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.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 662 | 664 | theorem ofNat_mul_im (n : β) [n.AtLeastTwo] (z : K) :
im (OfNat.ofNat n * z) = OfNat.ofNat n * im z := by |
rw [β ofReal_ofNat, im_ofReal_mul]
|
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 | 1,415 | 1,416 | theorem ContDiff.sub {f g : E β F} (hf : ContDiff π n f) (hg : ContDiff π n g) :
ContDiff π n fun x => f x - g x := by | simpa only [sub_eq_add_neg] using hf.add hg.neg
|
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 | 720 | 720 | theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 := by | rw [two_mul, cos_add, β sq, β sq]
|
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 | 897 | 898 | theorem isBigOWith_neg_right : (IsBigOWith c l f fun x => -g' x) β IsBigOWith c l f g' := by |
simp only [IsBigOWith_def, norm_neg]
|
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 | 177 | 180 | theorem pow_orderOf_eq_one (x : G) : x ^ orderOf x = 1 := by |
convert Eq.trans _ (isPeriodicPt_minimalPeriod (x * Β·) 1)
-- Porting note(#12129): additional beta reduction needed in the middle of the rewrite
rw [orderOf, mul_left_iterate]; beta_reduce; rw [mul_one]
|
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 90 | 99 | theorem dvd_norm [IsGalois K L] (x : π L) : x β£ algebraMap (π K) (π L) (norm K x) := by |
classical
have hint :
IsIntegral β€ (β Ο β univ.erase (AlgEquiv.refl : L ββ[K] L), Ο x) :=
IsIntegral.prod _ (fun Ο _ =>
((RingOfIntegers.isIntegral_coe x).map Ο))
refine β¨β¨_, hintβ©, ?_β©
ext
rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]
simp [β Finset.mul_prod_erase _ _ (mem_univ Al... |
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 | 876 | 881 | theorem digits_one (b n) (n0 : 0 < n) (nb : n < b) : Nat.digits b n = [n] β§ 1 < b β§ 0 < n := by |
have b2 : 1 < b :=
lt_iff_add_one_le.mpr (le_trans (add_le_add_right (lt_iff_add_one_le.mp n0) 1) nb)
refine β¨?_, b2, n0β©
rw [Nat.digits_def' b2 n0, Nat.mod_eq_of_lt nb,
(Nat.div_eq_zero_iff ((zero_le n).trans_lt nb)).2 nb, Nat.digits_zero]
|
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 672 | 673 | theorem inv_mem_nhds_one {S : Set G} (hS : S β (π 1 : Filter G)) : Sβ»ΒΉ β π (1 : G) := by |
rwa [β nhds_one_symm'] at hS
|
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,849 | 1,852 | theorem dvd_count_pow [Nontrivial Ξ±] [DecidableEq (Associates Ξ±)] {a : Associates Ξ±} (ha : a β 0)
{p : Associates Ξ±} (hp : Irreducible p) (k : β) : k β£ count p (a ^ k).factors := by |
rw [count_pow ha hp]
apply dvd_mul_right
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 606 | 619 | theorem xn_modEq_x2n_sub_lem {n j} (h : j β€ n) : xn a1 (2 * n - j) + xn a1 j β‘ 0 [MOD xn a1 n] := by |
have h1 : xz a1 n β£ d a1 * yz a1 n * yz a1 (n - j) + xz a1 j := by
rw [yz_sub _ h, mul_sub_left_distrib, sub_add_eq_add_sub]
exact
dvd_sub
(by
delta xz; delta yz
rw [mul_comm (xn _ _ : β€)]
exact mod_cast (xn_modEq_x2n_add_lem _ n j))
((dvd_mul_right _ _).mu... |
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.order from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset
variable {ΞΉ : Type*} {Ξ± : ΞΉ β Type*}
namespace DFinsupp
section Zero
variable [β i, Zero (Ξ± i)]
instance [β... | Mathlib/Data/DFinsupp/Order.lean | 320 | 323 | theorem support_sup : (f β g).support = f.support βͺ g.support := by |
ext
simp only [Finset.mem_union, mem_support_iff, sup_apply, Ne, β bot_eq_zero]
rw [_root_.sup_eq_bot_iff, not_and_or]
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open scoped Classical
open LinearMap Set Convex Pointwise
variable {π E F Ξ± Ξ² ΞΉ : Type*}
secti... | Mathlib/Analysis/Convex/Function.lean | 940 | 946 | theorem StrictConvexOn.translate_right (hf : StrictConvexOn π s f) (c : E) :
StrictConvexOn π ((fun z => c + z) β»ΒΉ' s) (f β fun z => c + z) :=
β¨hf.1.translate_preimage_right _, fun x hx y hy hxy a b ha hb hab =>
calc
f (c + (a β’ x + b β’ y)) = f (a β’ (c + x) + b β’ (c + y)) := by |
rw [smul_add, smul_add, add_add_add_comm, Convex.combo_self hab]
_ < a β’ f (c + x) + b β’ f (c + y) := hf.2 hx hy ((add_right_injective c).ne hxy) ha hb habβ©
|
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.Yoneda
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.limits.cones from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
-- morphism le... | Mathlib/CategoryTheory/Limits/Cones.lean | 181 | 184 | theorem Cocone.w {F : J β₯€ C} (c : Cocone F) {j j' : J} (f : j βΆ j') :
F.map f β« c.ΞΉ.app j' = c.ΞΉ.app j := by |
rw [c.ΞΉ.naturality f]
apply comp_id
|
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 150 | 154 | theorem tprod_inv : β' b, (f b)β»ΒΉ = (β' b, f b)β»ΒΉ := by |
by_cases hf : Multipliable f
Β· exact hf.hasProd.inv.tprod_eq
Β· simp [tprod_eq_one_of_not_multipliable hf,
tprod_eq_one_of_not_multipliable (mt Multipliable.of_inv hf)]
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876... | Mathlib/Analysis/Convex/Topology.lean | 213 | 218 | theorem Convex.add_smul_sub_mem_interior' {s : Set E} (hs : Convex π s) {x y : E}
(hx : x β closure s) (hy : y β interior s) {t : π} (ht : t β Ioc (0 : π) 1) :
x + t β’ (y - x) β interior s := by |
simpa only [sub_smul, smul_sub, one_smul, add_sub, add_comm] using
hs.combo_interior_closure_mem_interior hy hx ht.1 (sub_nonneg.mpr ht.2)
(add_sub_cancel _ _)
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units
import Mathlib.Data.List.Perm
import Mathlib.Data.List.ProdSigma
import Mathlib.Data.List.Range
import Mathlib.Data.List.Rotate
#align_im... | Mathlib/Algebra/BigOperators/Group/List.lean | 289 | 290 | theorem get?_zero_mul_tail_prod (l : List M) : (l.get? 0).getD 1 * l.tail.prod = l.prod := by |
cases l <;> simp
|
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 1,495 | 1,501 | theorem isOpen_iff_ultrafilter' [CompactSpace X] (U : Set X) :
IsOpen U β β F : Ultrafilter X, F.lim β U β U β F.1 := by |
rw [isOpen_iff_ultrafilter]
refine β¨fun h F hF => h F.lim hF F F.le_nhds_lim, ?_β©
intro cond x hx f h
rw [β Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx
exact cond _ hx
|
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
import Mathlib.Data.Set.Lattice
#align_import group_theory.sylow from "leanprove... | Mathlib/GroupTheory/Sylow.lean | 548 | 556 | theorem card_normalizer_modEq_card [Fintype G] {p : β} {n : β} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) β‘ card G [MOD p ^ (n + 1)] := by |
have : H.subgroupOf (normalizer H) β H := (subgroupOfEquivOfLe le_normalizer).toEquiv
simp only [β Nat.card_eq_fintype_card] at hH β’
rw [card_eq_card_quotient_mul_card_subgroup H,
card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Nat.card_congr this,
hH, pow_succ']
simp only [Nat.c... |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.nat.prime from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82... | Mathlib/Data/Nat/Prime.lean | 324 | 335 | theorem minFac_has_prop {n : β} (n1 : n β 1) : minFacProp n (minFac n) := by |
by_cases n0 : n = 0
Β· simp [n0, minFacProp, GE.ge]
have n2 : 2 β€ n := by
revert n0 n1
rcases n with (_ | _ | _) <;> simp [succ_le_succ]
simp only [minFac_eq, Nat.isUnit_iff]
by_cases d2 : 2 β£ n <;> simp [d2]
Β· exact β¨le_rfl, d2, fun k k2 _ => k2β©
Β· refine
minFacAux_has_prop n2 3 0 rfl fun m... |
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.RingTheory.Prime
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.eisenstein_criterion from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
open Polynomial Ideal.Quotient
v... | Mathlib/RingTheory/EisensteinCriterion.lean | 72 | 78 | theorem isUnit_of_natDegree_eq_zero_of_isPrimitive {p q : R[X]}
-- Porting note: stated using `IsPrimitive` which is defeq to old statement.
(hu : IsPrimitive (p * q)) (hpm : p.natDegree = 0) : IsUnit p := by |
rw [eq_C_of_degree_le_zero (natDegree_eq_zero_iff_degree_le_zero.1 hpm), isUnit_C]
refine hu _ ?_
rw [β eq_C_of_degree_le_zero (natDegree_eq_zero_iff_degree_le_zero.1 hpm)]
exact dvd_mul_right _ _
|
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 | 466 | 471 | theorem dense_of_exists_between [Nontrivial Ξ±] {s : Set Ξ±}
(h : β β¦a bβ¦, a < b β β c β s, a < c β§ c < b) : Dense s := by |
refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_
obtain β¨a, b, hab, Hβ© : β a b : Ξ±, a < b β§ Ioo a b β U := U_open.exists_Ioo_subset U_nonempty
obtain β¨x, xs, hxβ© : β x β s, a < x β§ x < b := h hab
exact β¨x, β¨H hx, xsβ©β©
|
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 1,144 | 1,150 | theorem mkHom_f_succ_succ (n : β) :
(mkHom P Q zero one one_zero_comm succ).f (n + 2) =
(succ n
β¨(mkHom P Q zero one one_zero_comm succ).f n,
(mkHom P Q zero one one_zero_comm succ).f (n + 1),
(mkHom P Q zero one one_zero_comm succ).comm n (n + 1)β©).1 := by |
dsimp [mkHom, mkHomAux]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {Ξ± : Type*} {Ξ² : Type*} {Ξ³ : Type*} {Ξ΄ : Type*}
variable [TopologicalSpace Ξ±]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 830 | 832 | theorem continuousWithinAt_compl_self {f : Ξ± β Ξ²} {a : Ξ±} :
ContinuousWithinAt f {a}αΆ a β ContinuousAt f a := by |
rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ]
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 802 | 803 | theorem floor_add_nat (a : Ξ±) (n : β) : βa + nβ = βaβ + n := by |
rw [β Int.cast_natCast, floor_add_int]
|
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theory.covering.density_theorem from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filt... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 146 | 151 | theorem ae_tendsto_measure_inter_div (S : Set Ξ±) (K : β) : βα΅ x βΞΌ.restrict S,
β {ΞΉ : Type*} {l : Filter ΞΉ} (w : ΞΉ β Ξ±) (Ξ΄ : ΞΉ β β) (Ξ΄lim : Tendsto Ξ΄ l (π[>] 0))
(xmem : βαΆ j in l, x β closedBall (w j) (K * Ξ΄ j)),
Tendsto (fun j => ΞΌ (S β© closedBall (w j) (Ξ΄ j)) / ΞΌ (closedBall (w j) (Ξ΄ j))) l (π 1) :... |
filter_upwards [(vitaliFamily ΞΌ K).ae_tendsto_measure_inter_div S] with x hx ΞΉ l w Ξ΄ Ξ΄lim
xmem using hx.comp (tendsto_closedBall_filterAt ΞΌ _ _ Ξ΄lim xmem)
|
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : β+) : β+ :=
β¨Nat.gcd (n : β) (m : β), Nat.gcd_pos_of_pos_left (m : β) n.posβ©
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 257 | 260 | theorem Coprime.symm {m n : β+} : m.Coprime n β n.Coprime m := by |
unfold Coprime
rw [gcd_comm]
simp
|
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 | 488 | 491 | theorem pow_of_dvd (h : IsPrimitiveRoot ΞΆ k) {p : β} (hp : p β 0) (hdiv : p β£ k) :
IsPrimitiveRoot (ΞΆ ^ p) (k / p) := by |
suffices orderOf (ΞΆ ^ p) = k / p by exact this βΈ IsPrimitiveRoot.orderOf (ΞΆ ^ p)
rw [orderOf_pow' _ hp, β eq_orderOf h, Nat.gcd_eq_right hdiv]
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 196 | 199 | theorem HasFiniteIntegral.add_measure {f : Ξ± β Ξ²} (hΞΌ : HasFiniteIntegral f ΞΌ)
(hΞ½ : HasFiniteIntegral f Ξ½) : HasFiniteIntegral f (ΞΌ + Ξ½) := by |
simp only [HasFiniteIntegral, lintegral_add_measure] at *
exact add_lt_top.2 β¨hΞΌ, hΞ½β©
|
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 295 | 298 | theorem submodule_comap_pow_reverse (p : Submodule R (CliffordAlgebra Q)) (n : β) :
(p ^ n).comap (reverse : CliffordAlgebra Q ββ[R] CliffordAlgebra Q) =
p.comap (reverse : CliffordAlgebra Q ββ[R] CliffordAlgebra Q) ^ n := by |
simp_rw [β submodule_map_reverse_eq_comap, submodule_map_pow_reverse]
|
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 217 | 219 | theorem toComplex_div_im (x y : β€[i]) : ((x / y : β€[i]) : β).im = round (x / y : β).im := by |
rw [div_def, β @Rat.round_cast β _ _, β @Rat.round_cast β _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, mul_add, add_mul]
|
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {Ξ± : Type*} [DecidableEq Ξ±]
namespace Finset
section CommGroup
variable [CommGroup Ξ±] (e : Ξ±) (x : F... | Mathlib/Combinatorics/Additive/ETransform.lean | 88 | 92 | theorem mulDysonETransform.smul_finset_snd_subset_fst :
e β’ (mulDysonETransform e x).2 β (mulDysonETransform e x).1 := by |
dsimp
rw [smul_finset_inter, smul_inv_smul, inter_comm]
exact inter_subset_union
|
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
import Mathlib.Topology.LocallyConstant.Algebra
import Mathlib.Init.Data.Bool.Lemmas
universe u
namespace Profinite
namespace NobelingProof
variable {I : Ty... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 129 | 139 | theorem proj_eq_of_subset (h : β i, J i β K i) : Ο (Ο C K) J = Ο C J := by |
ext x
refine β¨fun h β¦ ?_, fun h β¦ ?_β©
Β· obtain β¨y, β¨z, hz, rflβ©, rflβ© := h
refine β¨z, hz, (?_ : _ = (Proj J β Proj K) z)β©
rw [proj_comp_of_subset J K h]
Β· obtain β¨y, hy, rflβ© := h
dsimp [Ο]
rw [β Set.image_comp]
refine β¨y, hy, ?_β©
rw [proj_comp_of_subset J K h]
|
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.BigOperators.RingEquiv
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Star.BigOperators... | Mathlib/Data/Matrix/Basic.lean | 939 | 939 | theorem star_dotProduct : star v β¬α΅₯ w = star (star w β¬α΅₯ v) := by | simp [dotProduct]
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 763 | 765 | theorem UniformSpace.mem_nhds_iff {x : Ξ±} {s : Set Ξ±} : s β π x β β V β π€ Ξ±, ball x V β s := by |
rw [nhds_eq_comap_uniformity, mem_comap]
simp_rw [ball]
|
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 82 | 83 | theorem chain_append_singleton_iff_forallβ :
Chain R a (l ++ [b]) β Forallβ R (a :: l) (l ++ [b]) := by | simp [chain_iff_forallβ]
|
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
set_option autoImplicit true
open Set
def Matroid.ExchangeProperty {Ξ± : Type _} (P : Set Ξ± β Prop) : Prop :=
β X Y, P X β P Y β β a β X \ Y, β b β Y \ X, P (insert b (X \ {a}))
def Matroid.ExistsMaximalSubsetProperty {... | Mathlib/Data/Matroid/Basic.lean | 739 | 742 | theorem basis_iff' :
M.Basis I X β (M.Indep I β§ I β X β§ β J, M.Indep J β I β J β J β X β I = J) β§ X β M.E := by |
simp [Basis, mem_maximals_setOf_iff, and_assoc, and_congr_left_iff, and_imp,
and_congr_left_iff, and_congr_right_iff, @Imp.swap (_ β X)]
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 84 | 85 | theorem assoc_flip :
(M.X β M.mul) β« M.mul = (Ξ±_ M.X M.X M.X).inv β« (M.mul β· M.X) β« M.mul := by | simp
|
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 870 | 871 | theorem cast_inj [LinearOrderedSemiring Ξ±] {m n : Num} : (m : Ξ±) = n β m = n := by |
rw [β cast_to_nat m, β cast_to_nat n, Nat.cast_inj, to_nat_inj]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 1,075 | 1,087 | theorem eq_of_linearIndependent_of_span_subtype [Nontrivial R] {s t : Set M}
(hs : LinearIndependent R (fun x => x : s β M)) (h : t β s) (hst : s β span R t) : s = t := by |
let f : t βͺ s :=
β¨fun x => β¨x.1, h x.2β©, fun a b hab => Subtype.coe_injective (Subtype.mk.inj hab)β©
have h_surj : Surjective f := by
apply surjective_of_linearIndependent_of_span hs f _
convert hst <;> simp [f, comp]
show s = t
apply Subset.antisymm _ h
intro x hx
rcases h_surj β¨x, hxβ© with β¨y,... |
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
#align_import data.fin.basic from "leanprover-community/mathlib"@"3a2b5524a138b5d0b818b858b516d4ac8a484b03"
assert_not_exists Monoid
universe u v
open Fin Na... | Mathlib/Data/Fin/Basic.lean | 1,774 | 1,786 | theorem modNat_rev (i : Fin (m * n)) : i.rev.modNat = i.modNat.rev := by |
ext
have Hβ : i % n + 1 β€ n := i.modNat.is_lt
have Hβ : i / n < m := i.divNat.is_lt
simp only [coe_modNat, val_rev]
calc
(m * n - (i + 1)) % n = (m * n - ((i / n) * n + i % n + 1)) % n := by rw [Nat.div_add_mod']
_ = ((m - i / n - 1) * n + (n - (i % n + 1))) % n := by
rw [Nat.mul_sub_right_dist... |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {Ξ± : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 987 | 988 | theorem isClosed_iSup_iff {s : Set Ξ±} : IsClosed[β¨ i, t i] s β β i, IsClosed[t i] s := by |
simp [β @isOpen_compl_iff _ _ (β¨ i, t i), β @isOpen_compl_iff _ _ (t _), isOpen_iSup_iff]
|
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 | 720 | 721 | theorem trans_source' : (e.trans e').source = e.source β© e β»ΒΉ' (e.target β© e'.source) := by |
mfld_set_tac
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 464 | 468 | theorem prod_filter_mul_prod_filter_not
(s : Finset Ξ±) (p : Ξ± β Prop) [DecidablePred p] [β x, Decidable (Β¬p x)] (f : Ξ± β Ξ²) :
(β x β s.filter p, f x) * β x β s.filter fun x => Β¬p x, f x = β x β s, f x := by |
have := Classical.decEq Ξ±
rw [β prod_union (disjoint_filter_filter_neg s s p), filter_union_filter_neg_eq]
|
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 | 231 | 233 | theorem leadingCoeff_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) :
P.toPoly.leadingCoeff = P.d := by |
rw [of_c_eq_zero ha hb hc, leadingCoeff_C]
|
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ΞΉ : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 127 | 132 | theorem imageToKernel_comp_mono {D : V} (h : C βΆ D) [Mono h] (w) :
imageToKernel f (g β« h) w =
imageToKernel f g ((cancel_mono h).mp (by simpa using w : (f β« g) β« h = 0 β« h)) β«
(Subobject.isoOfEq _ _ (kernelSubobject_comp_mono g h)).inv := by |
ext
simp
|
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 620 | 623 | theorem collinear_insert_of_mem_affineSpan_pair {pβ pβ pβ : P} (h : pβ β line[k, pβ, pβ]) :
Collinear k ({pβ, pβ, pβ} : Set P) := by |
rw [collinear_insert_iff_of_mem_affineSpan h]
exact collinear_pair _ _ _
|
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 308 | 313 | theorem sameRay_of_mem_segment [StrictOrderedCommRing π] [AddCommGroup E] [Module π E] {x y z : E}
(h : x β [y -[π] z]) : SameRay π (x - y) (z - x) := by |
rw [segment_eq_image'] at h
rcases h with β¨ΞΈ, β¨hΞΈβ, hΞΈββ©, rflβ©
simpa only [add_sub_cancel_left, β sub_sub, sub_smul, one_smul] using
(SameRay.sameRay_nonneg_smul_left (z - y) hΞΈβ).nonneg_smul_right (sub_nonneg.2 hΞΈβ)
|
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 122 | 128 | theorem hasPDF_of_map_eq_withDensity {X : Ξ© β E} {β : Measure Ξ©} {ΞΌ : Measure E}
(hX : AEMeasurable X β) (f : E β ββ₯0β) (hf : AEMeasurable f ΞΌ) (h : map X β = ΞΌ.withDensity f) :
HasPDF X β ΞΌ := by |
refine β¨hX, ?_, ?_β© <;> rw [h]
Β· rw [withDensity_congr_ae hf.ae_eq_mk]
exact haveLebesgueDecomposition_withDensity ΞΌ hf.measurable_mk
Β· exact withDensity_absolutelyContinuous ΞΌ f
|
import Batteries.Classes.Order
import Batteries.Control.ForInStep.Basic
namespace Batteries
namespace BinomialHeap
namespace Imp
inductive HeapNode (Ξ± : Type u) where
| nil : HeapNode Ξ±
| node (a : Ξ±) (child sibling : HeapNode Ξ±) : HeapNode Ξ±
deriving Repr
@[simp] def HeapNode.realSize : HeapNode Ξ± β ... | .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean | 259 | 263 | theorem Heap.realSize_tail? {s : Heap Ξ±} : s.tail? le = some s' β
s.realSize = s'.realSize + 1 := by |
simp only [Heap.tail?]; intro eq
match eqβ : s.deleteMin le, eq with
| some (a, tl), rfl => exact realSize_deleteMin eqβ
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 103 | 107 | theorem coeff_hermite_self (n : β) : coeff (hermite n) n = 1 := by |
induction' n with n ih
Β· apply coeff_C
Β· rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero]
simp
|
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.NonemptyFinLinOrd
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import algebraic_topology.simplex_category from "leanprover-community/mathlib"@"e8ac6315bcfcbaf2d19a046719c3b5... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 658 | 668 | theorem iso_eq_iso_refl {x : SimplexCategory} (e : x β
x) : e = Iso.refl x := by |
have h : (Finset.univ : Finset (Fin (x.len + 1))).card = x.len + 1 := Finset.card_fin (x.len + 1)
have eqβ := Finset.orderEmbOfFin_unique' h fun i => Finset.mem_univ ((orderIsoOfIso e) i)
have eqβ :=
Finset.orderEmbOfFin_unique' h fun i => Finset.mem_univ ((orderIsoOfIso (Iso.refl x)) i)
-- Porting note: t... |
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 | 480 | 481 | theorem toIcoMod_add_right' (a b : Ξ±) : toIcoMod hp (a + p) b = toIcoMod hp a b + p := by |
simpa only [one_zsmul] using toIcoMod_add_zsmul' hp a b 1
|
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 250 | 252 | theorem sumCoords_self_apply : b.sumCoords (b i) = 1 := by |
simp only [Basis.sumCoords, LinearMap.id_coe, LinearEquiv.coe_coe, id, Basis.repr_self,
Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp, Finsupp.sum_single_index]
|
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
import Mathlib.Data.Set.Lattice
#align_import group_theory.sylow from "leanprove... | Mathlib/GroupTheory/Sylow.lean | 763 | 770 | theorem characteristic_of_normal {p : β} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
(h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by |
haveI := Sylow.unique_of_normal P h
rw [characteristic_iff_map_eq]
intro Ξ¦
show (Ξ¦ β’ P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (Ξ± : Type*) where... | Mathlib/Order/Filter/Basic.lean | 1,319 | 1,321 | theorem Eventually.and_frequently {p q : Ξ± β Prop} {f : Filter Ξ±} (hp : βαΆ x in f, p x)
(hq : βαΆ x in f, q x) : βαΆ x in f, p x β§ q x := by |
simpa only [and_comm] using hq.and_eventually hp
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 787 | 792 | theorem DirectSum.IsInternal.collectedOrthonormalBasis_mem [DecidableEq ΞΉ]
(h : DirectSum.IsInternal A) {Ξ± : ΞΉ β Type*} [β i, Fintype (Ξ± i)]
(hV : OrthogonalFamily π (fun i => A i) fun i => (A i).subtypeβα΅’)
(v : β i, OrthonormalBasis (Ξ± i) π (A i)) (a : Ξ£i, Ξ± i) :
h.collectedOrthonormalBasis hV v a β ... |
simp [DirectSum.IsInternal.collectedOrthonormalBasis]
|
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module β E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) β’ x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 48 | 52 | theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent β s) (hsb : IsVonNBounded β s)
(x : E) : gaugeRescale s s x = x := by |
rcases eq_or_ne x 0 with rfl | hx; Β· simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w vβ vβ uβ uβ
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 195 | 202 | theorem extend_agrees {x : FamilyOfElements P R} (t : x.Compatible) {f : Y βΆ X} (hf : R f) :
x.sieveExtend f (le_generate R Y hf) = x f hf := by |
have h := (le_generate R Y hf).choose_spec
unfold FamilyOfElements.sieveExtend
rw [t h.choose (π _) _ hf _]
Β· simp
Β· rw [id_comp]
exact h.choose_spec.choose_spec.2
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
#align_import ring_theory.subring.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
universe u v w
variable {R : Type u} {S : Type v} {T : Type w} [Ring R]
namespace Subring
instance ... | Mathlib/Algebra/Ring/Subring/Basic.lean | 919 | 946 | theorem mem_closure_iff {s : Set R} {x} :
x β closure s β x β AddSubgroup.closure (Submonoid.closure s : Set R) :=
β¨fun h =>
closure_induction h (fun x hx => AddSubgroup.subset_closure <| Submonoid.subset_closure hx)
(AddSubgroup.zero_mem _)
(AddSubgroup.subset_closure (Submonoid.one_mem (Submonoi... | simp
rw [f]; apply AddSubgroup.neg_mem _ hx)
(by rw [mul_zero x]; apply AddSubgroup.zero_mem _)
(fun qβ qβ ihqβ ihqβ => by rw [mul_add x qβ qβ]; apply AddSubgroup.add_mem _ ihqβ ihqβ)
fun z hz => by
have f : x * -z = -(x * z) := by simp
rw [f]; apply AddSubgroup.neg_... |
import Mathlib.Init.Data.Prod
import Mathlib.Data.Seq.WSeq
#align_import data.seq.parallel from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
universe u v
namespace Computation
open Stream'
variable {Ξ± : Type u} {Ξ² : Type v}
def parallel.aux2 : List (Computation Ξ±) β Sum Ξ± (List (Com... | Mathlib/Data/Seq/Parallel.lean | 189 | 266 | theorem exists_of_mem_parallel {S : WSeq (Computation Ξ±)} {a} (h : a β parallel S) :
β c β S, a β c := by |
suffices
β C, a β C β β (l : List (Computation Ξ±)) (S),
corec parallel.aux1 (l, S) = C β β c, (c β l β¨ c β S) β§ a β c from
let β¨c, h1, h2β© := this _ h [] S rfl
β¨c, h1.resolve_left <| List.not_mem_nil _, h2β©
let F : List (Computation Ξ±) β Sum Ξ± (List (Computation Ξ±)) β Prop := by
intro l a
... |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 867 | 871 | theorem cos_pi_div_thirty_two : cos (Ο / 32) = β(2 + β(2 + β(2 + β2))) / 2 := by |
trans cos (Ο / 2 ^ 5)
Β· congr
norm_num
Β· simp
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (Ξ± : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 232 | 236 | theorem left_cancel_inv (x : R) {y y' : R} : x ββ»ΒΉ y = x ββ»ΒΉ y' β y = y' := by |
constructor
Β· apply (act' x).symm.injective
rintro rfl
rfl
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 502 | 502 | theorem sin_add_pi_div_two (x : β) : sin (x + Ο / 2) = cos x := by | simp [sin_add]
|
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Finite
#align_import order.conditionally_complete_lattice.finset from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
open Set
variable {ΞΉ Ξ± Ξ² Ξ³ : Type*}
section ConditionallyCompleteLinearOrder
variable [... | Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | 33 | 35 | theorem Finset.Nonempty.csSup_mem {s : Finset Ξ±} (h : s.Nonempty) : sSup (s : Set Ξ±) β s := by |
rw [h.csSup_eq_max']
exact s.max'_mem _
|
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {Ξ± : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 58 | 62 | theorem exists_not_acc_lt_of_not_acc {a : Ξ±} {r} (h : Β¬Acc r a) : β b, Β¬Acc r b β§ r b a := by |
contrapose! h
refine β¨_, fun b hr => ?_β©
by_contra hb
exact h b hb hr
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover-community/mathlib"@"b1abe23ae96fef89ad30d9f4362c307f72a55010"
open Function Set
open OrderDual (toDual ofDual)
universe u v w x
variable {Ξ± : Type u} {Ξ² : Type v}... | Mathlib/Order/Bounds/Basic.lean | 496 | 499 | theorem bddAbove_iff_exists_ge [SemilatticeSup Ξ³] {s : Set Ξ³} (xβ : Ξ³) :
BddAbove s β β x, xβ β€ x β§ β y β s, y β€ x := by |
rw [bddAbove_def, exists_ge_and_iff_exists]
exact Monotone.ball fun x _ => monotone_le
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Algebra.Order.Nonneg.Floor
import Mathlib.Data.Real.Pointwise
import Mathlib.Order.ConditionallyCompleteLattice.Group
imp... | Mathlib/Data/Real/NNReal.lean | 550 | 551 | theorem coe_iInf {ΞΉ : Sort*} (s : ΞΉ β ββ₯0) : (β(β¨
i, s i) : β) = β¨
i, β(s i) := by |
rw [iInf, iInf, coe_sInf, β Set.range_comp]; rfl
|
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {π E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 367 | 369 | theorem StarConvex.neg (hs : StarConvex π x s) : StarConvex π (-x) (-s) := by |
rw [β image_neg]
exact hs.is_linear_image IsLinearMap.isLinearMap_neg
|
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 264 | 268 | theorem isNilpotent_toEnd_of_isNilpotent [IsNilpotent R L M] (x : L) :
_root_.IsNilpotent (toEnd R L M x) := by |
change β k, toEnd R L M x ^ k = 0
have := exists_forall_pow_toEnd_eq_zero R L M
tauto
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{π : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 460 | 462 | theorem tangentMapWithin_univ : tangentMapWithin I I' f univ = tangentMap I I' f := by |
ext p : 1
simp only [tangentMapWithin, tangentMap, mfld_simps]
|
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Nat
import Mathlib.GroupTheory.GroupAction.Defs
#align_import group_theory.submonoid.operations from "leanprover-community/mathlib"@"cf8e77c636317b059a8ce20807a29cf3772a0640"
assert_not_ex... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,211 | 1,214 | theorem submonoidMap_surjective (f : M β* N) (M' : Submonoid M) :
Function.Surjective (f.submonoidMap M') := by |
rintro β¨_, x, hx, rflβ©
exact β¨β¨x, hxβ©, rflβ©
|
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {Ξ± : Type u}
class OrderedAddCommGroup (Ξ± : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 816 | 816 | theorem div_le_iff_le_mul' : a / b β€ c β a β€ b * c := by | rw [div_le_iff_le_mul, mul_comm]
|
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 | 475 | 481 | theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by |
have h1 : x + 2 * x = 3 * x := by ring
rw [β h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
rw [h2, cosh_sq]
ring
|
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 234 | 237 | theorem evenOdd_map_involute (n : ZMod 2) :
(evenOdd Q n).map (involute : CliffordAlgebra Q ββ[R] CliffordAlgebra Q).toLinearMap =
evenOdd Q n := by |
simp_rw [evenOdd, Submodule.map_iSup, Submodule.map_pow, ΞΉ_range_map_involute]
|
import Mathlib.Data.Set.Basic
#align_import order.circular from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829ce42efec"
class Btw (Ξ± : Type*) where
btw : Ξ± β Ξ± β Ξ± β Prop
#align has_btw Btw
export Btw (btw)
class SBtw (Ξ± : Type*) where
sbtw : Ξ± β Ξ± β Ξ± β Prop
#align has_sbtw SBtw
... | Mathlib/Order/Circular.lean | 369 | 371 | theorem compl_cIcc {a b : Ξ±} : (cIcc a b)αΆ = cIoo b a := by |
ext
rw [Set.mem_cIoo, sbtw_iff_not_btw, cIcc, mem_compl_iff, mem_setOf]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 320 | 322 | theorem sum_smul_const_vsub_eq_neg_weightedVSub (w : ΞΉ β k) (pβ : ΞΉ β P) (pβ : P)
(h : β i β s, w i = 0) : (β i β s, w i β’ (pβ -α΅₯ pβ i)) = -s.weightedVSub pβ w := by |
rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, zero_sub]
|
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.quotient_operations from "leanprover-community/mathlib"@"b88d81c84530450a8989e918608e5960f015e6c8"
universe u v w
namespace Ideal
open Function RingHom
var... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 676 | 679 | theorem quotientEquivAlgOfEq_symm {I J : Ideal A} (h : I = J) :
(quotientEquivAlgOfEq Rβ h).symm = quotientEquivAlgOfEq Rβ h.symm := by |
ext
rfl
|
import Mathlib.Init.Control.Combinators
import Mathlib.Init.Function
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
#align_import control.basic from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
variable {Ξ± Ξ² Ξ³ : Type u}
section Monad
variable {m : Type u... | Mathlib/Control/Basic.lean | 83 | 85 | theorem map_bind (x : m Ξ±) {g : Ξ± β m Ξ²} {f : Ξ² β Ξ³} :
f <$> (x >>= g) = x >>= fun a => f <$> g a := by |
rw [β bind_pure_comp, bind_assoc]; simp [bind_pure_comp]
|
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 698 | 700 | theorem ofInjective_toHom [L.IsAlgebraic] {f : M β[L] N} (hf : Function.Injective f) :
(ofInjective hf).toHom = f := by |
ext; simp
|
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.group_with_zero.units.lemmas from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
assert_not_exis... | Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | 49 | 52 | theorem eq_on_invβ (f g : F') (h : f a = g a) : f aβ»ΒΉ = g aβ»ΒΉ := by |
rcases eq_or_ne a 0 with (rfl | ha)
Β· rw [inv_zero, map_zero, map_zero]
Β· exact (IsUnit.mk0 a ha).eq_on_inv f g h
|
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 202 | 205 | theorem lintegral_eq_measure_univ {X : Ξ© β E} [HasPDF X β ΞΌ] :
β«β» x, pdf X β ΞΌ x βΞΌ = β Set.univ := by |
rw [β set_lintegral_univ, β map_eq_set_lintegral_pdf X β ΞΌ MeasurableSet.univ,
map_apply_of_aemeasurable (HasPDF.aemeasurable X β ΞΌ) MeasurableSet.univ, Set.preimage_univ]
|
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u w v' u' w'
open ... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 101 | 105 | theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cα΅α΅ β₯€ A} (f : F βΆ G) :
IsLocallySurjective J f β (imagePresheaf (whiskerRight f (forget A))).sheafify J = β€ := by |
simp only [Subpresheaf.ext_iff, Function.funext_iff, Set.ext_iff, top_subpresheaf_obj,
Set.top_eq_univ, Set.mem_univ, iff_true_iff]
exact β¨fun H _ => H.imageSieve_mem, fun H => β¨H _β©β©
|
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
open Polynomial
open Polynomial
variable {R S T : Type*} [CommRing R] [Ring S] [Algebra R S]
variable {A B : Type*} [CommRing A] [CommRing B] [IsDomain B]... | Mathlib/RingTheory/PowerBasis.lean | 132 | 135 | theorem exists_eq_aeval' (pb : PowerBasis R S) (y : S) : β f : R[X], y = aeval pb.gen f := by |
nontriviality S
obtain β¨f, _, hfβ© := exists_eq_aeval pb y
exact β¨f, hfβ©
|
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w wβ wβ
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 318 | 326 | theorem LieSubmodule.lie_abelian_iff_lie_self_eq_bot : IsLieAbelian I β β
I, Iβ = β₯ := by |
simp only [_root_.eq_bot_iff, lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le,
LieSubmodule.bot_coe, Set.subset_singleton_iff, Set.mem_setOf_eq, exists_imp]
refine
β¨fun h z x y hz =>
hz.symm.trans
(((I : LieSubalgebra R L).coe_bracket x y).symm.trans
((coe_zero_iff_zero _ _).mpr (by ... |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 309 | 313 | theorem restrictScalars_one_smulRight' (x : E) :
ContinuousLinearMap.restrictScalars β ((1 : β βL[β] β).smulRight x : β βL[β] E) =
reCLM.smulRight x + I β’ imCLM.smulRight x := by |
ext β¨a, bβ©
simp [mk_eq_add_mul_I, mul_smul, smul_comm I b x]
|
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 325 | 325 | theorem one_div_pow {A : M} (n : β) : (1 / A) ^ n = 1 / A ^ n := by | simp only [one_div, inv_pow']
|
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 {Ξ± : Type u} (s : Stream' (Option Ξ±)) : Prop :=
β {n : β}, s n = none β s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 174 | 178 | theorem ge_stable (s : Seq Ξ±) {aβ : Ξ±} {n m : β} (m_le_n : m β€ n)
(s_nth_eq_some : s.get? n = some aβ) : β aβ : Ξ±, s.get? m = some aβ :=
have : s.get? n β none := by | simp [s_nth_eq_some]
have : s.get? m β none := mt (s.le_stable m_le_n) this
Option.ne_none_iff_exists'.mp this
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : β
β... | Mathlib/Topology/Basic.lean | 1,332 | 1,334 | theorem mem_closure_iff_ultrafilter :
x β closure s β β u : Ultrafilter X, s β u β§ βu β€ π x := by |
simp [closure_eq_cluster_pts, ClusterPt, β exists_ultrafilter_iff, and_comm]
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 624 | 625 | theorem preimage_mul_const_Ioc (a b : Ξ±) {c : Ξ±} (h : 0 < c) :
(fun x => x * c) β»ΒΉ' Ioc a b = Ioc (a / c) (b / c) := by | simp [β Ioi_inter_Iic, h]
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 82 | 97 | theorem coeff_mirror (n : β) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by |
by_cases h2 : p.natDegree < n
Β· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n β€ p.natDegree + p.natTrailingDegree
Β· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree]
exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _)
Β· rw [β revAtFun_eq, revAtFun, i... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 360 | 365 | theorem dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two (p1 p2 p3 : P) :
dist p1 p3 * dist p1 p3 = dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 β
β p1 p2 p3 = Ο / 2 := by |
erw [dist_comm p3 p2, dist_eq_norm_vsub V p1 p3, dist_eq_norm_vsub V p1 p2,
dist_eq_norm_vsub V p2 p3, β norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two,
vsub_sub_vsub_cancel_right p1, β neg_vsub_eq_vsub_rev p2 p3, norm_neg]
|
import Mathlib.Probability.Kernel.CondDistrib
#align_import probability.kernel.condexp from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheory
namespace ProbabilityTheory
variable {Ξ© F : Ty... | Mathlib/Probability/Kernel/Condexp.lean | 141 | 147 | theorem _root_.MeasureTheory.Integrable.norm_integral_condexpKernel [NormedSpace β F]
(hf_int : Integrable f ΞΌ) :
Integrable (fun Ο => ββ« y, f y βcondexpKernel ΞΌ m Οβ) ΞΌ := by |
rw [condexpKernel]
convert Integrable.norm_integral_condDistrib
(aemeasurable_id'' ΞΌ (inf_le_right : m β mΞ© β€ mΞ©)) aemeasurable_id
(hf_int.comp_snd_map_prod_id (inf_le_right : m β mΞ© β€ mΞ©)) using 1
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 600 | 603 | theorem iterate_pos_eq_iff {x : β} {m : β€} {n : β} (hn : 0 < n) :
f^[n] x = x + n * m β f x = x + m := by |
simpa only [nsmul_eq_mul, add_right_iterate] using
(f.commute_add_int m).iterate_pos_eq_iff_map_eq f.monotone (strictMono_id.add_const (m : β)) hn
|
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 141 | 149 | theorem exists_pos_odd_minimal {a b c : β€} (h : Fermat42 a b c) :
β a0 b0 c0, Minimal a0 b0 c0 β§ a0 % 2 = 1 β§ 0 < c0 := by |
obtain β¨a0, b0, c0, hf, hcβ© := exists_odd_minimal h
rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
Β· use a0, b0, c0
Β· exfalso
exact ne_zero hf.1 h1.symm
Β· use a0, b0, -c0, neg_of_minimal hf, hc
exact neg_pos.mpr h1
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Data.Finsupp.Antidiagonal
import Mathlib.Order.SymmDiff
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.mv_polynomial.basic fr... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,547 | 1,549 | theorem aeval_unique (Ο : MvPolynomial Ο R ββ[R] Sβ) : Ο = aeval (Ο β X) := by |
ext i
simp
|
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
section Subalgebra
variable {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]
def Subalgebra.toNonUnitalSubalgebra (S : Subalgebr... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 73 | 75 | theorem NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A)
(h1 : (1 : A) β S) : (NonUnitalSubalgebra.toSubalgebra S h1).toNonUnitalSubalgebra = S := by |
cases S; rfl
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Logic.Unique
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Lift
#align_import algebra.group.units from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
assert_not_exists Multiplicative
a... | Mathlib/Algebra/Group/Units.lean | 304 | 305 | theorem inv_mul_cancel_right (a : Ξ±) (b : Ξ±Λ£) : a * βbβ»ΒΉ * b = a := by |
rw [mul_assoc, inv_mul, mul_one]
|
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_limits_deriv from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Filter
open scoped uniformity Filter Topology
section d... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 556 | 564 | theorem hasDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
(hf : βαΆ n in l, β x : π, HasDerivAt (f n) (f' n x) x)
(hfg : β x : π, Tendsto (fun n => f n x) l (π (g x))) : β x : π, HasDerivAt g (g' x) x := by |
intro x
have hf : βαΆ n in l, β x : π, x β Set.univ β HasDerivAt (f n) (f' n x) x := by
filter_upwards [hf] with n h x _ using h x
have hfg : β x : π, x β Set.univ β Tendsto (fun n => f n x) l (π (g x)) := by simp [hfg]
have hf' : TendstoUniformlyOn f' g' l Set.univ := by rwa [tendstoUniformlyOn_univ]
... |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 123 | 125 | theorem invOneSubPow_val_zero_eq_invUnitSub_one :
(invOneSubPow 0).val = invUnitsSub (1 : SΛ£) := by |
simp [invOneSubPow, invUnitsSub]
|
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 | 453 | 455 | theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by |
rw [β cosh_sq_sub_sinh_sq x]
ring
|
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