exact hv hv'
Finite.subset (Filter.eventually_cofinite.mp (Associates.finite_factors hI)) h_subset
|
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 79 | 82 | theorem pentagon_inv_hom (W X Y Z : C) :
(α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) =
(α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) ≫ (α_ W (X ⊗ Y) Z).inv := by |
coherence
|
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 121 | 129 | theorem iff_card_le_one [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :
UniqueMul A B a0 b0 ↔ ((A ×ˢ B).filter (fun p ↦ p.1 * p.2 = a0 * b0)).card ≤ 1 := by |
simp_rw [card_le_one_iff, mem_filter, mem_product]
refine ⟨fun h p1 p2 ⟨⟨ha1, hb1⟩, he1⟩ ⟨⟨ha2, hb2⟩, he2⟩ ↦ ?_, fun h a b ha hb he ↦ ?_⟩
· have h1 := h ha1 hb1 he1; have h2 := h ha2 hb2 he2
ext
· rw [h1.1, h2.1]
· rw [h1.2, h2.2]
· exact Prod.ext_iff.1 (@h (a, b) (a0, b0) ⟨⟨ha, hb⟩, he⟩ ⟨⟨ha0, hb0... |
import Mathlib.Data.Set.Lattice
#align_import data.semiquot from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
-- Porting note: removed universe parameter
structure Semiquot (α : Type*) where mk' ::
s : Set α
val : Trunc s
#align semiquot Semiquot
namespace Semiquot
vari... | Mathlib/Data/Semiquot.lean | 136 | 137 | theorem mem_bind (q : Semiquot α) (f : α → Semiquot β) (b : β) :
b ∈ bind q f ↔ ∃ a ∈ q, b ∈ f a := by | simp_rw [← exists_prop]; exact Set.mem_iUnion₂
|
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 41 | 44 | theorem eval_one_cyclotomic_prime_pow {R : Type*} [CommRing R] {p : ℕ} (k : ℕ)
[hn : Fact p.Prime] : eval 1 (cyclotomic (p ^ (k + 1)) R) = p := by |
simp only [cyclotomic_prime_pow_eq_geom_sum hn.out, eval_X, one_pow, Finset.sum_const, eval_pow,
eval_finset_sum, Finset.card_range, smul_one_eq_cast]
|
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