Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.cont_mdiff_mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners Bundle
open sc... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 344 | 518 | theorem ContMDiffOn.contMDiffOn_tangentMapWithin (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n)
(hs : UniqueMDiffOn I s) :
ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
(π E (TangentSpace I) ⁻¹' s) := by |
/- The strategy of the proof is to avoid unfolding the definitions, and reduce by functoriality
to the case of functions on the model spaces, where we have already proved the result.
Let `l` and `r` be the charts to the left and to the right, so that we have
```
l^{-1} f r
H -------... |
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import category_theory.limits.shapes.biproducts from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa093722fa9b65d31"
noncomputab... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,753 | 1,755 | theorem biprod.isoCoprod_inv {X Y : C} [HasBinaryBiproduct X Y] :
(biprod.isoCoprod X Y).inv = coprod.desc biprod.inl biprod.inr := by |
ext <;> simp [biprod.isoCoprod]
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 119 | 122 | theorem rootsOfUnity_le_of_dvd (h : k ∣ l) : rootsOfUnity k M ≤ rootsOfUnity l M := by |
obtain ⟨d, rfl⟩ := h
intro ζ h
simp_all only [mem_rootsOfUnity, PNat.mul_coe, pow_mul, one_pow]
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 180 | 183 | theorem log_pos_of_lt_neg_one (hx : x < -1) : 0 < log x := by |
rw [← neg_neg x, log_neg_eq_log]
have : 1 < -x := by linarith
exact log_pos this
|
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions... | Mathlib/CategoryTheory/Extensive.lean | 455 | 460 | theorem FinitaryPreExtensive.isUniversal_finiteCoproducts [FinitaryPreExtensive C] {ι : Type*}
[Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) : IsUniversalColimit c := by |
obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin ι
apply (IsUniversalColimit.whiskerEquivalence_iff (Discrete.equivalence e).symm).mp
apply FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin
exact (IsColimit.whiskerEquivalenceEquiv (Discrete.equivalence e).symm) hc
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Synonym
import Mathlib.Order.Hom.Set
import Mathlib.Order.Bounds.Basic
#align_import order.galois_connection from "leanprover-community/mathlib"@"c5c7e2760814660967bc27f0de95d190a22297f3"
open Function OrderDual Set
universe u v w x
variable {α : Type u}... | Mathlib/Order/GaloisConnection.lean | 283 | 284 | theorem l_iSup₂ {f : ∀ i, κ i → α} : l (⨆ (i) (j), f i j) = ⨆ (i) (j), l (f i j) := by |
simp_rw [gc.l_iSup]
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 403 | 409 | theorem _root_.MeasurableEmbedding.integrableAtFilter_map_iff [MeasurableSpace β] {e : α → β}
(he : MeasurableEmbedding e) {f : β → E} :
IntegrableAtFilter f (l.map e) (μ.map e) ↔ IntegrableAtFilter (f ∘ e) l μ := by |
simp_rw [IntegrableAtFilter, he.integrableOn_map_iff]
constructor <;> rintro ⟨s, hs⟩
· exact ⟨_, hs⟩
· exact ⟨e '' s, by rwa [mem_map, he.injective.preimage_image]⟩
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 64 | 64 | theorem geom_sum_two {x : α} : ∑ i ∈ range 2, x ^ i = x + 1 := by | simp [geom_sum_succ']
|
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 344 | 346 | theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by |
rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl
|
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.PowerSeries.Order
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 140 | 147 | theorem coeff_inv (n) (φ : k⟦X⟧) :
coeff k n φ⁻¹ =
if n = 0 then (constantCoeff k φ)⁻¹
else
-(constantCoeff k φ)⁻¹ *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff k x.1 φ * coeff k x.2 φ⁻¹ else 0 := by |
rw [inv_eq_inv_aux, coeff_inv_aux n (constantCoeff k φ)⁻¹ φ]
|
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equiv from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Lean Meta Elab Tactic
def mfld_cfg : Simps.Config where
attrs :=... | Mathlib/Logic/Equiv/PartialEquiv.lean | 820 | 821 | theorem EqOnSource.target_eq {e e' : PartialEquiv α β} (h : e ≈ e') : e.target = e'.target := by |
simp only [← image_source_eq_target, ← source_eq h, h.2.image_eq]
|
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 114 | 116 | theorem lcm_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).lcm f = s.lcm (f ∘ g) := by |
classical induction' s using Finset.induction with c s _ ih <;> simp [*]
|
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Directed
#align_import data.set.Union_lift from "leanprover-community/mathlib"@"5a4ea8453f128345f73cc656e80a49de2a54f481"
variable {α : Type*} {ι β : Sort _}
namespace Set
variable {S : ι → Set α} {f : ∀ i, S i → β}
{hf : ∀ (i j) (x : α) (hxi : x ∈ S i) (hxj... | Mathlib/Data/Set/UnionLift.lean | 177 | 180 | theorem preimage_liftCover (t : Set β) : liftCover S f hf hS ⁻¹' t = ⋃ i, (↑) '' (f i ⁻¹' t) := by |
change (iUnionLift S f hf univ hS.symm.subset ∘ fun a => ⟨a, mem_univ a⟩) ⁻¹' t = _
rw [preimage_comp, preimage_iUnionLift]
ext; simp
|
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 453 | 503 | theorem gcd_props :
let d := gcdD a b
let w := gcdW a b
let x := gcdX a b
let y := gcdY a b
let z := gcdZ a b
let a' := gcdA' a b
let b' := gcdB' a b
w * z = succPNat (x * y) ∧
a = a' * d ∧
b = b' * d ∧
z * a' = succPNat (x * b') ∧
w * b' = succPNat (y... |
intros d w x y z a' b'
let u := XgcdType.start a b
let ur := u.reduce
have _ : d = ur.a := rfl
have hb : d = ur.b := u.reduce_isReduced'
have ha' : (a' : ℕ) = w + x := gcdA'_coe a b
have hb' : (b' : ℕ) = y + z := gcdB'_coe a b
have hdet : w * z = succPNat (x * y) := u.reduce_isSpecial' rfl
construct... |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 128 | 136 | theorem ext_iff (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, MeasurableSet i → v i = w i := by |
constructor
· rintro rfl _ _
rfl
· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
· exact h i hi
· simp_rw [not_measurable _ hi]
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 366 | 387 | theorem HasFTaylorSeriesUpToOn.compContinuousLinearMap (hf : HasFTaylorSeriesUpToOn n f p s)
(g : G →L[𝕜] E) :
HasFTaylorSeriesUpToOn n (f ∘ g) (fun x k => (p (g x) k).compContinuousLinearMap fun _ => g)
(g ⁻¹' s) := by |
let A : ∀ m : ℕ, (E[×m]→L[𝕜] F) → G[×m]→L[𝕜] F := fun m h => h.compContinuousLinearMap fun _ => g
have hA : ∀ m, IsBoundedLinearMap 𝕜 (A m) := fun m =>
isBoundedLinearMap_continuousMultilinearMap_comp_linear g
constructor
· intro x hx
simp only [(hf.zero_eq (g x) hx).symm, Function.comp_apply]
c... |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 480 | 488 | theorem subset_piiUnionInter {π : ι → Set (Set α)} {S : Set ι} {i : ι} (his : i ∈ S) :
π i ⊆ piiUnionInter π S := by |
have h_ss : {i} ⊆ S := by
intro j hj
rw [mem_singleton_iff] at hj
rwa [hj]
refine Subset.trans ?_ (piiUnionInter_mono_right h_ss)
rw [piiUnionInter_singleton]
exact subset_union_left
|
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import category_theory.limits.shapes.biproducts from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa093722fa9b65d31"
noncomputab... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,152 | 2,154 | theorem biprod.braiding_map_braiding {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) :
(biprod.braiding X W).hom ≫ biprod.map f g ≫ (biprod.braiding Y Z).hom = biprod.map g f := by |
aesop_cat
|
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.modeq from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
assert_not_exists Function.support
namespace Nat
def ModEq (n a b :... | Mathlib/Data/Nat/ModEq.lean | 517 | 518 | theorem odd_of_mod_four_eq_one {n : ℕ} : n % 4 = 1 → n % 2 = 1 := by |
simpa [ModEq, show 2 * 2 = 4 by norm_num] using @ModEq.of_mul_left 2 n 1 2
|
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 274 | 279 | theorem tail_iterate (f : α → α) (a : α) : tail (iterate f a) = iterate f (f a) := by |
ext n
rw [get_tail]
induction' n with n' ih
· rfl
· rw [get_succ_iterate', ih, get_succ_iterate']
|
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 | 158 | 162 | theorem jacobiSymNat.even_odd₃ (a b c : ℕ) (r : ℤ) (ha : a % 2 = 0) (hb : b % 8 = 3)
(hc : a / 2 = c) (hr : jacobiSymNat c b = r) : jacobiSymNat a b = -r := by |
simp only [jacobiSymNat, ← hr, ← hc, Int.ofNat_ediv, Nat.cast_ofNat]
rw [← jacobiSym.even_odd (mod_cast ha), if_pos (by simp [hb])]
rw [← Nat.mod_mod_of_dvd, hb]; norm_num
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 873 | 875 | theorem ncard_diff (h : s ⊆ t) (ht : t.Finite := by | toFinite_tac) :
(t \ s).ncard = t.ncard - s.ncard := by
rw [← ncard_diff_add_ncard_of_subset h ht, add_tsub_cancel_right]
|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 121 | 121 | theorem a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a := by | rw [← coeff_eq_a, h, coeff_eq_a]
|
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 | 619 | 622 | theorem algebraMap_apply {R : Type*} [CommSemiring R] [Algebra R K[X]] (x : R) :
algebraMap R (RatFunc K) x = algebraMap _ _ (algebraMap R K[X] x) / algebraMap K[X] _ 1 := by |
rw [← mk_eq_div]
rfl
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 169 | 170 | theorem midpoint_eq_left_iff {x y : P} : midpoint R x y = x ↔ x = y := by |
rw [midpoint_eq_iff, pointReflection_self]
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 839 | 842 | theorem ncard_union_add_ncard_inter (s t : Set α) (hs : s.Finite := by | toFinite_tac)
(ht : t.Finite := by toFinite_tac) : (s ∪ t).ncard + (s ∩ t).ncard = s.ncard + t.ncard := by
to_encard_tac; rw [hs.cast_ncard_eq, ht.cast_ncard_eq, (hs.union ht).cast_ncard_eq,
(hs.subset inter_subset_left).cast_ncard_eq, encard_union_add_encard_inter]
|
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib.Logic.Small.Defs
import Mathlib.Order.GameAdd
#align_import set_theory.game.pgame from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
set_option autoImplicit true
names... | Mathlib/SetTheory/Game/PGame.lean | 1,617 | 1,621 | theorem add_moveRight_inr (x : PGame) {y : PGame} (i) :
(x + y).moveRight (toRightMovesAdd (Sum.inr i)) = x + y.moveRight i := by |
cases x
cases y
rfl
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 133 | 135 | theorem factorization_eq_zero_iff (n p : ℕ) :
n.factorization p = 0 ↔ ¬p.Prime ∨ ¬p ∣ n ∨ n = 0 := by |
simp_rw [← not_mem_support_iff, support_factorization, mem_primeFactors, not_and_or, not_ne_iff]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 874 | 875 | theorem eq_zero_of_eq_zero (h : (0 : R) = (1 : R)) (p : R[X]) : p = 0 := by |
rw [← one_smul R p, ← h, zero_smul]
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 1,790 | 1,815 | theorem card_le_of_interleaved {s t : Finset α}
(h : ∀ᵉ (x ∈ s) (y ∈ s),
x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
s.card ≤ t.card + 1 := by |
replace h : ∀ᵉ (x ∈ s) (y ∈ s), x < y → ∃ z ∈ t, x < z ∧ z < y := by
intro x hx y hy hxy
rcases exists_next_right ⟨y, hy, hxy⟩ with ⟨a, has, hxa, ha⟩
rcases h x hx a has hxa fun z hzs hz => hz.2.not_le <| ha _ hzs hz.1 with ⟨b, hbt, hxb, hba⟩
exact ⟨b, hbt, hxb, hba.trans_le <| ha _ hy hxy⟩
set f :... |
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Ring.Subring.Basic
... | Mathlib/RingTheory/Perfection.lean | 420 | 427 | theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y := by |
have hx0 : x ≠ 0 := mt (by rintro rfl; rw [zero_mul]) hxy0
have hy0 : y ≠ 0 := mt (by rintro rfl; rw [mul_zero]) hxy0
obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
rw [← map_mul (Ideal.Quotient.mk (Ideal.span {↑p})) r s] at hxy0 ⊢
rw [preVal_mk hx0, pre... |
import Mathlib.Topology.Separation
#align_import topology.shrinking_lemma from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Function
open scoped Classical
noncomputable section
variable {ι X : Type*} [TopologicalSpace X] [NormalSpace X]
namespace ShrinkingLemma
-- the tr... | Mathlib/Topology/ShrinkingLemma.lean | 218 | 231 | theorem exists_subset_iUnion_closure_subset (hs : IsClosed s) (uo : ∀ i, IsOpen (u i))
(uf : ∀ x ∈ s, { i | x ∈ u i }.Finite) (us : s ⊆ ⋃ i, u i) :
∃ v : ι → Set X, s ⊆ iUnion v ∧ (∀ i, IsOpen (v i)) ∧ ∀ i, closure (v i) ⊆ u i := by |
haveI : Nonempty (PartialRefinement u s) := ⟨⟨u, ∅, uo, us, False.elim, fun _ => rfl⟩⟩
have : ∀ c : Set (PartialRefinement u s),
IsChain (· ≤ ·) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub :=
fun c hc ne => ⟨.chainSup c hc ne uf us, fun v hv => PartialRefinement.le_chainSup _ _ _ _ hv⟩
rcases zorn_nonempty_... |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Restrict
import Mathlib.Tactic.FailIfNoProgress
#align_import analysis.normed_space.affine_isomet... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 150 | 150 | theorem nndist_map (x y : P) : nndist (f x) (f y) = nndist x y := by | simp [nndist_dist]
|
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 490 | 493 | theorem toSheafify_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
η ≫ J.toSheafify _ = J.toSheafify _ ≫ J.sheafifyMap η := by |
dsimp [sheafifyMap, sheafify, toSheafify]
simp
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
variable {s s₁ s₂ : Set α} {t ... | Mathlib/Data/Set/Function.lean | 1,610 | 1,611 | theorem piecewise_insert_of_ne {i j : α} (h : i ≠ j) [∀ i, Decidable (i ∈ insert j s)] :
(insert j s).piecewise f g i = s.piecewise f g i := by | simp [piecewise, h]
|
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Tactic.FinCases
#align_import linear_algebra.matrix.block from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Finset Function OrderDual
open Matrix
universe v
v... | Mathlib/LinearAlgebra/Matrix/Block.lean | 175 | 177 | theorem equiv_block_det (M : Matrix m m R) {p q : m → Prop} [DecidablePred p] [DecidablePred q]
(e : ∀ x, q x ↔ p x) : (toSquareBlockProp M p).det = (toSquareBlockProp M q).det := by |
convert Matrix.det_reindex_self (Equiv.subtypeEquivRight e) (toSquareBlockProp M q)
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 825 | 826 | theorem mul_pos {a b : Ordinal} (h₁ : 0 < a) (h₂ : 0 < b) : 0 < a * b := by |
simpa only [mul_zero] using mul_lt_mul_of_pos_left h₂ h₁
|
import Mathlib.Data.Matroid.Restrict
variable {α : Type*} {M : Matroid α} {E B I X R J : Set α}
namespace Matroid
open Set
section FreeOn
def freeOn (E : Set α) : Matroid α := (loopyOn E)✶
@[simp] theorem freeOn_ground : (freeOn E).E = E := rfl
@[simp] theorem freeOn_dual_eq : (freeOn E)✶ = loopyOn E := by
... | Mathlib/Data/Matroid/Constructions.lean | 181 | 182 | theorem Indep.restrict_eq_freeOn (hI : M.Indep I) : M ↾ I = freeOn I := by |
rwa [restrict_eq_freeOn_iff]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 133 | 135 | theorem infEdist_le_edist_add_infEdist : infEdist x s ≤ edist x y + infEdist y s := by |
rw [add_comm]
exact infEdist_le_infEdist_add_edist
|
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 183 | 183 | theorem birthday_add_one : (a + 1).birthday = Order.succ a.birthday := by | simp
|
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (𝕜 : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 219 | 220 | theorem nnnorm_def (x : tsze R M) : ‖x‖₊ = ‖fst x‖₊ + ‖snd x‖₊ := by |
ext; simp [norm_def]
|
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 207 | 214 | theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes = {I.radical} := by |
ext J
constructor
· exact fun H =>
let e := H.1.1.radical_le_iff.mpr H.1.2
(H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
· rintro (rfl : J = I.radical)
exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
|
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,465 | 1,466 | theorem l_ciSup (gc : GaloisConnection l u) {f : ι → α} (hf : BddAbove (range f)) :
l (⨆ i, f i) = ⨆ i, l (f i) := by | rw [iSup, gc.l_csSup (range_nonempty _) hf, iSup_range']
|
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 133 | 137 | theorem map_map_permutations'Aux (f : α → β) (t : α) (ts : List α) :
map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts) := by |
induction' ts with a ts ih
· rfl
· simp only [permutations'Aux, map_cons, map_map, ← ih, cons.injEq, true_and, Function.comp_def]
|
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 | 157 | 159 | theorem liftRight_inv_mul {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :
↑(liftRight f g h x)⁻¹ * f x = 1 := by |
rw [Units.inv_mul_eq_iff_eq_mul, mul_one, coe_liftRight]
|
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 33 | 41 | theorem tendsto_Icc_vitaliFamily_right (x : ℝ) :
Tendsto (fun y => Icc x y) (𝓝[>] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
· intro ε εpos
have : x ∈ Ico x (x + ε) := ⟨le_refl _, by linarith⟩
filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy
rw [closedBall_eq_Icc]
exact I... |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 75 | 76 | theorem preimage_coe_Iio_top : (some : α → WithTop α) ⁻¹' Iio ⊤ = univ := by |
rw [← range_coe, preimage_range]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 303 | 310 | theorem digits_eq_nil_iff_eq_zero {b n : ℕ} : digits b n = [] ↔ n = 0 := by |
constructor
· intro h
have : ofDigits b (digits b n) = ofDigits b [] := by rw [h]
convert this
rw [ofDigits_digits]
· rintro rfl
simp
|
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,200 | 2,204 | theorem isBigOWith_pi {ι : Type*} [Fintype ι] {E' : ι → Type*} [∀ i, NormedAddCommGroup (E' i)]
{f : α → ∀ i, E' i} {C : ℝ} (hC : 0 ≤ C) :
IsBigOWith C l f g' ↔ ∀ i, IsBigOWith C l (fun x => f x i) g' := by |
have : ∀ x, 0 ≤ C * ‖g' x‖ := fun x => mul_nonneg hC (norm_nonneg _)
simp only [isBigOWith_iff, pi_norm_le_iff_of_nonneg (this _), eventually_all]
|
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
open Function
namespace SimpleGraph
variable {V W X : Type*} (G : SimpleGraph V) (G' : SimpleGraph W) {u v : V}
protected def map (f : V ↪ W) (G : SimpleGraph V) : SimpleGraph W where
Adj := Relation.Map G.Adj f f
symm a b... | Mathlib/Combinatorics/SimpleGraph/Maps.lean | 123 | 125 | theorem comap_monotone (f : V ↪ W) : Monotone (SimpleGraph.comap f) := by |
intro G G' h _ _ ha
exact h ha
|
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 | 73 | 80 | theorem count_apply_infinite (hs : s.Infinite) : count s = ∞ := by |
refine top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => ?_)
rcases hs.exists_subset_card_eq n with ⟨t, ht, rfl⟩
calc
(t.card : ℝ≥0∞) = ∑ i ∈ t, 1 := by simp
_ = ∑' i : (t : Set α), 1 := (t.tsum_subtype 1).symm
_ ≤ count (t : Set α) := le_count_apply
_ ≤ count s := measure_mono ht
|
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
open Bornology Filter Set Function
open scoped Topology
namespace Bornology.IsVonNBounded
variable {ι 𝕜 F : Type*} {E : ι → Type*} [NormedField 𝕜]
[∀ i, AddCommGroup (E i)] [∀ i, Module 𝕜 (E i)] [∀ i, Topol... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 90 | 96 | theorem image_multilinear [ContinuousSMul 𝕜 F] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := by |
cases isEmpty_or_nonempty ι with
| inl h =>
exact (isBounded_iff_isVonNBounded _).1 <|
@Set.Finite.isBounded _ (vonNBornology 𝕜 F) _ (s.toFinite.image _)
| inr h => exact hs.image_multilinear' f
|
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.Init.Data.Prod
import Mathlib.RingTheory.OreLocalization.Basic
#align_import group_theory.monoid_localization from "leanprover-community/... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,598 | 1,602 | theorem ofMulEquivOfDom_id :
f.ofMulEquivOfDom
(show S.map (MulEquiv.refl M).toMonoidHom = S from
Submonoid.ext fun x ↦ ⟨fun ⟨_, hy, h⟩ ↦ h ▸ hy, fun h ↦ ⟨x, h, rfl⟩⟩) = f := by |
ext; rfl
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 121 | 123 | theorem sum_incMatrix_apply [Fintype (neighborSet G a)] :
∑ e, G.incMatrix R a e = G.degree a := by |
classical simp [incMatrix_apply', sum_boole, Set.filter_mem_univ_eq_toFinset]
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,382 | 1,388 | theorem setToFun_neg (hT : DominatedFinMeasAdditive μ T C) (f : α → E) :
setToFun μ T hT (-f) = -setToFun μ T hT f := by |
by_cases hf : Integrable f μ
· rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,
(L1.setToL1 hT).map_neg]
· rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
rwa [← integrable_neg_iff] at hf
|
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 98 | 100 | theorem eventually_cocardinal_ne (x : α) : ∀ᶠ a in cocardinal α hreg, a ≠ x := by |
simp [Set.finite_singleton x]
exact hreg.nat_lt 1
|
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 362 | 372 | theorem fixInduction'_stop {C : α → Sort*} {f : α →. Sum β α} {b : β} {a : α} (h : b ∈ f.fix a)
(fa : Sum.inl b ∈ f a) (hbase : ∀ a_final : α, Sum.inl b ∈ f a_final → C a_final)
(hind : ∀ a₀ a₁ : α, b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀) :
@fixInduction' _ _ C _ _ _ h hbase hind = hbase a fa := by |
unfold fixInduction'
rw [fixInduction_spec]
-- Porting note: the explicit motive required because `simp` behaves differently
refine Eq.rec (motive := fun x e ↦
Sum.casesOn x ?_ ?_ (Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) = hbase a fa) ?_
(Part.get_eq_of_mem fa (dom_of_mem_fix h)).symm
... |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 266 | 272 | theorem integrable_cexp_neg_sum_mul_add {ι : Type*} [Fintype ι] {b : ι → ℂ}
(hb : ∀ i, 0 < (b i).re) (c : ι → ℂ) :
Integrable (fun (v : ι → ℝ) ↦ cexp (- ∑ i, b i * (v i : ℂ) ^ 2 + ∑ i, c i * v i)) := by |
simp_rw [← Finset.sum_neg_distrib, ← Finset.sum_add_distrib, Complex.exp_sum, ← neg_mul]
apply Integrable.fintype_prod (f := fun i (v : ℝ) ↦ cexp (-b i * v^2 + c i * v)) (fun i ↦ ?_)
convert integrable_cexp_quadratic (hb i) (c i) 0 using 3 with x
simp only [add_zero]
|
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathlib.Topology.ShrinkingLemma
#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 634 | 638 | theorem exists_isSubordinate_chartAt_source_of_isClosed {s : Set M} (hs : IsClosed s) :
∃ f : SmoothPartitionOfUnity s I M s,
f.IsSubordinate (fun x ↦ (chartAt H (x : M)).source) := by |
apply exists_isSubordinate _ hs _ (fun i ↦ (chartAt H _).open_source) (fun x hx ↦ ?_)
exact mem_iUnion_of_mem ⟨x, hx⟩ (mem_chart_source H x)
|
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 313 | 314 | theorem leftDual_ρ [LeftRigidCategory V] (h : H) : (ᘁX).ρ h = ᘁX.ρ (h⁻¹ : H) := by |
rw [← SingleObj.inv_as_inv]; rfl
|
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 278 | 288 | theorem smul_Iic : r • Iic a = Iic (r • a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Iio]
constructor
· rintro ⟨a_w, a_h_left, rfl⟩
exact (mul_le_mul_left hr).mpr a_h_left
· rintro h
use x / r
constructor
· exact (div_le_iff' hr).mpr h
· exact mul_div_cancel₀ _ (ne_of_gt hr)
|
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 356 | 359 | theorem memℒp_const_iff {p : ℝ≥0∞} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
Memℒp (fun _ : α => c) p μ ↔ c = 0 ∨ μ Set.univ < ∞ := by |
rw [← snorm_const_lt_top_iff hp_ne_zero hp_ne_top]
exact ⟨fun h => h.2, fun h => ⟨aestronglyMeasurable_const, h⟩⟩
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 55 | 69 | theorem traceAux_eq : traceAux R b = traceAux R c :=
LinearMap.ext fun f =>
calc
Matrix.trace (LinearMap.toMatrix b b f) =
Matrix.trace (LinearMap.toMatrix b b ((LinearMap.id.comp f).comp LinearMap.id)) := by |
rw [LinearMap.id_comp, LinearMap.comp_id]
_ = Matrix.trace (LinearMap.toMatrix c b LinearMap.id * LinearMap.toMatrix c c f *
LinearMap.toMatrix b c LinearMap.id) := by
rw [LinearMap.toMatrix_comp _ c, LinearMap.toMatrix_comp _ c]
_ = Matrix.trace (LinearMap.toMatrix c c f * Linear... |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 374 | 392 | theorem hausdorffEdist_triangle : hausdorffEdist s u ≤ hausdorffEdist s t + hausdorffEdist t u := by |
rw [hausdorffEdist_def]
simp only [sup_le_iff, iSup_le_iff]
constructor
· show ∀ x ∈ s, infEdist x u ≤ hausdorffEdist s t + hausdorffEdist t u
exact fun x xs =>
calc
infEdist x u ≤ infEdist x t + hausdorffEdist t u :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ hausdorffEdis... |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set Function
structure Complex : Type where... | Mathlib/Data/Complex/Basic.lean | 761 | 761 | theorem I_sq : I ^ 2 = -1 := by | rw [sq, I_mul_I]
|
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 | 561 | 563 | theorem toIocMod_neg (a b : α) : toIocMod hp a (-b) = p - toIcoMod hp (-a) b := by |
simp_rw [toIocMod, toIcoMod, toIocDiv_neg, neg_smul, add_smul]
abel
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 121 | 129 | theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.1 ≤ log b o := by |
refine CNFRec b ?_ (fun o ho H ↦ ?_) o
· rw [CNF_zero]
intro contra; contradiction
· rw [CNF_ne_zero ho, mem_cons]
rintro (rfl | h)
· exact le_rfl
· exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le)
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousFunction.Polynomial
import Mathlib.Topology.ContinuousFunction.Compact
#align_import analysis.special_functions.bernstein from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba14... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 170 | 172 | theorem apply (n : ℕ) (f : C(I, ℝ)) (x : I) :
bernsteinApproximation n f x = ∑ k : Fin (n + 1), f k/ₙ * bernstein n k x := by |
simp [bernsteinApproximation]
|
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 253 | 254 | theorem unfold_sub {α} [SubtractionMonoid α] (a b c : α) (h : a + -b = c) : a - b = c := by |
rw [sub_eq_add_neg, h]
|
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order.ring.lemmas from "leanprover-community/mathlib"@"44e29dbcff83ba7114a464d592b8c3743987c1e5"
variable (α : Type*)
set_option quotPrecheck false in
notation "α≥... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 1,022 | 1,025 | theorem exists_square_le' [PosMulStrictMono α] (a0 : 0 < a) : ∃ b : α, b * b ≤ a := by |
obtain ha | ha := lt_or_le a 1
· exact ⟨a, (mul_lt_of_lt_one_right a0 ha).le⟩
· exact ⟨1, by rwa [mul_one]⟩
|
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
variable [∀ a, DecidableEq (N a)]
@[simp]
theorem neLocus_add_left [∀ a, AddLe... | Mathlib/Data/DFinsupp/NeLocus.lean | 175 | 176 | theorem neLocus_self_sub_left : neLocus (f - g) f = g.support := by |
rw [neLocus_comm, neLocus_self_sub_right]
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 88 | 90 | theorem _root_.IsOpen.ae_eq_empty_iff_eq (hU : IsOpen U) :
U =ᵐ[μ] (∅ : Set X) ↔ U = ∅ := by |
rw [ae_eq_empty, hU.measure_zero_iff_eq_empty]
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 769 | 780 | theorem count_map_roots [IsDomain A] [DecidableEq B] {p : A[X]} {f : A →+* B} (hmap : map f p ≠ 0)
(b : B) :
(p.roots.map f).count b ≤ rootMultiplicity b (p.map f) := by |
rw [le_rootMultiplicity_iff hmap, ← Multiset.prod_replicate, ←
Multiset.map_replicate fun a => X - C a]
rw [← Multiset.filter_eq]
refine
(Multiset.prod_dvd_prod_of_le <| Multiset.map_le_map <| Multiset.filter_le (Eq b) _).trans ?_
convert Polynomial.map_dvd f p.prod_multiset_X_sub_C_dvd
simp only [Po... |
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathlib.Topology.ShrinkingLemma
#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 697 | 706 | theorem Emetric.exists_smooth_forall_closedBall_subset {M} [EMetricSpace M] [ChartedSpace H M]
[SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K : ι → Set M} {U : ι → Set M}
(hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i)
(hfin : LocallyFinite K) :
∃ δ : C^∞⟮I, M; 𝓘(ℝ,... |
simpa only [mem_inter_iff, forall_and, mem_preimage, mem_iInter, @forall_swap ι M]
using exists_smooth_forall_mem_convex_of_local_const I
EMetric.exists_forall_closedBall_subset_aux₂
(EMetric.exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
|
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.OrderClosed
#align_import topology.order.basic from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafe... | Mathlib/Topology/Order/Basic.lean | 91 | 93 | theorem isOpen_iff_generate_intervals {s : Set α} :
IsOpen s ↔ GenerateOpen { s | ∃ a, s = Ioi a ∨ s = Iio a } s := by |
rw [t.topology_eq_generate_intervals]; rfl
|
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 565 | 592 | theorem localization_finiteType : RingHom.LocalizationPreserves @RingHom.FiniteType := by |
introv R hf
-- mirrors the proof of `localization_map_finite`
letI := f.toAlgebra
letI := ((algebraMap S S').comp f).toAlgebra
let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
letI := f'.toAlgebra
haveI : IsScalarTower R R' S' :=
IsScalarTower.of_algebraMap_eq' (IsLocalization... |
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.OrderClosed
#align_import topology.order.basic from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafe... | Mathlib/Topology/Order/Basic.lean | 212 | 231 | theorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
[OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)
(H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)
(H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :
induced f ‹TopologicalSpace β›... |
let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩
refine le_antisymm (induced_topology_le_preorder hf) ?_
refine le_of_nhds_le_nhds fun a => ?_
simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal]
refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_)
·... |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 297 | 299 | theorem rootMultiplicity_expand :
(expand R p f).rootMultiplicity r = p * f.rootMultiplicity (r ^ p) := by |
rw [← pow_one p, rootMultiplicity_expand_pow]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 408 | 415 | theorem submartingale_of_setIntegral_le_succ [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
(hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
(hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f (i + 1) ω ∂μ) :
Submartingale f 𝒢 μ := by |
refine submartingale_of_setIntegral_le hadp hint fun i j hij s hs => ?_
induction' hij with k hk₁ hk₂
· exact le_rfl
· exact le_trans hk₂ (hf k s (𝒢.mono hk₁ _ hs))
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 158 | 161 | theorem single_mul_single_of_ne [Fintype n] [DecidableEq n] [DecidableEq k] [DecidableEq m]
[Semiring α] {b₁ b₂ : n} (hb : b₁ ≠ b₂) (a : m) (c : k) :
(single a b₁).toMatrix * (single b₂ c).toMatrix = (0 : Matrix _ _ α) := by |
rw [← toMatrix_trans, single_trans_single_of_ne hb, toMatrix_bot]
|
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 | 212 | 218 | theorem continuous_circleMap_inv {R : ℝ} {z w : ℂ} (hw : w ∈ ball z R) :
Continuous fun θ => (circleMap z R θ - w)⁻¹ := by |
have : ∀ θ, circleMap z R θ - w ≠ 0 := by
simp_rw [sub_ne_zero]
exact fun θ => circleMap_ne_mem_ball hw θ
-- Porting note: was `continuity`
exact Continuous.inv₀ (by continuity) this
|
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 58 | 59 | theorem toList_ofList (l : List α) : DList.toList (DList.ofList l) = l := by |
cases l; rfl; simp only [DList.toList, DList.ofList, List.cons_append, List.append_nil]
|
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 | 485 | 486 | theorem toIocMod_add_right (a b : α) : toIocMod hp a (b + p) = toIocMod hp a b := by |
simpa only [one_zsmul] using toIocMod_add_zsmul hp a b 1
|
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
open CategoryTheory
open CategoryTheory.Category
open CategoryTheory.Limits
open CategoryTheory.Preadditive
open Opposite
namespace Catego... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 198 | 204 | theorem isIdempotentComplete_iff_opposite : IsIdempotentComplete Cᵒᵖ ↔ IsIdempotentComplete C := by |
constructor
· exact isIdempotentComplete_of_isIdempotentComplete_opposite
· intro h
apply isIdempotentComplete_of_isIdempotentComplete_opposite
rw [isIdempotentComplete_iff_of_equivalence (opOpEquivalence C)]
exact h
|
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section LocallySmall
variable {C : Type v₁} [Category.{v₁} C] {D... | Mathlib/CategoryTheory/Filtered/Final.lean | 225 | 229 | theorem Functor.initial_iff_isCofiltered_costructuredArrow [IsCofilteredOrEmpty C] :
Initial F ↔ ∀ d, IsCofiltered (CostructuredArrow F d) := by |
refine ⟨?_, fun h => initial_of_isCofiltered_costructuredArrow F⟩
rw [initial_iff_of_isCofiltered]
exact fun h => isCofiltered_costructuredArrow_of_isCofiltered_of_exists F h.1 h.2
|
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {α β : Type*} {n : ℕ}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 55 | 58 | theorem Iio_castSucc (i : Fin n) : Iio (castSucc i) = (Iio i).map Fin.castSuccEmb := by |
apply Finset.map_injective Fin.valEmbedding
rw [Finset.map_map, Fin.map_valEmbedding_Iio]
exact (Fin.map_valEmbedding_Iio i).symm
|
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 505 | 507 | theorem HasFDerivAtFilter.sub (hf : HasFDerivAtFilter f f' x L) (hg : HasFDerivAtFilter g g' x L) :
HasFDerivAtFilter (fun x => f x - g x) (f' - g') x L := by |
simpa only [sub_eq_add_neg] using hf.add hg.neg
|
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {α : Type*} {β : α →... | Mathlib/Control/LawfulFix.lean | 177 | 188 | theorem fix_eq : Part.fix f = f (Part.fix f) := by |
rw [fix_eq_ωSup f, hc]
apply le_antisymm
· apply ωSup_le_ωSup_of_le _
intro i
exists i
intro x
-- intros x y hx,
apply le_f_of_mem_approx _ ⟨i, rfl⟩
· apply ωSup_le_ωSup_of_le _
intro i
exists i.succ
|
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
variable {R : Type*} {R₂ : Type*} {R₃ : Type*}
variable {K : Type*} {K₂ : Type*}
variable {M : Type*} {M₂ : Type*} {M₃ : Type*}
variable {V : ... | Mathlib/Algebra/Module/Submodule/Range.lean | 379 | 384 | theorem ker_eq_bot_of_cancel {f : M →ₛₗ[τ₁₂] M₂}
(h : ∀ u v : ker f →ₗ[R] M, f.comp u = f.comp v → u = v) : ker f = ⊥ := by |
have h₁ : f.comp (0 : ker f →ₗ[R] M) = 0 := comp_zero _
rw [← Submodule.range_subtype (ker f),
← h 0 f.ker.subtype (Eq.trans h₁ (comp_ker_subtype f).symm)]
exact range_zero
|
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 | 317 | 320 | theorem isReflection_of_mem_leftInvSeq (ω : List B) {t : W} (ht : t ∈ lis ω) :
cs.IsReflection t := by |
simp only [leftInvSeq_eq_reverse_rightInvSeq_reverse, mem_reverse] at ht
exact cs.isReflection_of_mem_rightInvSeq ω.reverse ht
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
variable {s s₁ s₂ : Set α} {t ... | Mathlib/Data/Set/Function.lean | 1,362 | 1,363 | theorem invFunOn_neg (h : ¬∃ a ∈ s, f a = b) : invFunOn f s b = Classical.choice ‹Nonempty α› := by |
rw [invFunOn, dif_neg h]
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 736 | 736 | theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by | rw [← sin_sq_add_cos_sq x, add_sub_cancel_left]
|
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqFocus
namespace String
attribute [ext] ext
theorem lt_trans {s₁ s₂ s₃ : String} : s₁ < s₂ → s₂ < s₃ → s₁ < s₃ :=
List.lt_trans' (α := Char) Nat.lt_trans
... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 360 | 363 | theorem extract.go₁_cons_addChar (c : Char) (cs : List Char) (b e : Pos) :
go₁ (c :: cs) 0 (b + c) (e + c) = go₁ cs 0 b e := by |
simp [go₁, Pos.ext_iff, Nat.ne_of_lt add_csize_pos]
apply go₁_add_right_cancel
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,395 | 2,400 | theorem not_monotoneOn_not_antitoneOn_iff_exists_le_le :
¬MonotoneOn f s ∧ ¬AntitoneOn f s ↔
∃ᵉ (a ∈ s) (b ∈ s) (c ∈ s), a ≤ b ∧ b ≤ c ∧
(f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c) := by |
simp [monotoneOn_iff_monotone, antitoneOn_iff_antitone, and_assoc, exists_and_left,
not_monotone_not_antitone_iff_exists_le_le, @and_left_comm (_ ∈ s)]
|
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 | 122 | 122 | theorem inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b := by | rw [inv_mul_lt_iff h, mul_comm]
|
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 92 | 94 | theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
∫ x, f x ∂μ = 0 := by |
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_left_eq_self]
|
import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
import Mathlib.Topology.Homotopy.Contractible
import Mathlib.CategoryTheory.PUnit
import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit
#align_import algebraic_topology.fundamental_groupoid.simply_connected from "leanprover-community/mathlib"@"3834... | Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean | 42 | 48 | theorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :
SimplyConnectedSpace X ↔
Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) := by |
simp only [simply_connected_def, equiv_punit_iff_unique,
FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]
intros
exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩
|
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Tactic.AdaptationNote
#align_import geometry.euclidean.inversion from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Metric Function AffineMap Set Aff... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 186 | 203 | theorem mul_dist_le_mul_dist_add_mul_dist (a b c d : P) :
dist a c * dist b d ≤ dist a b * dist c d + dist b c * dist a d := by |
-- If one of the points `b`, `c`, `d` is equal to `a`, then the inequality is trivial.
rcases eq_or_ne b a with (rfl | hb)
· rw [dist_self, zero_mul, zero_add]
rcases eq_or_ne c a with (rfl | hc)
· rw [dist_self, zero_mul]
apply_rules [add_nonneg, mul_nonneg, dist_nonneg]
rcases eq_or_ne d a with (rfl ... |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDomain
#align_import data.polynomial.field_division from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Polynomial
namespace Polynomial
u... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 124 | 130 | theorem one_lt_rootMultiplicity_iff_isRoot
{p : R[X]} {t : R} (h : p ≠ 0) :
1 < p.rootMultiplicity t ↔ p.IsRoot t ∧ (derivative p).IsRoot t := by |
rw [one_lt_rootMultiplicity_iff_isRoot_iterate_derivative h]
refine ⟨fun h ↦ ⟨h 0 (by norm_num), h 1 (by norm_num)⟩, fun ⟨h0, h1⟩ m hm ↦ ?_⟩
obtain (_|_|m) := m
exacts [h0, h1, by omega]
|
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.DiscreteQuotient
import Mathlib.Topology.Category.TopCat.Limits.Cofiltered
import Mathlib.Topology.Category.TopCat.Limits.Konig
#align_import topology.category.Profinite.cofiltered_limit from "leanpr... | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | 165 | 194 | theorem exists_locallyConstant_finite_nonempty {α : Type*} [Finite α] [Nonempty α]
(hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _) := by |
inhabit α
obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f
let ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
let σ : (α → Fin 2) → α := fun f => if h : ∃ a : α, ι a = f then h.choose else default
refine ⟨j, gg.map σ, ?_⟩
ext x
simp only [Functor.const_obj_obj, LocallyConstant.coe_co... |
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