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.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
open Finset Matrix
namespace SimpleGraph
variable {V : Type*} (R : Type*)
variable [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj]
def degMatrix [AddMonoidWithOne R] : Matrix V V R := Matrix.diago... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 52 | 54 | theorem lapMatrix_mulVec_apply [NonAssocRing R] (v : V) (vec : V → R) :
(G.lapMatrix R *ᵥ vec) v = G.degree v * vec v - ∑ u ∈ G.neighborFinset v, vec u := by |
simp_rw [lapMatrix, sub_mulVec, Pi.sub_apply, degMatrix_mulVec_apply, adjMatrix_mulVec_apply]
|
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 | 110 | 111 | theorem gradient_eq_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : ∇ f x = 0 := by |
rw [gradient, fderiv_zero_of_not_differentiableAt h, map_zero]
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 93 | 93 | theorem div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a := by | rw [mul_comm, div_lt_iff hc]
|
import Mathlib.Topology.ContinuousOn
import Mathlib.Order.Filter.SmallSets
#align_import topology.locally_finite from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
-- locally finite family [General Topology (Bourbaki, 1995)]
open Set Function Filter Topology
variable {ι ι' α X Y : Type... | Mathlib/Topology/LocallyFinite.lean | 91 | 101 | theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f)
(hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) :
ContinuousOn g (⋃ i, f i) := by |
rintro x -
rw [ContinuousWithinAt, hf.nhdsWithin_iUnion, tendsto_iSup]
intro i
by_cases hx : x ∈ closure (f i)
· exact hc i _ hx
· rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx
rw [hx]
exact tendsto_bot
|
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
∀ x ∈ s, convexHull ℝ (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 65 | 74 | theorem union {s t : Set F} (hs : AmpleSet s) (ht : AmpleSet t) : AmpleSet (s ∪ t) := by |
intro x hx
rcases hx with (h | h) <;>
-- The connected component of `x ∈ s` in `s ∪ t` contains the connected component of `x` in `s`,
-- hence is also full; similarly for `t`.
[have hx := hs x h; have hx := ht x h] <;>
rw [← Set.univ_subset_iff, ← hx] <;>
apply convexHull_mono <;>
apply connectedCompo... |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 92 | 97 | theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
evariance X μ < ∞ ↔ Memℒp X 2 μ := by |
refine ⟨?_, MeasureTheory.Memℒp.evariance_lt_top⟩
contrapose
rw [not_lt, top_le_iff]
exact evariance_eq_top hX
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 146 | 161 | theorem acc_iff_wellFoundedOn {α} {r : α → α → Prop} {a : α} :
TFAE [Acc r a,
WellFoundedOn { b | ReflTransGen r b a } r,
WellFoundedOn { b | TransGen r b a } r] := by |
tfae_have 1 → 2
· refine fun h => ⟨fun b => InvImage.accessible _ ?_⟩
rw [← acc_transGen_iff] at h ⊢
obtain h' | h' := reflTransGen_iff_eq_or_transGen.1 b.2
· rwa [h'] at h
· exact h.inv h'
tfae_have 2 → 3
· exact fun h => h.subset fun _ => TransGen.to_reflTransGen
tfae_have 3 → 1
· refine ... |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ ... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 177 | 179 | theorem card_edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset.card = G.edgeFinset.card + 1 := by |
rw [G.edgeFinset_sup_edge hn h, card_cons]
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Index
import Mathlib.Order.Interval.Finset.Nat
import Mat... | Mathlib/GroupTheory/OrderOfElement.lean | 107 | 110 | theorem Submonoid.isOfFinOrder_coe {H : Submonoid G} {x : H} :
IsOfFinOrder (x : G) ↔ IsOfFinOrder x := by |
rw [isOfFinOrder_iff_pow_eq_one, isOfFinOrder_iff_pow_eq_one]
norm_cast
|
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
set_option autoImplicit true
variable {C : Levenshtein.Cost α β δ} [CanonicallyLinearOrderedAddCommMonoid δ]
theorem suffixLevenshtein_minimum_le_levenshtein... | Mathlib/Data/List/EditDistance/Bounds.lean | 81 | 87 | theorem suffixLevenshtein_minimum_le_levenshtein_append (xs ys₁ ys₂) :
(suffixLevenshtein C xs ys₂).1.minimum ≤ levenshtein C xs (ys₁ ++ ys₂) := by |
cases ys₁ with
| nil => exact List.minimum_le_of_mem' (List.get_mem _ _ _)
| cons y ys₁ =>
exact (le_suffixLevenshtein_append_minimum _ _ _).trans
(suffixLevenshtein_minimum_le_levenshtein_cons _ _ _)
|
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 37 | 37 | theorem tfae_singleton (p) : TFAE [p] := by | simp [TFAE, -eq_iff_iff]
|
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 | 107 | 107 | theorem supported_empty : supported R (∅ : Set σ) = ⊥ := by | simp [supported_eq_adjoin_X]
|
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 120 | 121 | theorem tendsto_of_ne_zero {γ : Γ₀} (h : γ ≠ 0) : Tendsto f l (𝓝 γ) ↔ ∀ᶠ x in l, f x = γ := by |
rw [nhds_of_ne_zero h, tendsto_pure]
|
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
open Finset Matrix
namespace SimpleGraph
variable {V : Type*} (R : Type*)
variable [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj]
def degMatrix [AddMonoidWithOne R] : Matrix V V R := Matrix.diago... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 56 | 59 | theorem lapMatrix_mulVec_const_eq_zero [Ring R] : mulVec (G.lapMatrix R) (fun _ ↦ 1) = 0 := by |
ext1 i
rw [lapMatrix_mulVec_apply]
simp
|
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 134 | 137 | theorem map₂_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
map₂ f a (map₂ g b c) = map₂ g' b (map₂ f' a c) := by |
cases a <;> cases b <;> cases c <;> simp [h_left_comm]
|
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 123 | 127 | theorem eventuallyEq_of_toReal_eventuallyEq {l : Filter α} {f g : α → ℝ≥0∞}
(hfi : ∀ᶠ x in l, f x ≠ ∞) (hgi : ∀ᶠ x in l, g x ≠ ∞)
(hfg : (fun x => (f x).toReal) =ᶠ[l] fun x => (g x).toReal) : f =ᶠ[l] g := by |
filter_upwards [hfi, hgi, hfg] with _ hfx hgx _
rwa [← ENNReal.toReal_eq_toReal hfx hgx]
|
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 61 | 64 | theorem Reflexive.rel_of_ne_imp (h : Reflexive r) {x y : α} (hr : x ≠ y → r x y) : r x y := by |
by_cases hxy : x = y
· exact hxy ▸ h x
· exact hr hxy
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 162 | 163 | theorem pbind_eq_bind (f : α → Option β) (x : Option α) : (x.pbind fun a _ ↦ f a) = x.bind f := by |
cases x <;> simp only [pbind, none_bind', some_bind']
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 133 | 134 | theorem toIcoMod_sub_self (a b : α) : toIcoMod hp a b - b = -toIcoDiv hp a b • p := by |
rw [toIcoMod, sub_sub_cancel_left, neg_smul]
|
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
variable {n : ℕ}
variable {E : Type*} [NormedAddCommGroup E]
noncomputa... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 166 | 167 | theorem torusIntegral_neg (f : ℂⁿ → E) (c : ℂⁿ) (R : ℝⁿ) :
(∯ x in T(c, R), -f x) = -∯ x in T(c, R), f x := by | simp [torusIntegral, integral_neg]
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 584 | 588 | theorem oangle_right_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃) := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
|
import Mathlib.Topology.Instances.Int
#align_import topology.instances.nat from "leanprover-community/mathlib"@"620af85adf5cd4282f962eb060e6e562e3e0c0ba"
noncomputable section
open Metric Set Filter
namespace Nat
noncomputable instance : Dist ℕ :=
⟨fun x y => dist (x : ℝ) y⟩
theorem dist_eq (x y : ℕ) : dist ... | Mathlib/Topology/Instances/Nat.lean | 55 | 63 | theorem closedBall_eq_Icc (x : ℕ) (r : ℝ) : closedBall x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊ := by |
rcases le_or_lt 0 r with (hr | hr)
· rw [← preimage_closedBall, Real.closedBall_eq_Icc, preimage_Icc]
exact add_nonneg (cast_nonneg x) hr
· rw [closedBall_eq_empty.2 hr, Icc_eq_empty_of_lt]
calc ⌊(x : ℝ) + r⌋₊ ≤ ⌊(x : ℝ)⌋₊ := floor_mono <| by linarith
_ < ⌈↑x - r⌉₊ := by
rw [floor_natCast, Nat.... |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 106 | 109 | theorem powHalf_le_one (n : ℕ) : powHalf n ≤ 1 := by |
induction' n with n hn
· exact le_rfl
· exact (powHalf_succ_le_powHalf n).trans hn
|
import Mathlib.Topology.Order.ExtendFrom
import Mathlib.Topology.Algebra.Order.Compact
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.T5
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set Topology
variabl... | Mathlib/Topology/Algebra/Order/Rolle.lean | 37 | 55 | theorem exists_Ioo_extr_on_Icc (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c := by |
have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)
-- Consider absolute min and max points
obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x :=
isCompact_Icc.exists_isMinOn ne hfc
obtain ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f C :=
isCompact_Icc.exists_isMaxOn ne... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 86 | 89 | theorem derivative_monomial (a : R) (n : ℕ) :
derivative (monomial n a) = monomial (n - 1) (a * n) := by |
rw [derivative_apply, sum_monomial_index, C_mul_X_pow_eq_monomial]
simp
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 133 | 134 | theorem log_one_right (b : ℕ) : log b (1 : R) = 0 := by |
rw [log_of_one_le_right _ le_rfl, Nat.floor_one, Nat.log_one_right, Int.ofNat_zero]
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
suppress_compilation
set_option linter.uppercaseLean3 false
open Metric
open scoped Classical NNReal Topology Uniformity
variable {𝕜 E : Type*} [NontriviallyNormedField 𝕜]
section SemiNormed
variable [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
... | Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 243 | 246 | theorem opNorm_lsmul_le : ‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ ≤ 1 := by |
refine ContinuousLinearMap.opNorm_le_bound _ zero_le_one fun x => ?_
simp_rw [one_mul]
exact opNorm_lsmul_apply_le _
|
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
universe u
open LawfulTraversable
open Function hiding comp
open Functor
attribute [functor_norm] LawfulTraversabl... | Mathlib/Control/Traversable/Lemmas.lean | 123 | 126 | theorem traverse_map' (g : α → β) (h : β → G γ) :
traverse (h ∘ g) = (traverse h ∘ map g : t α → G (t γ)) := by |
ext
rw [comp_apply, traverse_map]
|
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic
na... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 87 | 100 | theorem EqualizerCondition.bijective_mapToEqualizer_pullback (P : Cᵒᵖ ⥤ Type*)
(hP : EqualizerCondition P) : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [HasPullback π π],
Function.Bijective
(MapToEqualizer P π (pullback.fst (f := π) (g := π)) (pullback.snd (f := π) (g := π))
pullback.condition) := b... |
intro X B π _ _
specialize hP π _ (pullbackIsPullback π π)
rw [Types.type_equalizer_iff_unique] at hP
rw [Function.bijective_iff_existsUnique]
intro ⟨b, hb⟩
obtain ⟨a, ha₁, ha₂⟩ := hP b hb
refine ⟨a, ?_, ?_⟩
· simpa [MapToEqualizer] using ha₁
· simpa [MapToEqualizer] using ha₂
|
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Logic.UnivLE
import Mathlib.Logic.Small.Basic
#align_import category_theory.essentially_small from "leanprover-community/mathlib"@"f7707875544ef1f81b32cb68c79e0e24e45a0e76"
universe w v v' u u'
open CategoryTheory
... | Mathlib/CategoryTheory/EssentiallySmall.lean | 71 | 77 | theorem essentiallySmall_congr {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
(e : C ≌ D) : EssentiallySmall.{w} C ↔ EssentiallySmall.{w} D := by |
fconstructor
· rintro ⟨S, 𝒮, ⟨f⟩⟩
exact EssentiallySmall.mk' (e.symm.trans f)
· rintro ⟨S, 𝒮, ⟨f⟩⟩
exact EssentiallySmall.mk' (e.trans f)
|
import Mathlib.Topology.MetricSpace.PiNat
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Gluing
import Mathlib.Topology.Sets.Opens
import Mathlib.Analysis.Normed.Field.Basic
#align_import topology.metric_space.polish from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78... | Mathlib/Topology/MetricSpace/Polish.lean | 155 | 163 | theorem _root_.ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β] [PolishSpace β]
{f : α → β} (hf : ClosedEmbedding f) : PolishSpace α := by |
letI := upgradePolishSpace β
letI : MetricSpace α := hf.toEmbedding.comapMetricSpace f
haveI : SecondCountableTopology α := hf.toEmbedding.secondCountableTopology
have : CompleteSpace α := by
rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]
exact hf.isClosed_range.isCo... |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 88 | 96 | theorem cotangent_subsingleton_iff : Subsingleton I.Cotangent ↔ IsIdempotentElem I := by |
constructor
· intro H
refine (pow_two I).symm.trans (le_antisymm (Ideal.pow_le_self two_ne_zero) ?_)
exact fun x hx => (I.toCotangent_eq_zero ⟨x, hx⟩).mp (Subsingleton.elim _ _)
· exact fun e =>
⟨fun x y =>
Quotient.inductionOn₂' x y fun x y =>
I.toCotangent_eq.mpr <| ((pow_two I)... |
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis
#align_import algebra.direct_sum.module from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v w u₁
namespace DirectSum
open DirectSum
section General
variable {... | Mathlib/Algebra/DirectSum/Module.lean | 164 | 168 | theorem linearEquivFunOnFintype_lof [Fintype ι] [DecidableEq ι] (i : ι) (m : M i) :
(linearEquivFunOnFintype R ι M) (lof R ι M i m) = Pi.single i m := by |
ext a
change (DFinsupp.equivFunOnFintype (lof R ι M i m)) a = _
convert _root_.congr_fun (DFinsupp.equivFunOnFintype_single i m) a
|
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 122 | 126 | theorem empty_of_count_eq_zero [MeasurableSingletonClass α] (hsc : count s = 0) : s = ∅ := by |
have hs : s.Finite := by
rw [← count_apply_lt_top, hsc]
exact WithTop.zero_lt_top
simpa [count_apply_finite _ hs] using hsc
|
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
import Mathlib.Order.SetNotation
#align_import order.complete_lattice from "leanprover-community/mathlib"@"5709b0d8725255e76f47debca6400c07b5c2... | Mathlib/Order/CompleteLattice.lean | 180 | 181 | theorem iInf_le_iff {s : ι → α} : iInf s ≤ a ↔ ∀ b, (∀ i, b ≤ s i) → b ≤ a := by |
simp [iInf, sInf_le_iff, lowerBounds]
|
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 80 | 83 | theorem toMatrix_self [DecidableEq ι] : e.toMatrix e = 1 := by |
unfold Basis.toMatrix
ext i j
simp [Basis.equivFun, Matrix.one_apply, Finsupp.single_apply, eq_comm]
|
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 132 | 133 | theorem toNNReal_strict_mono (hb : b ≠ ∞) (h : a < b) : a.toNNReal < b.toNNReal := by |
simpa [← ENNReal.coe_lt_coe, hb, h.ne_top]
|
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 255 | 257 | theorem iInf_ker_lapply_le_bot : ⨅ a, ker (lapply a : (α →₀ M) →ₗ[R] M) ≤ ⊥ := by |
simp only [SetLike.le_def, mem_iInf, mem_ker, mem_bot, lapply_apply]
exact fun a h => Finsupp.ext h
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 124 | 126 | theorem IsCoprime.mul_right (H1 : IsCoprime x y) (H2 : IsCoprime x z) : IsCoprime x (y * z) := by |
rw [isCoprime_comm] at H1 H2 ⊢
exact H1.mul_left H2
|
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 59 | 69 | theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
{ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
(hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine b.ext_elem fun i ↦ ?_
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
exact
(Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
⟨x, hx, rfl⟩
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 62 | 64 | theorem one_sub_K_ne_top : (1 : ℝ≥0∞) - K ≠ ∞ := by |
norm_cast
exact ENNReal.coe_ne_top
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 134 | 147 | theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ cylinder y n = cylinder x n := by |
constructor
· intro hy
apply Subset.antisymm
· intro z hz i hi
rw [← hy i hi]
exact hz i hi
· intro z hz i hi
rw [hy i hi]
exact hz i hi
· intro h
rw [← h]
exact self_mem_cylinder _ _
|
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Galois
universe u v w
open scoped Classical Polynomial
open Polynomial
variable (k : Type u) [Field k] (K : Type v) [Field K]
class IsSepClosed : Prop where
splits_of_separable : ∀ p : k[X], p.Separable → (p.Splits <| RingHom.... | Mathlib/FieldTheory/IsSepClosed.lean | 122 | 129 | theorem roots_eq_zero_iff [IsSepClosed k] {p : k[X]} (hsep : p.Separable) :
p.roots = 0 ↔ p = Polynomial.C (p.coeff 0) := by |
refine ⟨fun h => ?_, fun hp => by rw [hp, roots_C]⟩
rcases le_or_lt (degree p) 0 with hd | hd
· exact eq_C_of_degree_le_zero hd
· obtain ⟨z, hz⟩ := IsSepClosed.exists_root p hd.ne' hsep
rw [← mem_roots (ne_zero_of_degree_gt hd), h] at hz
simp at hz
|
import Mathlib.Data.Int.GCD
import Mathlib.Tactic.NormNum
namespace Tactic
namespace NormNum
theorem int_gcd_helper' {d : ℕ} {x y : ℤ} (a b : ℤ) (h₁ : (d : ℤ) ∣ x) (h₂ : (d : ℤ) ∣ y)
(h₃ : x * a + y * b = d) : Int.gcd x y = d := by
refine Nat.dvd_antisymm ?_ (Int.natCast_dvd_natCast.1 (Int.dvd_gcd h₁ h₂))
... | Mathlib/Tactic/NormNum/GCD.lean | 64 | 66 | theorem int_gcd_helper {x y : ℤ} {x' y' d : ℕ}
(hx : x.natAbs = x') (hy : y.natAbs = y') (h : Nat.gcd x' y' = d) :
Int.gcd x y = d := by | subst_vars; rw [Int.gcd_def]
|
import Mathlib.Order.PrimeIdeal
import Mathlib.Order.Zorn
universe u
variable {α : Type*}
open Order Ideal Set
variable [DistribLattice α] [BoundedOrder α]
variable {F : PFilter α} {I : Ideal α}
namespace DistribLattice
lemma mem_ideal_sup_principal (a b : α) (J : Ideal α) : b ∈ J ⊔ principal a ↔ ∃ j ∈ J, ... | Mathlib/Order/PrimeSeparator.lean | 46 | 143 | theorem prime_ideal_of_disjoint_filter_ideal (hFI : Disjoint (F : Set α) (I : Set α)) :
∃ J : Ideal α, (IsPrime J) ∧ I ≤ J ∧ Disjoint (F : Set α) J := by |
-- Let S be the set of ideals containing I and disjoint from F.
set S : Set (Set α) := { J : Set α | IsIdeal J ∧ I ≤ J ∧ Disjoint (F : Set α) J }
-- Then I is in S...
have IinS : ↑I ∈ S := by
refine ⟨Order.Ideal.isIdeal I, by trivial⟩
-- ...and S contains upper bounds for any non-empty chains.
have ... |
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Topology
section omegaLimit
variable {τ : Type*} {α : Type*} {β : Type*} {ι : Type*}
def omegaLimit [Topol... | Mathlib/Dynamics/OmegaLimit.lean | 70 | 74 | theorem omegaLimit_subset_of_tendsto {m : τ → τ} {f₁ f₂ : Filter τ} (hf : Tendsto m f₁ f₂) :
ω f₁ (fun t x ↦ ϕ (m t) x) s ⊆ ω f₂ ϕ s := by |
refine iInter₂_mono' fun u hu ↦ ⟨m ⁻¹' u, tendsto_def.mp hf _ hu, ?_⟩
rw [← image2_image_left]
exact closure_mono (image2_subset (image_preimage_subset _ _) Subset.rfl)
|
import Mathlib.Topology.Order.Basic
open Set Filter OrderDual
open scoped Topology
section OrderClosedTopology
variable {α : Type*} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α}
@[simp] theorem nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) := isOpen_Ioi.nhdsSet_eq
@[simp] theorem nhdsSet... | Mathlib/Topology/Order/NhdsSet.lean | 36 | 37 | theorem nhdsSet_Ici : 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a) := by |
rw [← Ioi_insert, nhdsSet_insert, nhdsSet_Ioi]
|
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
import Mathlib.Topology.Category.Stonean.Limits
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
universe u
open CategoryTheory Limits
namespace Stonean
noncomputable
def struct {B X : St... | Mathlib/Topology/Category/Stonean/EffectiveEpi.lean | 62 | 75 | theorem effectiveEpi_tfae
{B X : Stonean.{u}} (π : X ⟶ B) :
TFAE
[ EffectiveEpi π
, Epi π
, Function.Surjective π
] := by |
tfae_have 1 → 2
· intro; infer_instance
tfae_have 2 ↔ 3
· exact epi_iff_surjective π
tfae_have 3 → 1
· exact fun hπ ↦ ⟨⟨struct π hπ⟩⟩
tfae_finish
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 48 | 53 | theorem integralNormalization_coeff {f : R[X]} {i : ℕ} :
(integralNormalization f).coeff i =
if f.degree = i then 1 else coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) := by |
have : f.coeff i = 0 → f.degree ≠ i := fun hc hd => coeff_ne_zero_of_eq_degree hd hc
simp (config := { contextual := true }) [integralNormalization, coeff_monomial, this,
mem_support_iff]
|
import Mathlib.Algebra.Periodic
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.periodic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Nat
open Nat Function
theorem periodic_gcd (a : ℕ) : P... | Mathlib/Data/Nat/Periodic.lean | 29 | 30 | theorem periodic_coprime (a : ℕ) : Periodic (Coprime a) a := by |
simp only [coprime_add_self_right, forall_const, iff_self_iff, eq_iff_iff, Periodic]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 95 | 98 | theorem toMeasurable_iff {X : Ω → E} {s : Set E} :
IsUniform X (toMeasurable μ s) ℙ μ ↔ IsUniform X s ℙ μ := by |
unfold IsUniform
rw [ProbabilityTheory.cond_toMeasurable_eq]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Sym.Card
open Finset Function
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V) {e : Sym2 V}
section EdgeFinset
variable {G₁ G₂ : SimpleGraph V} [Fintype G.edgeSet] [Fintype G₁.edgeSet] [Finty... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 90 | 90 | theorem edgeFinset_bot : (⊥ : SimpleGraph V).edgeFinset = ∅ := by | simp [edgeFinset]
|
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 114 | 114 | theorem re_exp (q : ℍ[ℝ]) : (exp ℝ q).re = exp ℝ q.re * Real.cos ‖q - q.re‖ := by | simp [exp_eq]
|
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[Topolo... | Mathlib/Topology/Inseparable.lean | 95 | 96 | theorem ker_nhds_eq_specializes : (𝓝 x).ker = {y | y ⤳ x} := by |
ext; simp [specializes_iff_pure, le_def]
|
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
noncomputable section
namespace Finsupp
variable {n : ℕ} (i : Fin n) {M : Type*} [Zero M] (y : M) (t : Fin (n + 1) →₀ M) (s : Fin n →₀ M)
def tail (s : Fin (n + 1) →₀ ... | Mathlib/Data/Finsupp/Fin.lean | 83 | 86 | theorem cons_ne_zero_of_right (h : s ≠ 0) : cons y s ≠ 0 := by |
contrapose! h with c
ext a
simp [← cons_succ a y s, c]
|
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antid... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 44 | 46 | theorem coeff_trunc (m) (n) (φ : R⟦X⟧) :
(trunc n φ).coeff m = if m < n then coeff R m φ else 0 := by |
simp [trunc, Polynomial.coeff_sum, Polynomial.coeff_monomial, Nat.lt_succ_iff]
|
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 84 | 86 | theorem prod_empty : s ×ˢ (∅ : Set β) = ∅ := by |
ext
exact and_false_iff _
|
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {α ι ι' : Type*}
instance [DecidableEq α] {r : α → α → Prop} [DecidableRel r] {s : Finset α} :
Decidable ((s : Set α).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 27 | 30 | theorem Finset.pairwiseDisjoint_range_singleton :
(Set.range (singleton : α → Finset α)).PairwiseDisjoint id := by |
rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ h
exact disjoint_singleton.2 (ne_of_apply_ne _ h)
|
import Mathlib.Algebra.Order.Ring.Int
#align_import data.int.least_greatest from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
namespace Int
def leastOfBdd {P : ℤ → Prop} [DecidablePred P] (b : ℤ) (Hb : ∀ z : ℤ, P z → b ≤ z)
(Hinh : ∃ z : ℤ, P z) : { lb : ℤ // P lb ∧ ∀ z : ℤ, P z... | Mathlib/Data/Int/LeastGreatest.lean | 96 | 103 | theorem exists_greatest_of_bdd
{P : ℤ → Prop}
(Hbdd : ∃ b : ℤ , ∀ z : ℤ , P z → z ≤ b)
(Hinh : ∃ z : ℤ , P z) : ∃ ub : ℤ , P ub ∧ ∀ z : ℤ , P z → z ≤ ub := by |
classical
let ⟨b, Hb⟩ := Hbdd
let ⟨lb, H⟩ := greatestOfBdd b Hb Hinh
exact ⟨lb, H⟩
|
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 109 | 114 | theorem dependent_pair_iff_eq (u v : ℙ K V) : Dependent ![u, v] ↔ u = v := by |
rw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2,
Function.comp_apply, Matrix.cons_val_one, Matrix.head_cons, Ne]
simp only [Matrix.cons_val_zero, not_and, not_forall, Classical.not_not, Function.comp_apply,
← mk_eq_mk_iff' K _ _ (rep_nonzero u) (rep_nonzero v), mk_rep, Classical.... |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 110 | 111 | theorem count_succ_eq_count_iff {n : ℕ} : count p (n + 1) = count p n ↔ ¬p n := by |
by_cases h : p n <;> simp [h, count_succ]
|
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 131 | 133 | theorem Matrix.Represents.add {A A' : Matrix ι ι R} {f f' : Module.End R M} (h : A.Represents b f)
(h' : Matrix.Represents b A' f') : (A + A').Represents b (f + f') := by |
delta Matrix.Represents at h h' ⊢; rw [map_add, map_add, h, h']
|
import Mathlib.Algebra.Group.Center
#align_import group_theory.subsemigroup.centralizer from "leanprover-community/mathlib"@"cc67cd75b4e54191e13c2e8d722289a89e67e4fa"
variable {M : Type*} {S T : Set M}
namespace Set
variable (S)
@[to_additive addCentralizer " The centralizer of a subset of an additive magma. ... | Mathlib/Algebra/Group/Centralizer.lean | 102 | 105 | theorem div_mem_centralizer₀ [GroupWithZero M] (ha : a ∈ centralizer S) (hb : b ∈ centralizer S) :
a / b ∈ centralizer S := by |
rw [div_eq_mul_inv]
exact mul_mem_centralizer ha (inv_mem_centralizer₀ hb)
|
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.zmod from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
theorem ZMod.ker_intCastRingHom (n : ℕ) :
RingHom.ker (Int.castRingHom (ZMod n)) =... | Mathlib/RingTheory/ZMod.lean | 42 | 46 | theorem isReduced_zmod {n : ℕ} : IsReduced (ZMod n) ↔ Squarefree n ∨ n = 0 := by |
rw [← RingHom.ker_isRadical_iff_reduced_of_surjective
(ZMod.ringHom_surjective <| Int.castRingHom <| ZMod n),
ZMod.ker_intCastRingHom, ← isRadical_iff_span_singleton, isRadical_iff_squarefree_or_zero,
Int.squarefree_natCast, Nat.cast_eq_zero]
|
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegrallyClosed
#align_import ring_theory.polynomial.gauss_lemma from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
open... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 115 | 121 | theorem IsPrimitive.isUnit_iff_isUnit_map_of_injective : IsUnit f ↔ IsUnit (map φ f) := by |
refine ⟨(mapRingHom φ).isUnit_map, fun h => ?_⟩
rcases isUnit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩
have hdeg := degree_C u.ne_zero
rw [hu, degree_map_eq_of_injective hinj] at hdeg
rw [eq_C_of_degree_eq_zero hdeg] at hf ⊢
exact isUnit_C.mpr (isPrimitive_iff_isUnit_of_C_dvd.mp hf (f.coeff 0) dvd_rfl)
|
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071ab828a4ee6e26a52afe4"
section Lemmas
namespace Mathlib.Meta.NormNum
def jacobiSymNat (a b : ℕ) : ℤ :=
jacobiSym a b
#align norm_num.jacobi_sym_... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 98 | 100 | theorem JacobiSym.mod_left (a : ℤ) (b ab' : ℕ) (ab r b' : ℤ) (hb' : (b : ℤ) = b')
(hab : a % b' = ab) (h : (ab' : ℤ) = ab) (hr : jacobiSymNat ab' b = r) : jacobiSym a b = r := by |
rw [← hr, jacobiSymNat, jacobiSym.mod_left, hb', hab, ← h]
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 93 | 98 | theorem interedges_disjoint_right (s : Finset α) {t t' : Finset β} (ht : Disjoint t t') :
Disjoint (interedges r s t) (interedges r s t') := by |
rw [Finset.disjoint_left] at ht ⊢
intro _ hx hy
rw [mem_interedges_iff] at hx hy
exact ht hx.2.1 hy.2.1
|
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 120 | 124 | theorem coeff_sum [Semiring S] (n : ℕ) (f : ℕ → R → S[X]) :
coeff (p.sum f) n = p.sum fun a b => coeff (f a b) n := by |
rcases p with ⟨⟩
-- porting note (#10745): was `simp [Polynomial.sum, support, coeff]`.
simp [Polynomial.sum, support_ofFinsupp, coeff_ofFinsupp]
|
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 31 | 34 | theorem sq_add_mul_sq_mul_sq_add_mul_sq :
(x₁ ^ 2 + n * x₂ ^ 2) * (y₁ ^ 2 + n * y₂ ^ 2) =
(x₁ * y₁ - n * x₂ * y₂) ^ 2 + n * (x₁ * y₂ + x₂ * y₁) ^ 2 := by |
ring
|
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 | 62 | 64 | theorem equitableOn_iff_exists_eq_eq_add_one {s : Set α} {f : α → ℕ} :
s.EquitableOn f ↔ ∃ b, ∀ a ∈ s, f a = b ∨ f a = b + 1 := by |
simp_rw [equitableOn_iff_exists_le_le_add_one, Nat.le_and_le_add_one_iff]
|
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.quasi_compact from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
noncomputable section
open CategoryTheory CategoryT... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 109 | 111 | theorem QuasiCompact.affineProperty_toProperty {X Y : Scheme} (f : X ⟶ Y) :
(QuasiCompact.affineProperty : _).toProperty f ↔ IsAffine Y ∧ CompactSpace X.carrier := by |
delta AffineTargetMorphismProperty.toProperty QuasiCompact.affineProperty; simp
|
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 49 | 52 | theorem hasMFDerivWithinAt_iff_hasFDerivWithinAt {f'} :
HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x f' ↔ HasFDerivWithinAt f f' s x := by |
simpa only [HasMFDerivWithinAt, and_iff_right_iff_imp, mfld_simps] using
HasFDerivWithinAt.continuousWithinAt
|
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 133 | 134 | theorem symmDiff_eq_bot {a b : α} : a ∆ b = ⊥ ↔ a = b := by |
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
|
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 353 | 357 | theorem cases_head_iff : ReflTransGen r a b ↔ a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by |
use cases_head
rintro (rfl | ⟨c, hac, hcb⟩)
· rfl
· exact head hac hcb
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 44 | 45 | theorem integralNormalization_zero : integralNormalization (0 : R[X]) = 0 := by |
simp [integralNormalization]
|
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 86 | 89 | theorem degree_list_prod_le (l : List S[X]) : degree l.prod ≤ (l.map degree).sum := by |
induction' l with hd tl IH
· simp
· simpa using (degree_mul_le _ _).trans (add_le_add_left IH _)
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 105 | 108 | theorem mul_logb {a b c : ℝ} (h₁ : b ≠ 0) (h₂ : b ≠ 1) (h₃ : b ≠ -1) :
logb a b * logb b c = logb a c := by |
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr ⟨h₁, h₂, h₃⟩)]
|
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 113 | 123 | theorem exists_linearIndependent_cons_of_lt_rank [StrongRankCondition R] {n : ℕ} {v : Fin n → M}
(hv : LinearIndependent R v) (h : n < Module.rank R M) :
∃ (x : M), LinearIndependent R (Fin.cons x v) := by |
obtain ⟨t, h₁, h₂, h₃⟩ := exists_linearIndependent_of_lt_rank hv.to_subtype_range
have : range v ≠ t := by
refine fun e ↦ h.ne ?_
rw [← e, ← lift_injective.eq_iff, mk_range_eq_of_injective hv.injective] at h₂
simpa only [mk_fintype, Fintype.card_fin, lift_natCast, lift_id'] using h₂
obtain ⟨x, hx, hx... |
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 | 298 | 301 | theorem finsuppTensorFinsuppLid_single_tmul_single (a : ι) (b : κ) (r : R) (n : N) :
finsuppTensorFinsuppLid R N ι κ (Finsupp.single a r ⊗ₜ[R] Finsupp.single b n) =
Finsupp.single (a, b) (r • n) := by |
simp [finsuppTensorFinsuppLid]
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 123 | 127 | theorem liftOn_ofFractionRing_mk {P : Sort v} (n : K[X]) (d : K[X]⁰) (f : K[X] → K[X] → P)
(H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q') :
RatFunc.liftOn (ofFractionRing (Localization.mk n d)) f @H = f n d := by |
rw [RatFunc.liftOn]
exact Localization.liftOn_mk _ _ _ _
|
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE
#align_import algebraic_geometry.morphisms.basic from "leanprover-community/mathlib"@"434e2fd21c1900747afc6d13d8be7f4eedba7218"
set_option lin... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 104 | 106 | theorem affine_cancel_right_isIso {P : AffineTargetMorphismProperty} (hP : P.toProperty.RespectsIso)
{X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] [IsAffine Z] [IsAffine Y] :
P (f ≫ g) ↔ P f := by | rw [← P.toProperty_apply, ← P.toProperty_apply, hP.cancel_right_isIso]
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 336 | 337 | theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by |
rw [average, measure_univ, inv_one, one_smul]
|
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 115 | 121 | theorem genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain R] :
genericPoint (Scheme.Spec.obj <| op R).carrier = (⟨0, Ideal.bot_prime⟩ : PrimeSpectrum R) := by |
apply (genericPoint_spec (Scheme.Spec.obj <| op R).carrier).eq
rw [isGenericPoint_def]
rw [← PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure, PrimeSpectrum.vanishingIdeal_singleton]
rw [Set.top_eq_univ, ← PrimeSpectrum.zeroLocus_singleton_zero]
simp_rw [Submodule.zero_eq_bot, Submodule.bot_coe]
|
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprover-community/mathlib"@"dd1f8496baa505636a82748e6b652165ea888733"
noncomputable section
open CategoryTheory CategoryTheory.Category Category... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 77 | 84 | theorem ext (A₁ A₂ : IndexSet Δ) (h₁ : A₁.1 = A₂.1) (h₂ : A₁.e ≫ eqToHom (by rw [h₁]) = A₂.e) :
A₁ = A₂ := by |
rcases A₁ with ⟨Δ₁, ⟨α₁, hα₁⟩⟩
rcases A₂ with ⟨Δ₂, ⟨α₂, hα₂⟩⟩
simp only at h₁
subst h₁
simp only [eqToHom_refl, comp_id, IndexSet.e] at h₂
simp only [h₂]
|
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 122 | 127 | theorem ConcaveOn.average_mem_hypograph [IsFiniteMeasure μ] [NeZero μ] (hg : ConcaveOn ℝ s g)
(hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
(⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1} := by |
simpa only [mem_setOf_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
hg.neg.average_mem_epigraph hgc.neg hsc hfs hfi hgi.neg
|
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 108 | 116 | theorem Subgroup.commProb_subgroup_le : commProb H ≤ commProb G * (H.index : ℚ) ^ 2 := by |
/- After rewriting with `commProb_def`, we reduce to showing that `G` has at least as many
commuting pairs as `H`. -/
rw [commProb_def, commProb_def, div_le_iff, mul_assoc, ← mul_pow, ← Nat.cast_mul,
mul_comm H.index, H.card_mul_index, div_mul_cancel₀, Nat.cast_le]
· refine Finite.card_le_of_injective ... |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 289 | 289 | theorem mk_mem_row_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.row i ↔ (i, j) ∈ μ := by | simp [row]
|
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation
open scoped Topology
variable {X Y A} [TopologicalSpace X] [TopologicalSpace A]
theorem embedding_toPullbackDiag (f : X → Y) : Embedding (toPullbackDiag f) :=
Embedding.mk' _ (injective_toPullbackDiag f) fun x ↦ by
rw [toPullbackDiag,... | Mathlib/Topology/SeparatedMap.lean | 89 | 92 | theorem isSeparatedMap_iff_closedEmbedding {f : X → Y} :
IsSeparatedMap f ↔ ClosedEmbedding (toPullbackDiag f) := by |
rw [isSeparatedMap_iff_isClosed_diagonal, ← range_toPullbackDiag]
exact ⟨fun h ↦ ⟨embedding_toPullbackDiag f, h⟩, fun h ↦ h.isClosed_range⟩
|
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 141 | 142 | theorem cramer_subsingleton_apply [Subsingleton n] (A : Matrix n n α) (b : n → α) (i : n) :
cramer A b i = b i := by | rw [cramer_apply, det_eq_elem_of_subsingleton _ i, updateColumn_self]
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 74 | 77 | theorem apply_firstDiff_ne {x y : ∀ n, E n} (h : x ≠ y) :
x (firstDiff x y) ≠ y (firstDiff x y) := by |
rw [firstDiff_def, dif_pos h]
exact Nat.find_spec (ne_iff.1 h)
|
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 46 | 48 | theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b := by |
cases a <;> rfl
|
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Conj
#align_import category_theory.adjunction.mates from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
universe v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄
namespace CategoryTheory
open Category
variable {C : Type u₁} {D : Typ... | Mathlib/CategoryTheory/Adjunction/Mates.lean | 111 | 115 | theorem transferNatTrans_counit (f : G ⋙ L₂ ⟶ L₁ ⋙ H) (Y : D) :
L₂.map ((transferNatTrans adj₁ adj₂ f).app _) ≫ adj₂.counit.app _ =
f.app _ ≫ H.map (adj₁.counit.app Y) := by |
erw [Functor.map_comp]
simp
|
import Mathlib.MeasureTheory.Integral.Bochner
open MeasureTheory Filter
open scoped ENNReal NNReal BoundedContinuousFunction Topology
namespace BoundedContinuousFunction
section NNRealValued
lemma apply_le_nndist_zero {X : Type*} [TopologicalSpace X] (f : X →ᵇ ℝ≥0) (x : X) :
f x ≤ nndist 0 f := by
convert ... | Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 66 | 71 | theorem toReal_lintegral_coe_eq_integral (f : X →ᵇ ℝ≥0) (μ : Measure X) :
(∫⁻ x, (f x : ℝ≥0∞) ∂μ).toReal = ∫ x, (f x : ℝ) ∂μ := by |
rw [integral_eq_lintegral_of_nonneg_ae _ (by simpa [Function.comp_apply] using
(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable)]
· simp only [ENNReal.ofReal_coe_nnreal]
· exact eventually_of_forall (by simp only [Pi.zero_apply, NNReal.zero_le_coe, imp_true_iff])
|
import Mathlib.Topology.ContinuousOn
import Mathlib.Data.Set.BoolIndicator
open Set Filter Topology TopologicalSpace Classical
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Clopen
protected theorem IsClopen.isOpen (hs : IsClo... | Mathlib/Topology/Clopen.lean | 113 | 120 | theorem isClopen_inter_of_disjoint_cover_clopen {s a b : Set X} (h : IsClopen s) (cover : s ⊆ a ∪ b)
(ha : IsOpen a) (hb : IsOpen b) (hab : Disjoint a b) : IsClopen (s ∩ a) := by |
refine ⟨?_, IsOpen.inter h.2 ha⟩
have : IsClosed (s ∩ bᶜ) := IsClosed.inter h.1 (isClosed_compl_iff.2 hb)
convert this using 1
refine (inter_subset_inter_right s hab.subset_compl_right).antisymm ?_
rintro x ⟨hx₁, hx₂⟩
exact ⟨hx₁, by simpa [not_mem_of_mem_compl hx₂] using cover hx₁⟩
|
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 65 | 70 | theorem HasBasis.lift {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α} (hf : f.HasBasis p s)
{β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ} {g : Set α → Filter γ}
(hg : ∀ i, (g (s i)).HasBasis (pg i) (sg i)) (gm : Monotone g) :
(f.lift g).HasBasis (fun i : Σi, β i => p i.1 ∧ pg i.1 i.2) fun... |
refine ⟨fun t => (hf.mem_lift_iff hg gm).trans ?_⟩
simp [Sigma.exists, and_assoc, exists_and_left]
|
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749fc4661828cba18cd0f4e3c5eb66a8e80598"
open scoped Pointwise
universe u₁ u₂ u₃
namespace MonoidAlgebra
open Finset Finsupp
variable {k : Type u₁} ... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 55 | 62 | theorem support_mul_single_eq_image [DecidableEq G] [Mul G] (f : MonoidAlgebra k G) {r : k}
(hr : ∀ y, y * r = 0 ↔ y = 0) {x : G} (rx : IsRightRegular x) :
(f * single x r).support = Finset.image (· * x) f.support := by |
refine subset_antisymm (support_mul_single_subset f _ _) fun y hy => ?_
obtain ⟨y, yf, rfl⟩ : ∃ a : G, a ∈ f.support ∧ a * x = y := by
simpa only [Finset.mem_image, exists_prop] using hy
simp only [mul_apply, mem_support_iff.mp yf, hr, mem_support_iff, sum_single_index,
Finsupp.sum_ite_eq', Ne, not_false... |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 107 | 119 | theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by |
induction l with
| nil => apply Nat.le_refl
| cons a l ihl =>
rw [forall_mem_cons] at h
have ⟨ha, hl⟩ := h
simp [countP_cons]
cases h : p a
. simp
apply Nat.le_trans ?_ (Nat.le_add_right _ _)
apply ihl hl
. simp [ha h]
apply ihl hl
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 106 | 109 | theorem isPiSystem_Ici_rat : IsPiSystem (⋃ a : ℚ, {Ici (a : ℝ)}) := by |
convert isPiSystem_image_Ici (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Finsupp
#align_import algebra.big_operators.associated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β γ δ : Type*}
-- the same local notation used in `Algebra.Associated`
local infixl:50 " ~ᵤ " => ... | Mathlib/Algebra/BigOperators/Associated.lean | 29 | 36 | theorem exists_mem_multiset_dvd {s : Multiset α} : p ∣ s.prod → ∃ a ∈ s, p ∣ a :=
Multiset.induction_on s (fun h => (hp.not_dvd_one h).elim) fun a s ih h =>
have : p ∣ a * s.prod := by | simpa using h
match hp.dvd_or_dvd this with
| Or.inl h => ⟨a, Multiset.mem_cons_self a s, h⟩
| Or.inr h =>
let ⟨a, has, h⟩ := ih h
⟨a, Multiset.mem_cons_of_mem has, h⟩
|
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
universe u v w
def IsSeq {α : Type u} (s : Stream' (Option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 160 | 163 | theorem le_stable (s : Seq α) {m n} (h : m ≤ n) : s.get? m = none → s.get? n = none := by |
cases' s with f al
induction' h with n _ IH
exacts [id, fun h2 => al (IH h2)]
|
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