Context stringlengths 285 157k | 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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 662 | 664 | theorem ofNat_mul_im (n : ℕ) [n.AtLeastTwo] (z : K) :
im (OfNat.ofNat n * z) = OfNat.ofNat n * im z := by |
rw [← ofReal_ofNat, im_ofReal_mul]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,415 | 1,416 | theorem ContDiff.sub {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
ContDiff 𝕜 n fun x => f x - g x := by | simpa only [sub_eq_add_neg] using hf.add hg.neg
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | 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, ← sq, ← sq]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 897 | 898 | theorem isBigOWith_neg_right : (IsBigOWith c l f fun x => -g' x) ↔ IsBigOWith c l f g' := by |
simp only [IsBigOWith_def, norm_neg]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 177 | 180 | theorem pow_orderOf_eq_one (x : G) : x ^ orderOf x = 1 := by |
convert Eq.trans _ (isPeriodicPt_minimalPeriod (x * ·) 1)
-- Porting note(#12129): additional beta reduction needed in the middle of the rewrite
rw [orderOf, mul_left_iterate]; beta_reduce; rw [mul_one]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "... | Mathlib/NumberTheory/NumberField/Norm.lean | 90 | 99 | theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) := by |
classical
have hint :
IsIntegral ℤ (∏ σ ∈ univ.erase (AlgEquiv.refl : L ≃ₐ[K] L), σ x) :=
IsIntegral.prod _ (fun σ _ =>
((RingOfIntegers.isIntegral_coe x).map σ))
refine ⟨⟨_, hint⟩, ?_⟩
ext
rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]
simp [← Finset.mul_prod_erase _ _ (mem_univ Al... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 876 | 881 | theorem digits_one (b n) (n0 : 0 < n) (nb : n < b) : Nat.digits b n = [n] ∧ 1 < b ∧ 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 ⟨?_, b2, n0⟩
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]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 672 | 673 | theorem inv_mem_nhds_one {S : Set G} (hS : S ∈ (𝓝 1 : Filter G)) : S⁻¹ ∈ 𝓝 (1 : G) := by |
rwa [← nhds_one_symm'] at hS
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,849 | 1,852 | theorem dvd_count_pow [Nontrivial α] [DecidableEq (Associates α)] {a : Associates α} (ha : a ≠ 0)
{p : Associates α} (hp : Irreducible p) (k : ℕ) : k ∣ count p (a ^ k).factors := by |
rw [count_pow ha hp]
apply dvd_mul_right
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.Numb... | Mathlib/NumberTheory/PellMatiyasevic.lean | 606 | 619 | theorem xn_modEq_x2n_sub_lem {n j} (h : j ≤ n) : xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n] := by |
have h1 : xz a1 n ∣ 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 _ _ : ℤ)]
exact mod_cast (xn_modEq_x2n_add_lem _ n j))
((dvd_mul_right _ _).mu... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.order from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5... | Mathlib/Data/DFinsupp/Order.lean | 320 | 323 | theorem support_sup : (f ⊔ g).support = f.support ∪ g.support := by |
ext
simp only [Finset.mem_union, mem_support_iff, sup_apply, Ne, ← bot_eq_zero]
rw [_root_.sup_eq_bot_iff, not_and_or]
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 940 | 946 | theorem StrictConvexOn.translate_right (hf : StrictConvexOn 𝕜 s f) (c : E) :
StrictConvexOn 𝕜 ((fun z => c + z) ⁻¹' s) (f ∘ fun z => c + z) :=
⟨hf.1.translate_preimage_right _, fun x hx y hy hxy a b ha hb hab =>
calc
f (c + (a • x + b • y)) = f (a • (c + x) + b • (c + y)) := by |
rw [smul_add, smul_add, add_add_add_comm, Convex.combo_self hab]
_ < a • f (c + x) + b • f (c + y) := hf.2 hx hy ((add_right_injective c).ne hxy) ha hb hab⟩
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.Yoneda
impo... | Mathlib/CategoryTheory/Limits/Cones.lean | 181 | 184 | theorem Cocone.w {F : J ⥤ C} (c : Cocone F) {j j' : J} (f : j ⟶ j') :
F.map f ≫ c.ι.app j' = c.ι.app j := by |
rw [c.ι.naturality f]
apply comp_id
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 150 | 154 | theorem tprod_inv : ∏' b, (f b)⁻¹ = (∏' b, f b)⁻¹ := by |
by_cases hf : Multipliable f
· exact hf.hasProd.inv.tprod_eq
· simp [tprod_eq_one_of_not_multipliable hf,
tprod_eq_one_of_not_multipliable (mt Multipliable.of_inv hf)]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.... | Mathlib/Analysis/Convex/Topology.lean | 213 | 218 | theorem Convex.add_smul_sub_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E}
(hx : x ∈ closure s) (hy : y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) :
x + t • (y - x) ∈ 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 _ _)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | 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
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 1,495 | 1,501 | theorem isOpen_iff_ultrafilter' [CompactSpace X] (U : Set X) :
IsOpen U ↔ ∀ F : Ultrafilter X, F.lim ∈ U → U ∈ F.1 := by |
rw [isOpen_iff_ultrafilter]
refine ⟨fun h F hF => h F.lim hF F F.le_nhds_lim, ?_⟩
intro cond x hx f h
rw [← Ultrafilter.lim_eq_iff_le_nhds.2 h] at hx
exact cond _ hx
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Sylow.lean | 548 | 556 | theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
(hH : Fintype.card H = p ^ n) : card (normalizer H) ≡ card G [MOD p ^ (n + 1)] := by |
have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
simp only [← Nat.card_eq_fintype_card] at hH ⊢
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... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 324 | 335 | theorem minFac_has_prop {n : ℕ} (n1 : n ≠ 1) : minFacProp n (minFac n) := by |
by_cases n0 : n = 0
· simp [n0, minFacProp, GE.ge]
have n2 : 2 ≤ n := by
revert n0 n1
rcases n with (_ | _ | _) <;> simp [succ_le_succ]
simp only [minFac_eq, Nat.isUnit_iff]
by_cases d2 : 2 ∣ n <;> simp [d2]
· exact ⟨le_rfl, d2, fun k k2 _ => k2⟩
· refine
minFacAux_has_prop n2 3 0 rfl fun m... |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.RingTheory.Prime
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.Ideal.Quotient
#align_import ... | 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 [← eq_C_of_degree_le_zero (natDegree_eq_zero_iff_degree_le_zero.1 hpm)]
exact dvd_mul_right _ _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 466 | 471 | theorem dense_of_exists_between [Nontrivial α] {s : Set α}
(h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by |
refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_
obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty
obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab
exact ⟨x, ⟨H hx, xs⟩⟩
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 1,144 | 1,150 | theorem mkHom_f_succ_succ (n : ℕ) :
(mkHom P Q zero one one_zero_comm succ).f (n + 2) =
(succ n
⟨(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)⟩).1 := by |
dsimp [mkHom, mkHomAux]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 830 | 832 | theorem continuousWithinAt_compl_self {f : α → β} {a : α} :
ContinuousWithinAt f {a}ᶜ a ↔ ContinuousAt f a := by |
rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 802 | 803 | theorem floor_add_nat (a : α) (n : ℕ) : ⌊a + n⌋ = ⌊a⌋ + n := by |
rw [← Int.cast_natCast, floor_add_int]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theo... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 146 | 151 | theorem ae_tendsto_measure_inter_div (S : Set α) (K : ℝ) : ∀ᵐ x ∂μ.restrict S,
∀ {ι : Type*} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
(xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)),
Tendsto (fun j => μ (S ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :... |
filter_upwards [(vitaliFamily μ K).ae_tendsto_measure_inter_div S] with x hx ι l w δ δlim
xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f0242... | Mathlib/Data/PNat/Prime.lean | 257 | 260 | theorem Coprime.symm {m n : ℕ+} : m.Coprime n → n.Coprime m := by |
unfold Coprime
rw [gcd_comm]
simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 488 | 491 | theorem pow_of_dvd (h : IsPrimitiveRoot ζ k) {p : ℕ} (hp : p ≠ 0) (hdiv : p ∣ k) :
IsPrimitiveRoot (ζ ^ p) (k / p) := by |
suffices orderOf (ζ ^ p) = k / p by exact this ▸ IsPrimitiveRoot.orderOf (ζ ^ p)
rw [orderOf_pow' _ hp, ← eq_orderOf h, Nat.gcd_eq_right hdiv]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 196 | 199 | theorem HasFiniteIntegral.add_measure {f : α → β} (hμ : HasFiniteIntegral f μ)
(hν : HasFiniteIntegral f ν) : HasFiniteIntegral f (μ + ν) := by |
simp only [HasFiniteIntegral, lintegral_add_measure] at *
exact add_lt_top.2 ⟨hμ, hν⟩
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 295 | 298 | theorem submodule_comap_pow_reverse (p : Submodule R (CliffordAlgebra Q)) (n : ℕ) :
(p ^ n).comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
p.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) ^ n := by |
simp_rw [← submodule_map_reverse_eq_comap, submodule_map_pow_reverse]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_imp... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 217 | 219 | theorem toComplex_div_im (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).im = round (x / y : ℂ).im := by |
rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, mul_add, add_mul]
|
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
/-!... | Mathlib/Combinatorics/Additive/ETransform.lean | 88 | 92 | theorem mulDysonETransform.smul_finset_snd_subset_fst :
e • (mulDysonETransform e x).2 ⊆ (mulDysonETransform e x).1 := by |
dsimp
rw [smul_finset_inter, smul_inv_smul, inter_comm]
exact inter_subset_union
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 129 | 139 | theorem proj_eq_of_subset (h : ∀ i, J i → K i) : π (π C K) J = π C J := by |
ext x
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· obtain ⟨y, ⟨z, hz, rfl⟩, rfl⟩ := h
refine ⟨z, hz, (?_ : _ = (Proj J ∘ Proj K) z)⟩
rw [proj_comp_of_subset J K h]
· obtain ⟨y, hy, rfl⟩ := h
dsimp [π]
rw [← Set.image_comp]
refine ⟨y, hy, ?_⟩
rw [proj_comp_of_subset J K h]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 939 | 939 | theorem star_dotProduct : star v ⬝ᵥ w = star (star w ⬝ᵥ v) := by | simp [dotProduct]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 763 | 765 | theorem UniformSpace.mem_nhds_iff {x : α} {s : Set α} : s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, ball x V ⊆ s := by |
rw [nhds_eq_comap_uniformity, mem_comap]
simp_rw [ball]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import ... | Mathlib/Data/List/Chain.lean | 82 | 83 | theorem chain_append_singleton_iff_forall₂ :
Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b]) := by | simp [chain_iff_forall₂]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 739 | 742 | theorem basis_iff' :
M.Basis I X ↔ (M.Indep I ∧ I ⊆ X ∧ ∀ J, M.Indep J → I ⊆ J → J ⊆ X → I = J) ∧ X ⊆ 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 (_ ⊆ X)]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mat... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 84 | 85 | theorem assoc_flip :
(M.X ◁ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ▷ M.X) ≫ M.mul := by | simp
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
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... | Mathlib/Data/Num/Lemmas.lean | 870 | 871 | theorem cast_inj [LinearOrderedSemiring α] {m n : Num} : (m : α) = n ↔ m = n := by |
rw [← cast_to_nat m, ← cast_to_nat n, Nat.cast_inj, to_nat_inj]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | 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 → M)) (h : t ⊆ s) (hst : s ⊆ span R t) : s = t := by |
let f : t ↪ s :=
⟨fun x => ⟨x.1, h x.2⟩, fun a b hab => Subtype.coe_injective (Subtype.mk.inj hab)⟩
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 ⟨x, hx⟩ with ⟨y,... |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | 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 H₁ : i % n + 1 ≤ n := i.modNat.is_lt
have H₂ : 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... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 987 | 988 | theorem isClosed_iSup_iff {s : Set α} : IsClosed[⨆ i, t i] s ↔ ∀ i, IsClosed[t i] s := by |
simp [← @isOpen_compl_iff _ _ (⨆ i, t i), ← @isOpen_compl_iff _ _ (t _), isOpen_iSup_iff]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 720 | 721 | theorem trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) := by |
mfld_set_tac
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 464 | 468 | theorem prod_filter_mul_prod_filter_not
(s : Finset α) (p : α → Prop) [DecidablePred p] [∀ x, Decidable (¬p x)] (f : α → β) :
(∏ x ∈ s.filter p, f x) * ∏ x ∈ s.filter fun x => ¬p x, f x = ∏ x ∈ s, f x := by |
have := Classical.decEq α
rw [← prod_union (disjoint_filter_filter_neg s s p), filter_union_filter_neg_eq]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | 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]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 127 | 132 | theorem imageToKernel_comp_mono {D : V} (h : C ⟶ D) [Mono h] (w) :
imageToKernel f (g ≫ h) w =
imageToKernel f g ((cancel_mono h).mp (by simpa using w : (f ≫ g) ≫ h = 0 ≫ h)) ≫
(Subobject.isoOfEq _ _ (kernelSubobject_comp_mono g h)).inv := by |
ext
simp
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 620 | 623 | theorem collinear_insert_of_mem_affineSpan_pair {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) :
Collinear k ({p₁, p₂, p₃} : Set P) := by |
rw [collinear_insert_iff_of_mem_affineSpan h]
exact collinear_pair _ _ _
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 308 | 313 | theorem sameRay_of_mem_segment [StrictOrderedCommRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {x y z : E}
(h : x ∈ [y -[𝕜] z]) : SameRay 𝕜 (x - y) (z - x) := by |
rw [segment_eq_image'] at h
rcases h with ⟨θ, ⟨hθ₀, hθ₁⟩, rfl⟩
simpa only [add_sub_cancel_left, ← sub_sub, sub_smul, one_smul] using
(SameRay.sameRay_nonneg_smul_left (z - y) hθ₀).nonneg_smul_right (sub_nonneg.2 hθ₁)
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import proba... | Mathlib/Probability/Density.lean | 122 | 128 | theorem hasPDF_of_map_eq_withDensity {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E}
(hX : AEMeasurable X ℙ) (f : E → ℝ≥0∞) (hf : AEMeasurable f μ) (h : map X ℙ = μ.withDensity f) :
HasPDF X ℙ μ := by |
refine ⟨hX, ?_, ?_⟩ <;> rw [h]
· rw [withDensity_congr_ae hf.ae_eq_mk]
exact haveLebesgueDecomposition_withDensity μ hf.measurable_mk
· exact withDensity_absolutelyContinuous μ f
|
/-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jannis Limperg, Mario Carneiro
-/
import Batteries.Classes.Order
import Batteries.Control.ForInStep.Basic
namespace Batteries
namespace BinomialHeap
namespac... | .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean | 259 | 263 | theorem Heap.realSize_tail? {s : Heap α} : s.tail? le = some s' →
s.realSize = s'.realSize + 1 := by |
simp only [Heap.tail?]; intro eq
match eq₂ : s.deleteMin le, eq with
| some (a, tl), rfl => exact realSize_deleteMin eq₂
|
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.h... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 103 | 107 | theorem coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 := by |
induction' n with n ih
· apply coeff_C
· rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero]
simp
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 658 | 668 | theorem iso_eq_iso_refl {x : SimplexCategory} (e : x ≅ 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 eq₁ := Finset.orderEmbOfFin_unique' h fun i => Finset.mem_univ ((orderIsoOfIso e) i)
have eq₂ :=
Finset.orderEmbOfFin_unique' h fun i => Finset.mem_univ ((orderIsoOfIso (Iso.refl x)) i)
-- Porting note: t... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 480 | 481 | theorem toIcoMod_add_right' (a b : α) : toIcoMod hp (a + p) b = toIcoMod hp a b + p := by |
simpa only [one_zsmul] using toIcoMod_add_zsmul' hp a b 1
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | 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]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Sylow.lean | 763 | 770 | theorem characteristic_of_normal {p : ℕ} [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 Φ
show (Φ • P).toSubgroup = P.toSubgroup
congr
simp [eq_iff_true_of_subsingleton]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,319 | 1,321 | theorem Eventually.and_frequently {p q : α → Prop} {f : Filter α} (hp : ∀ᶠ x in f, p x)
(hq : ∃ᶠ x in f, q x) : ∃ᶠ x in f, p x ∧ q x := by |
simpa only [and_comm] using hq.and_eventually hp
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 787 | 792 | theorem DirectSum.IsInternal.collectedOrthonormalBasis_mem [DecidableEq ι]
(h : DirectSum.IsInternal A) {α : ι → Type*} [∀ i, Fintype (α i)]
(hV : OrthogonalFamily 𝕜 (fun i => A i) fun i => (A i).subtypeₗᵢ)
(v : ∀ i, OrthonormalBasis (α i) 𝕜 (A i)) (a : Σi, α i) :
h.collectedOrthonormalBasis hV v a ∈ ... |
simp [DirectSum.IsInternal.collectedOrthonormalBasis]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neumann bou... | Mathlib/Analysis/Convex/GaugeRescale.lean | 48 | 52 | theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by |
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 195 | 202 | theorem extend_agrees {x : FamilyOfElements P R} (t : x.Compatible) {f : Y ⟶ 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 (𝟙 _) _ hf _]
· simp
· rw [id_comp]
exact h.choose_spec.choose_spec.2
|
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
#align_import ring_theory.subring.basic from "leanprover-community/math... | Mathlib/Algebra/Ring/Subring/Basic.lean | 919 | 946 | theorem mem_closure_iff {s : Set R} {x} :
x ∈ closure s ↔ x ∈ AddSubgroup.closure (Submonoid.closure s : Set R) :=
⟨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 q₁ q₂ ihq₁ ihq₂ => by rw [mul_add x q₁ q₂]; apply AddSubgroup.add_mem _ ihq₁ ihq₂)
fun z hz => by
have f : x * -z = -(x * z) := by simp
rw [f]; apply AddSubgroup.neg_... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Prod
import Mathlib.Data.Seq.WSeq
#align_import data.seq.parallel from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7... | Mathlib/Data/Seq/Parallel.lean | 189 | 266 | theorem exists_of_mem_parallel {S : WSeq (Computation α)} {a} (h : a ∈ parallel S) :
∃ c ∈ S, a ∈ c := by |
suffices
∀ C, a ∈ C → ∀ (l : List (Computation α)) (S),
corec parallel.aux1 (l, S) = C → ∃ c, (c ∈ l ∨ c ∈ S) ∧ a ∈ c from
let ⟨c, h1, h2⟩ := this _ h [] S rfl
⟨c, h1.resolve_left <| List.not_mem_nil _, h2⟩
let F : List (Computation α) → Sum α (List (Computation α)) → Prop := by
intro l a
... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 867 | 871 | theorem cos_pi_div_thirty_two : cos (π / 32) = √(2 + √(2 + √(2 + √2))) / 2 := by |
trans cos (π / 2 ^ 5)
· congr
norm_num
· simp
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
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 "lean... | Mathlib/Algebra/Quandle.lean | 232 | 236 | theorem left_cancel_inv (x : R) {y y' : R} : x ◃⁻¹ y = x ◃⁻¹ y' ↔ y = y' := by |
constructor
· apply (act' x).symm.injective
rintro rfl
rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 502 | 502 | theorem sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x := by | simp [sin_add]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Finite
#align_import order.conditionally_complete_lattice.finset from "leanprov... | Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | 33 | 35 | theorem Finset.Nonempty.csSup_mem {s : Finset α} (h : s.Nonempty) : sSup (s : Set α) ∈ s := by |
rw [h.csSup_eq_max']
exact s.max'_mem _
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | Mathlib/Order/OrderIsoNat.lean | 58 | 62 | theorem exists_not_acc_lt_of_not_acc {a : α} {r} (h : ¬Acc r a) : ∃ b, ¬Acc r b ∧ r b a := by |
contrapose! h
refine ⟨_, fun b hr => ?_⟩
by_contra hb
exact h b hb hr
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 496 | 499 | theorem bddAbove_iff_exists_ge [SemilatticeSup γ] {s : Set γ} (x₀ : γ) :
BddAbove s ↔ ∃ x, x₀ ≤ x ∧ ∀ y ∈ s, y ≤ x := by |
rw [bddAbove_def, exists_ge_and_iff_exists]
exact Monotone.ball fun x _ => monotone_le
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 550 | 551 | theorem coe_iInf {ι : Sort*} (s : ι → ℝ≥0) : (↑(⨅ i, s i) : ℝ) = ⨅ i, ↑(s i) := by |
rw [iInf, iInf, coe_sInf, ← Set.range_comp]; rfl
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 367 | 369 | theorem StarConvex.neg (hs : StarConvex 𝕜 x s) : StarConvex 𝕜 (-x) (-s) := by |
rw [← image_neg]
exact hs.is_linear_image IsLinearMap.isLinearMap_neg
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | 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 ∃ k, toEnd R L M x ^ k = 0
have := exists_forall_pow_toEnd_eq_zero R L M
tauto
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | 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]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,211 | 1,214 | theorem submonoidMap_surjective (f : M →* N) (M' : Submonoid M) :
Function.Surjective (f.submonoidMap M') := by |
rintro ⟨_, x, hx, rfl⟩
exact ⟨⟨x, hx⟩, rfl⟩
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 816 | 816 | theorem div_le_iff_le_mul' : a / b ≤ c ↔ a ≤ b * c := by | rw [div_le_iff_le_mul, mul_comm]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | 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 [← 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
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 234 | 237 | theorem evenOdd_map_involute (n : ZMod 2) :
(evenOdd Q n).map (involute : CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q).toLinearMap =
evenOdd Q n := by |
simp_rw [evenOdd, Submodule.map_iSup, Submodule.map_pow, ι_range_map_involute]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Basic
#align_import order.circular from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829ce42efec"
/-!
# Circular order hierarchy... | Mathlib/Order/Circular.lean | 369 | 371 | theorem compl_cIcc {a b : α} : (cIcc a b)ᶜ = cIoo b a := by |
ext
rw [Set.mem_cIoo, sbtw_iff_not_btw, cIcc, mem_compl_iff, mem_setOf]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 320 | 322 | theorem sum_smul_const_vsub_eq_neg_weightedVSub (w : ι → k) (p₂ : ι → P) (p₁ : P)
(h : ∑ i ∈ s, w i = 0) : (∑ i ∈ s, w i • (p₁ -ᵥ p₂ i)) = -s.weightedVSub p₂ w := by |
rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, zero_sub]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.q... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 676 | 679 | theorem quotientEquivAlgOfEq_symm {I J : Ideal A} (h : I = J) :
(quotientEquivAlgOfEq R₁ h).symm = quotientEquivAlgOfEq R₁ h.symm := by |
ext
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Init.Function
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
#align_import control.basic from "l... | Mathlib/Control/Basic.lean | 83 | 85 | theorem map_bind (x : m α) {g : α → m β} {f : β → γ} :
f <$> (x >>= g) = x >>= fun a => f <$> g a := by |
rw [← bind_pure_comp, bind_assoc]; simp [bind_pure_comp]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import m... | Mathlib/ModelTheory/Basic.lean | 698 | 700 | theorem ofInjective_toHom [L.IsAlgebraic] {f : M →[L] N} (hf : Function.Injective f) :
(ofInjective hf).toHom = f := by |
ext; simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.GroupTheory.GroupAction.Un... | Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | 49 | 52 | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ := by |
rcases eq_or_ne a 0 with (rfl | ha)
· rw [inv_zero, map_zero, map_zero]
· exact (IsUnit.mk0 a ha).eq_on_inv f g h
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import proba... | Mathlib/Probability/Density.lean | 202 | 205 | theorem lintegral_eq_measure_univ {X : Ω → E} [HasPDF X ℙ μ] :
∫⁻ x, pdf X ℙ μ x ∂μ = ℙ Set.univ := by |
rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ MeasurableSet.univ,
map_apply_of_aemeasurable (HasPDF.aemeasurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
#... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 101 | 105 | theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ (imagePresheaf (whiskerRight f (forget A))).sheafify J = ⊤ := 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 ⟨fun H _ => H.imageSieve_mem, fun H => ⟨H _⟩⟩
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
/-!
# Power ... | Mathlib/RingTheory/PowerBasis.lean | 132 | 135 | theorem exists_eq_aeval' (pb : PowerBasis R S) (y : S) : ∃ f : R[X], y = aeval pb.gen f := by |
nontriviality S
obtain ⟨f, _, hf⟩ := exists_eq_aeval pb y
exact ⟨f, hf⟩
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2d... | Mathlib/Algebra/Lie/Abelian.lean | 318 | 326 | theorem LieSubmodule.lie_abelian_iff_lie_self_eq_bot : IsLieAbelian I ↔ ⁅I, I⁆ = ⊥ := 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
⟨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 ... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib... | Mathlib/Analysis/Complex/Basic.lean | 309 | 313 | theorem restrictScalars_one_smulRight' (x : E) :
ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] E) =
reCLM.smulRight x + I • imCLM.smulRight x := by |
ext ⟨a, b⟩
simp [mk_eq_add_mul_I, mul_smul, smul_comm I b x]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.m... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 325 | 325 | theorem one_div_pow {A : M} (n : ℕ) : (1 / A) ^ n = 1 / A ^ n := by | simp only [one_div, inv_pow']
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b34... | Mathlib/Data/Seq/Seq.lean | 174 | 178 | theorem ge_stable (s : Seq α) {aₙ : α} {n m : ℕ} (m_le_n : m ≤ n)
(s_nth_eq_some : s.get? n = some aₙ) : ∃ aₘ : α, s.get? m = some aₘ :=
have : s.get? n ≠ none := by | simp [s_nth_eq_some]
have : s.get? m ≠ none := mt (s.le_stable m_le_n) this
Option.ne_none_iff_exists'.mp this
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 1,332 | 1,334 | theorem mem_closure_iff_ultrafilter :
x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x := by |
simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 624 | 625 | theorem preimage_mul_const_Ioc (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Ioc a b = Ioc (a / c) (b / c) := by | simp [← Ioi_inter_Iic, h]
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/... | Mathlib/Algebra/Polynomial/Mirror.lean | 82 | 97 | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by |
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree]
exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _)
· rw [← revAtFun_eq, revAtFun, i... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | 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 ↔
∠ p1 p2 p3 = π / 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, ← norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two,
vsub_sub_vsub_cancel_right p1, ← neg_vsub_eq_vsub_rev p2 p3, norm_neg]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.CondDistrib
#align_import probability.kernel.condexp from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
/-... | Mathlib/Probability/Kernel/Condexp.lean | 141 | 147 | theorem _root_.MeasureTheory.Integrable.norm_integral_condexpKernel [NormedSpace ℝ F]
(hf_int : Integrable f μ) :
Integrable (fun ω => ‖∫ y, f y ∂condexpKernel μ m ω‖) μ := by |
rw [condexpKernel]
convert Integrable.norm_integral_condDistrib
(aemeasurable_id'' μ (inf_le_right : m ⊓ mΩ ≤ mΩ)) aemeasurable_id
(hf_int.comp_snd_map_prod_id (inf_le_right : m ⊓ mΩ ≤ mΩ)) using 1
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 600 | 603 | theorem iterate_pos_eq_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
f^[n] x = x + n * m ↔ 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 : ℝ)) hn
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombinat... | Mathlib/NumberTheory/FLT/Four.lean | 141 | 149 | theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 := by |
obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h
rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)
· use a0, b0, c0
· exfalso
exact ne_zero hf.1 h1.symm
· use a0, b0, -c0, neg_of_minimal hf, hc
exact neg_pos.mpr h1
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,547 | 1,549 | theorem aeval_unique (φ : MvPolynomial σ R →ₐ[R] S₁) : φ = aeval (φ ∘ X) := by |
ext i
simp
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 73 | 75 | theorem NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra (S : NonUnitalSubalgebra R A)
(h1 : (1 : A) ∈ S) : (NonUnitalSubalgebra.toSubalgebra S h1).toNonUnitalSubalgebra = S := by |
cases S; rfl
|
/-
Copyright (c) 2017 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker, Jon Eugster
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Logic.Uni... | Mathlib/Algebra/Group/Units.lean | 304 | 305 | theorem inv_mul_cancel_right (a : α) (b : αˣ) : a * ↑b⁻¹ * b = a := by |
rw [mul_assoc, inv_mul, mul_one]
|
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_lim... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 556 | 564 | theorem hasDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
(hf : ∀ᶠ n in l, ∀ x : 𝕜, HasDerivAt (f n) (f' n x) x)
(hfg : ∀ x : 𝕜, Tendsto (fun n => f n x) l (𝓝 (g x))) : ∀ x : 𝕜, HasDerivAt g (g' x) x := by |
intro x
have hf : ∀ᶠ n in l, ∀ x : 𝕜, x ∈ Set.univ → HasDerivAt (f n) (f' n x) x := by
filter_upwards [hf] with n h x _ using h x
have hfg : ∀ x : 𝕜, x ∈ Set.univ → Tendsto (fun n => f n x) l (𝓝 (g x)) := by simp [hfg]
have hf' : TendstoUniformlyOn f' g' l Set.univ := by rwa [tendstoUniformlyOn_univ]
... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSe... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 123 | 125 | theorem invOneSubPow_val_zero_eq_invUnitSub_one :
(invOneSubPow 0).val = invUnitsSub (1 : Sˣ) := by |
simp [invOneSubPow, invUnitsSub]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 453 | 455 | theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by |
rw [← cosh_sq_sub_sinh_sq x]
ring
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.