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 |
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/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556... | Mathlib/SetTheory/Game/Impartial.lean | 173 | 176 | theorem equiv_iff_add_equiv_zero' (H : PGame) : (G ≈ H) ↔ (G + H ≈ 0) := by |
rw [Game.PGame.equiv_iff_game_eq, ← @add_left_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add,
Game.PGame.equiv_iff_game_eq]
exact ⟨Eq.symm, Eq.symm⟩
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 715 | 715 | theorem star_add_self' : star a + a = ↑(2 * a.re) := by | rw [add_comm, self_add_star']
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-communi... | Mathlib/Algebra/DirectSum/Decomposition.lean | 136 | 137 | theorem decompose_of_mem_same {x : M} {i : ι} (hx : x ∈ ℳ i) : (decompose ℳ x i : M) = x := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_same, Subtype.coe_mk]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 520 | 521 | theorem card_units (p : ℕ) [Fact p.Prime] : Fintype.card (ZMod p)ˣ = p - 1 := by |
rw [Fintype.card_units, card]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yaël Dillies
-/
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.Filter.NAry
import Mathlib.Order.Filter.Ultrafilter
#align_import order.filter.pointwise from... | Mathlib/Order/Filter/Pointwise.lean | 709 | 710 | theorem bot_pow {n : ℕ} (hn : n ≠ 0) : (⊥ : Filter α) ^ n = ⊥ := by |
rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero hn), pow_succ', bot_mul]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 356 | 359 | theorem fourierCoeffOn.const_smul {a b : ℝ} (f : ℝ → E) (c : ℂ) (n : ℤ) (hab : a < b) :
fourierCoeffOn hab (c • f) n = c • fourierCoeffOn hab f n := by |
haveI := Fact.mk (by linarith : 0 < b - a)
apply fourierCoeff.const_smul
|
/-
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.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,085 | 1,089 | theorem neg_right {N : Type*} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N]
{v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ≪ᵥ w) : v ≪ᵥ -w := by |
intro s hs
rw [neg_apply, neg_eq_zero] at hs
exact h hs
|
/-
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.Limits.Types
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Limits.FunctorCategory
#align_import category... | Mathlib/CategoryTheory/Limits/ColimitLimit.lean | 89 | 93 | theorem ι_colimitLimitToLimitColimit_π (j) (k) :
colimit.ι _ k ≫ colimitLimitToLimitColimit F ≫ limit.π _ j =
limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k := by |
dsimp [colimitLimitToLimitColimit]
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 196 | 197 | theorem range_eq_bot {f : M →ₛₗ[τ₁₂] M₂} : range f = ⊥ ↔ f = 0 := by |
rw [← range_le_bot_iff, le_bot_iff]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Convex.Slope
/-!
# Convexity of functions and derivatives
... | Mathlib/Analysis/Convex/Deriv.lean | 111 | 121 | theorem StrictMonoOn.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) := by |
have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
(differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
exists_deriv_eq_slope f hxy hf A
rcases nonempty_Ioo.2 hxa with ⟨b, ⟨hxb, hba⟩⟩
refine ⟨b, ⟨hxb... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Grou... | Mathlib/RingTheory/Coprime/Basic.lean | 336 | 338 | theorem add_mul_right_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + z * x) := by |
rw [isCoprime_comm]
exact h.symm.add_mul_right_left z
|
/-
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.Order.Filter.Basic
import Mathlib.Data.Set.Countable
#align_import order.filter.countable_Inter from "leanprover-community/mathlib"@"b9e46fe10... | Mathlib/Order/Filter/CountableInter.lean | 116 | 121 | theorem EventuallyLE.countable_bInter {ι : Type*} {S : Set ι} (hS : S.Countable)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋂ i ∈ S, s i ‹_› ≤ᶠ[l] ⋂ i ∈ S, t i ‹_› := by |
simp only [biInter_eq_iInter]
haveI := hS.toEncodable
exact EventuallyLE.countable_iInter fun i => h i i.2
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 290 | 303 | theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular] {p : ℝ} (hp : 0 < p)
{f : α → E} (hf : Memℒp f (ENNReal.ofReal p) μ) {ε : ℝ} (hε : 0 < ε) :
∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Memℒp g (ENNReal.ofReal p) μ := by |
have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _
have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by
simp only [Ne, ENNReal.ofReal_eq_zero, not_le, I]
have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hp
rcases hf.exists_boundedContinuous_snorm_sub_le ENNReal.coe_ne... |
/-
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.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 219 | 224 | theorem functorCategoryMonoidalEquivalence.μIso_inv_app (A B : Action V G) :
((functorCategoryMonoidalEquivalence V G).μIso A B).inv.app PUnit.unit = 𝟙 _ := by |
rw [← NatIso.app_inv, ← IsIso.Iso.inv_hom]
refine IsIso.inv_eq_of_hom_inv_id ?_
rw [Category.comp_id, NatIso.app_hom, MonoidalFunctor.μIso_hom,
functorCategoryMonoidalEquivalence.μ_app]
|
/-
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 | 327 | 329 | theorem le_isGLB_iff (h : IsGLB s a) : b ≤ a ↔ b ∈ lowerBounds s := by |
rw [h.lowerBounds_eq]
rfl
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 2,324 | 2,329 | theorem stk_nth_val {K : Type*} {Γ : K → Type*} {L : ListBlank (∀ k, Option (Γ k))} {k S} (n)
(hL : ListBlank.map (proj k) L = ListBlank.mk (List.map some S).reverse) :
L.nth n k = S.reverse.get? n := by |
rw [← proj_map_nth, hL, ← List.map_reverse, ListBlank.nth_mk, List.getI_eq_iget_get?,
List.get?_map]
cases S.reverse.get? n <;> rfl
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 298 | 311 | theorem IsBlock.of_subgroup_of_conjugate {B : Set X} {H : Subgroup G} (hB : IsBlock H B) (g : G) :
IsBlock (Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H) (g • B) := by |
rw [IsBlock.def_one]
intro h'
obtain ⟨h, hH, hh⟩ := Subgroup.mem_map.mp (SetLike.coe_mem h')
simp only [MulEquiv.coe_toMonoidHom, MulAut.conj_apply] at hh
suffices h' • g • B = g • h • B by
simp only [this]
apply (hB.smul_eq_or_disjoint ⟨h, hH⟩).imp
· intro; congr
· exact Set.disjoint_image_o... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 467 | 469 | theorem card_support_eq_one {f : α →₀ M} :
card f.support = 1 ↔ ∃ a, f a ≠ 0 ∧ f = single a (f a) := by |
simp only [card_eq_one, support_eq_singleton]
|
/-
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.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Tactic.Ring
#align_import data.fintype.perm from "leanprover-community/mathlib"... | Mathlib/Data/Fintype/Perm.lean | 102 | 128 | theorem nodup_permsOfList : ∀ {l : List α}, l.Nodup → (permsOfList l).Nodup
| [], _ => by simp [permsOfList]
| a :: l, hl => by
have hl' : l.Nodup := hl.of_cons
have hln' : (permsOfList l).Nodup := nodup_permsOfList hl'
have hmeml : ∀ {f : Perm α}, f ∈ permsOfList l → f a = a := fun {f} hf =>
not_... |
rw [← hf.2, mul_apply, hmeml hf.1, swap_apply_left]
have hiy : x a = List.get l j := by
rw [← hg.2, mul_apply, hmeml hg.1, swap_apply_left]
have hieqj : i = j := nodup_iff_injective_get.1 hl' (hix.symm.trans hiy)
exact absurd hieqj (_root_.ne_of_lt hij)
· intros f hf₁ hf₂
le... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.Norm... | Mathlib/Analysis/NormedSpace/PiLp.lean | 185 | 187 | theorem edist_eq_iSup (f g : PiLp ∞ β) : edist f g = ⨆ i, edist (f i) (g i) := by |
dsimp [edist]
exact if_neg ENNReal.top_ne_zero
|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheor... | Mathlib/Algebra/Module/Torsion.lean | 296 | 300 | theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) :
torsionBy R M a ≤ torsionBy R M b := by |
rw [← torsionBySet_span_singleton_eq, ← torsionBySet_singleton_eq]
apply torsionBySet_le_torsionBySet_of_subset
rintro c (rfl : c = b); exact Ideal.mem_span_singleton.mpr dvd
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheo... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 440 | 449 | theorem genEigenspace_restrict (f : End R M) (p : Submodule R M) (k : ℕ) (μ : R)
(hfp : ∀ x : M, x ∈ p → f x ∈ p) :
genEigenspace (LinearMap.restrict f hfp) μ k =
Submodule.comap p.subtype (f.genEigenspace μ k) := by |
simp only [genEigenspace, OrderHom.coe_mk, ← LinearMap.ker_comp]
induction' k with k ih
· rw [pow_zero, pow_zero, LinearMap.one_eq_id]
apply (Submodule.ker_subtype _).symm
· erw [pow_succ, pow_succ, LinearMap.ker_comp, LinearMap.ker_comp, ih, ← LinearMap.ker_comp,
LinearMap.comp_assoc]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.DFinsupp.Basic
#align_import algebra.direct_sum.basic from "leanprover-community/mathlib"@"f7fc89d5d5ff1d... | Mathlib/Algebra/DirectSum/Basic.lean | 376 | 381 | theorem coe_of_apply {M S : Type*} [DecidableEq ι] [AddCommMonoid M] [SetLike S M]
[AddSubmonoidClass S M] {A : ι → S} (i j : ι) (x : A i) :
(of (fun i ↦ {x // x ∈ A i}) i x j : M) = if i = j then x else 0 := by |
obtain rfl | h := Decidable.eq_or_ne i j
· rw [DirectSum.of_eq_same, if_pos rfl]
· rw [DirectSum.of_eq_of_ne _ _ _ _ h, if_neg h, ZeroMemClass.coe_zero, ZeroMemClass.coe_zero]
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 329 | 330 | theorem IsWide.eqToHom_mem {S : Subgroupoid C} (Sw : S.IsWide) {c d : C} (h : c = d) :
eqToHom h ∈ S.arrows c d := by | cases h; simp only [eqToHom_refl]; apply Sw.id_mem c
|
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-communi... | Mathlib/Data/Real/Hyperreal.lean | 761 | 765 | theorem infinite_of_infinitesimal_inv {x : ℝ*} (h0 : x ≠ 0) (hi : Infinitesimal x⁻¹) :
Infinite x := by |
cases' lt_or_gt_of_ne h0 with hn hp
· exact Or.inr (infiniteNeg_iff_infinitesimal_inv_neg.mpr ⟨hi, inv_lt_zero.mpr hn⟩)
· exact Or.inl (infinitePos_iff_infinitesimal_inv_pos.mpr ⟨hi, inv_pos.mpr hp⟩)
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 210 | 212 | theorem logb_lt_logb_iff (hx : 0 < x) (hy : 0 < y) : logb b x < logb b y ↔ x < y := by |
rw [logb, logb, div_lt_div_right (log_pos hb)]
exact log_lt_log_iff hx hy
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 280 | 294 | theorem sum_basis (hvs : Set.InjOn v s) (hs : s.Nonempty) :
∑ j ∈ s, Lagrange.basis s v j = 1 := by |
refine eq_of_degrees_lt_of_eval_index_eq s hvs (lt_of_le_of_lt (degree_sum_le _ _) ?_) ?_ ?_
· rw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe s.card)]
intro i hi
rw [degree_basis hvs hi, Nat.cast_withBot, WithBot.coe_lt_coe]
exact Nat.pred_lt (card_ne_zero_of_mem hi)
· rw [degree_one, ← ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 188 | 192 | theorem geom_lt {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n)
(h : ∀ k < n, c * u k < u (k + 1)) : c ^ n * u 0 < u n := by |
apply (monotone_mul_left_of_nonneg hc).seq_pos_lt_seq_of_le_of_lt hn _ _ h
· simp
· simp [_root_.pow_succ', mul_assoc, le_refl]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"900... | Mathlib/Data/Finset/Sigma.lean | 91 | 94 | theorem sigma_eq_biUnion [DecidableEq (Σi, α i)] (s : Finset ι) (t : ∀ i, Finset (α i)) :
s.sigma t = s.biUnion fun i => (t i).map <| Embedding.sigmaMk i := by |
ext ⟨x, y⟩
simp [and_left_comm]
|
/-
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.RingTheory.FiniteType
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.finite... | Mathlib/RingTheory/FinitePresentation.lean | 192 | 212 | theorem mvPolynomial_of_finitePresentation [FinitePresentation.{w₁, w₂} R A]
(ι : Type v) [Finite ι] :
FinitePresentation.{w₁, max v w₂} R (MvPolynomial ι A) := by |
have hfp : FinitePresentation.{w₁, w₂} R A := inferInstance
rw [iff_quotient_mvPolynomial'] at hfp ⊢
classical
-- Porting note: use the same universe level
obtain ⟨(ι' : Type v), _, f, hf_surj, hf_ker⟩ := hfp
let g := (MvPolynomial.mapAlgHom f).comp (MvPolynomial.sumAlgEquiv R ι ι').toAlgHom
cases nonemp... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,064 | 1,067 | theorem continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
ContinuousWithinAt f s a ↔
∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε := by |
rw [ContinuousWithinAt, tendsto_nhdsWithin_nhds]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 184 | 195 | theorem content_eq_gcd_range_of_lt (p : R[X]) (n : ℕ) (h : p.natDegree < n) :
p.content = (Finset.range n).gcd p.coeff := by |
apply dvd_antisymm_of_normalize_eq normalize_content Finset.normalize_gcd
· rw [Finset.dvd_gcd_iff]
intro i _
apply content_dvd_coeff _
· apply Finset.gcd_mono
intro i
simp only [Nat.lt_succ_iff, mem_support_iff, Ne, Finset.mem_range]
contrapose!
intro h1
apply coeff_eq_zero_of_natDeg... |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 859 | 867 | theorem iso_symm_apply_aux (m : M') :
(iso S f).symm m =
LocalizedModule.mk (IsLocalizedModule.surj S f m).choose.1
(IsLocalizedModule.surj S f m).choose.2 := by |
-- Porting note: We remove `generalize_proofs _ h2`.
apply_fun iso S f using LinearEquiv.injective (iso S f)
rw [LinearEquiv.apply_symm_apply]
simp only [iso_apply, LinearMap.toFun_eq_coe, fromLocalizedModule_mk]
erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff', (surj' _).choose_spec]
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset... | Mathlib/GroupTheory/Exponent.lean | 316 | 323 | theorem _root_.Commute.exists_orderOf_eq_lcm {x y : G} (h : Commute x y) :
∃ z ∈ closure {x, y}, orderOf z = Nat.lcm (orderOf x) (orderOf y) := by |
by_cases hx : orderOf x = 0 <;> by_cases hy : orderOf y = 0
· exact ⟨x, subset_closure (by simp), by simp [hx]⟩
· exact ⟨x, subset_closure (by simp), by simp [hx]⟩
· exact ⟨y, subset_closure (by simp), by simp [hy]⟩
· exact ⟨_, mul_mem (pow_mem (subset_closure (by simp)) _) (pow_mem (subset_closure (by simp)... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 36 | 42 | theorem normalize_eq {num den} (den_nz) : normalize num den den_nz =
{ num := num / num.natAbs.gcd den
den := den / num.natAbs.gcd den
den_nz := normalize.den_nz den_nz rfl
reduced := normalize.reduced' den_nz rfl } := by |
simp only [normalize, maybeNormalize_eq,
Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,037 | 1,054 | theorem le_range_of_union_finset_eq_top {α β : Type*} [Infinite β] (f : α → Finset β)
(w : ⋃ a, (f a : Set β) = ⊤) : #β ≤ #(range f) := by |
have k : _root_.Infinite (range f) := by
rw [infinite_coe_iff]
apply mt (union_finset_finite_of_range_finite f)
rw [w]
exact infinite_univ
by_contra h
simp only [not_le] at h
let u : ∀ b, ∃ a, b ∈ f a := fun b => by simpa using (w.ge : _) (Set.mem_univ b)
let u' : β → range f := fun b => ⟨f (... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 1,206 | 1,208 | theorem exists_mem_insert {P : α → Prop} {a : α} {s : Set α} :
(∃ x ∈ insert a s, P x) ↔ (P a ∨ ∃ x ∈ s, P x) := by |
simp [mem_insert_iff, or_and_right, exists_and_left, exists_or]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 392 | 396 | theorem update_eq_add_sub_coeff {R : Type*} [Ring R] (p : R[X]) (n : ℕ) (a : R) :
p.update n a = p + Polynomial.C (a - p.coeff n) * Polynomial.X ^ n := by |
ext
rw [coeff_update_apply, coeff_add, coeff_C_mul_X_pow]
split_ifs with h <;> simp [h]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-... | Mathlib/Data/TypeVec.lean | 60 | 62 | theorem Arrow.ext {α β : TypeVec n} (f g : α ⟹ β) :
(∀ i, f i = g i) → f = g := by |
intro h; funext i; apply 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.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 491 | 493 | theorem toJordanDecomposition_eq {s : SignedMeasure α} {j : JordanDecomposition α}
(h : s = j.toSignedMeasure) : s.toJordanDecomposition = j := by |
rw [h, toJordanDecomposition_toSignedMeasure]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.Analysis.BoxIntegral.Partition.Additive
import Mathlib.Analysis.Calculus.FDeriv.Prod
#align_import analysi... | Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean | 264 | 278 | theorem hasIntegral_GP_divergence_of_forall_hasDerivWithinAt
(f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E)
(f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → E))
(s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable)
(Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x)
(Hd : ∀ x ∈ (Box.Icc I) \ s, H... |
refine HasIntegral.sum fun i _ => ?_
simp only [hasFDerivWithinAt_pi', continuousWithinAt_pi] at Hd Hs
exact hasIntegral_GP_pderiv I _ _ s hs (fun x hx => Hs x hx i) (fun x hx => Hd x hx i) i
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes... | Mathlib/CategoryTheory/Limits/Opposites.lean | 837 | 842 | theorem pushoutIsoUnopPullback_inv_fst {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y) [HasPushout f g]
[HasPullback f.op g.op] :
(pushoutIsoUnopPullback f g).inv.op ≫ pullback.fst = pushout.inl.op := by |
apply Quiver.Hom.unop_inj
dsimp
rw [← pushoutIsoUnopPullback_inl_hom, Category.assoc, Iso.hom_inv_id, Category.comp_id]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 199 | 233 | theorem ae_lineDeriv_sum_eq
(hf : LipschitzWith C f) {ι : Type*} (s : Finset ι) (a : ι → ℝ) (v : ι → E) :
∀ᵐ x ∂μ, lineDeriv ℝ f x (∑ i ∈ s, a i • v i) = ∑ i ∈ s, a i • lineDeriv ℝ f x (v i) := by |
/- Clever argument by Morrey: integrate against a smooth compactly supported function `g`, switch
the derivative to `g` by integration by parts, and use the linearity of the derivative of `g` to
conclude that the initial integrals coincide. -/
apply ae_eq_of_integral_contDiff_smul_eq (hf.locallyIntegrable_line... |
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 158 | 162 | theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by |
obtain rfl | hμ := eq_or_ne μ 0
· simp
· rw [laverage_eq]
exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
|
/-
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.Data.Real.Cardinality
import Mathlib.Topology.Separation
import Mathlib.Topology.TietzeExtension
/-!
# Not normal topological spaces
In this file we... | Mathlib/Topology/Separation/NotNormal.lean | 26 | 53 | theorem IsClosed.mk_lt_continuum [NormalSpace X] {s : Set X} (hs : IsClosed s)
[DiscreteTopology s] : #s < 𝔠 := by |
-- Proof by contradiction: assume `𝔠 ≤ #s`
by_contra! h
-- Choose a countable dense set `t : Set X`
rcases exists_countable_dense X with ⟨t, htc, htd⟩
haveI := htc.to_subtype
-- To obtain a contradiction, we will prove `2 ^ 𝔠 ≤ 𝔠`.
refine (Cardinal.cantor 𝔠).not_le ?_
calc
-- Any function `s → ... |
/-
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 | 368 | 370 | theorem orderOf_injective {H : Type*} [Monoid H] (f : G →* H) (hf : Function.Injective f) (x : G) :
orderOf (f x) = orderOf x := by |
simp_rw [orderOf_eq_orderOf_iff, ← f.map_pow, ← f.map_one, hf.eq_iff, forall_const]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 100 | 100 | theorem eval₂_bit0 : (bit0 p).eval₂ f x = bit0 (p.eval₂ f x) := by | rw [bit0, eval₂_add, bit0]
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee35... | Mathlib/RingTheory/WittVector/Domain.lean | 79 | 85 | theorem eq_iterate_verschiebung {x : 𝕎 R} {n : ℕ} (h : ∀ i < n, x.coeff i = 0) :
x = verschiebung^[n] (x.shift n) := by |
induction' n with k ih
· cases x; simp [shift]
· dsimp; rw [verschiebung_shift]
· exact ih fun i hi => h _ (hi.trans (Nat.lt_succ_self _))
· exact h
|
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,352 | 2,353 | theorem erase_sdiff_distrib (s t : Finset α) (a : α) : (s \ t).erase a = s.erase a \ t.erase a := by |
simp_rw [erase_eq, sdiff_sdiff, sup_sdiff_eq_sup le_rfl, sup_comm]
|
/-
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.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.con... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 287 | 339 | theorem ContMDiffOn.contMDiffOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
(hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n) (hs : UniqueMDiffOn I s) :
ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
(π E (TangentSpace I) ⁻¹' s) := by |
have m_le_n : m ≤ n := (le_add_right le_rfl).trans hmn
have one_le_n : 1 ≤ n := (le_add_left le_rfl).trans hmn
have U' : UniqueDiffOn 𝕜 (range I ∩ I.symm ⁻¹' s) := fun y hy ↦ by
simpa only [UniqueMDiffOn, UniqueMDiffWithinAt, hy.1, inter_comm, mfld_simps]
using hs (I.symm y) hy.2
rw [contMDiffOn_iff... |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 600 | 602 | theorem equivPair_head_smul_equivPair_tail {i : ι} (w : Word M) :
of (equivPair i w).head • (equivPair i w).tail = w := by |
rw [← rcons_eq_smul, ← equivPair_symm, Equiv.symm_apply_apply]
|
/-
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 | 1,832 | 1,835 | theorem vecMul_vecMul [Fintype n] [Fintype m] (v : m → α) (M : Matrix m n α) (N : Matrix n o α) :
v ᵥ* M ᵥ* N = v ᵥ* (M * N) := by |
ext
apply dotProduct_assoc
|
/-
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 | 994 | 996 | theorem symm_comp_self_toHom (f : M ≃[L] N) :
f.symm.toHom.comp f.toHom = Hom.id L M := by |
rw [← comp_toHom, symm_comp_self, refl_toHom]
|
/-
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 | 835 | 839 | theorem cos_pi_div_eight : cos (π / 8) = √(2 + √2) / 2 := by |
trans cos (π / 2 ^ 3)
· congr
norm_num
· simp
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ModularLattice
import Mathlib.Order.WellFounded
import Mathlib.Tactic.Nontriviality
#align_import order.atoms fr... | Mathlib/Order/Atoms.lean | 844 | 846 | theorem isAtom_iff' [OrderBot α] [IsAtomic α] [OrderBot β] {l : α → β} {u : β → α}
(gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) (h_atom : ∀ a, IsAtom a → u (l a) = a) (b : β) :
IsAtom (u b) ↔ IsAtom b := by | rw [← gi.isAtom_iff hbot h_atom, gi.l_u_eq]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 679 | 688 | theorem IsComplement'.isCompl (h : IsComplement' H K) : IsCompl H K := by |
refine
⟨disjoint_iff_inf_le.mpr fun g ⟨p, q⟩ =>
let x : H × K := ⟨⟨g, p⟩, 1⟩
let y : H × K := ⟨1, g, q⟩
Subtype.ext_iff.mp
(Prod.ext_iff.mp (show x = y from h.1 ((mul_one g).trans (one_mul g).symm))).1,
codisjoint_iff_le_sup.mpr fun g _ => ?_⟩
obtain ⟨⟨h, k⟩, rfl⟩ := h.2... |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Eric Wieser
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.Star
import Mathlib.LinearAlgebra.... | Mathlib/Topology/Instances/Matrix.lean | 308 | 312 | theorem Matrix.transpose_tsum [T2Space R] {f : X → Matrix m n R} : (∑' x, f x)ᵀ = ∑' x, (f x)ᵀ := by |
by_cases hf : Summable f
· exact hf.hasSum.matrix_transpose.tsum_eq.symm
· have hft := summable_matrix_transpose.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, transpose_zero]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 370 | 372 | theorem weightedHomogeneousComponent_C_mul (n : M) (r : R) :
weightedHomogeneousComponent w n (C r * φ) = C r * weightedHomogeneousComponent w n φ := by |
simp only [C_mul', LinearMap.map_smul]
|
/-
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
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 839 | 840 | theorem forall_apply_eq_imp_iff' {f : α → β} {p : β → Prop} :
(∀ a b, f a = b → p b) ↔ ∀ a, p (f a) := by | simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 385 | 396 | theorem prod_subset_prod_iff : s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ := by |
rcases (s ×ˢ t).eq_empty_or_nonempty with h | h
· simp [h, prod_eq_empty_iff.1 h]
have st : s.Nonempty ∧ t.Nonempty := by rwa [prod_nonempty_iff] at h
refine ⟨fun H => Or.inl ⟨?_, ?_⟩, ?_⟩
· have := image_subset (Prod.fst : α × β → α) H
rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this... |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Integral/Periodic.lean | 39 | 46 | theorem isAddFundamentalDomain_Ioc {T : ℝ} (hT : 0 < T) (t : ℝ)
(μ : Measure ℝ := by | volume_tac) :
IsAddFundamentalDomain (AddSubgroup.zmultiples T) (Ioc t (t + T)) μ := by
refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strict... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
impo... | Mathlib/RingTheory/Ideal/Cotangent.lean | 163 | 170 | theorem cotangentEquivIdeal_symm_apply (x : R) (hx : x ∈ I) :
-- Note: #8386 had to specify `(R₂ := R)` because `I.toCotangent` suggested `R ⧸ I^2` instead
I.cotangentEquivIdeal.symm ⟨(I ^ 2).mkQ x, Submodule.mem_map_of_mem (R₂ := R) hx⟩ =
I.toCotangent ⟨x, hx⟩ := by |
apply I.cotangentEquivIdeal.injective
rw [I.cotangentEquivIdeal.apply_symm_apply]
ext
rfl
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 78 | 79 | theorem tendsto_zero : Tendsto f l (𝓝 (0 : Γ₀)) ↔ ∀ (γ₀) (_ : γ₀ ≠ 0), ∀ᶠ x in l, f x < γ₀ := by |
simp [nhds_zero]
|
/-
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 | 1,030 | 1,031 | theorem toIcoMod_eq_fract_mul (b : α) : toIcoMod hp 0 b = Int.fract (b / p) * p := by |
simp [toIcoMod_eq_add_fract_mul]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 105 | 112 | theorem IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X}
{l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x := by |
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _
(comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _)
exact ⟨b, b.prop, h⟩
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro
-/
import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.cat... | Mathlib/Topology/Category/TopCat/Basic.lean | 206 | 212 | theorem openEmbedding_iff_isIso_comp {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] :
OpenEmbedding (f ≫ g) ↔ OpenEmbedding g := by |
constructor
· intro h
convert h.comp (TopCat.homeoOfIso (asIso f).symm).openEmbedding
exact congrArg _ (IsIso.inv_hom_id_assoc f g).symm
· exact fun h => h.comp (TopCat.homeoOfIso (asIso f)).openEmbedding
|
/-
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.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.GradedObject
#align_import algebra.hom... | Mathlib/Algebra/Homology/Homology.lean | 212 | 213 | theorem cycles'Map_arrow (f : C₁ ⟶ C₂) (i : ι) :
cycles'Map f i ≫ (C₂.cycles' i).arrow = (C₁.cycles' i).arrow ≫ f.f i := by | simp
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Anal... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 96 | 105 | theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀)^[p.order] f z := by |
have hq := hasFPowerSeriesAt_iff'.mp (has_fpower_series_iterate_dslope_fslope p.order hp)
filter_upwards [hq, hasFPowerSeriesAt_iff'.mp hp] with x hx1 hx2
have : ∀ k < p.order, p.coeff k = 0 := fun k hk => by
simpa [coeff_eq_zero] using apply_eq_zero_of_lt_order hk
obtain ⟨s, hs1, hs2⟩ := HasSum.exists_has... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#a... | Mathlib/Data/Rat/Floor.lean | 56 | 66 | theorem floor_int_div_nat_eq_div {n : ℤ} {d : ℕ} : ⌊(↑n : ℚ) / (↑d : ℚ)⌋ = n / (↑d : ℤ) := by |
rw [Rat.floor_def]
obtain rfl | hd := @eq_zero_or_pos _ _ d
· simp
set q := (n : ℚ) / d with q_eq
obtain ⟨c, n_eq_c_mul_num, d_eq_c_mul_denom⟩ : ∃ c, n = c * q.num ∧ (d : ℤ) = c * q.den := by
rw [q_eq]
exact mod_cast @Rat.exists_eq_mul_div_num_and_eq_mul_div_den n d (mod_cast hd.ne')
rw [n_eq_c_mul... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 473 | 475 | theorem even_cons_cons (a₁ a₂ : α) (s : Stream' α) : even (a₁::a₂::s) = a₁::even s := by |
unfold even
rw [corec_eq]; rfl
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 756 | 757 | theorem integral_cos_sq : ∫ x in a..b, cos x ^ 2 = (cos b * sin b - cos a * sin a + b - a) / 2 := by |
field_simp [integral_cos_pow, add_sub_assoc]
|
/-
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 | 225 | 227 | theorem hasFiniteIntegral_zero_measure {m : MeasurableSpace α} (f : α → β) :
HasFiniteIntegral f (0 : Measure α) := by |
simp only [HasFiniteIntegral, lintegral_zero_measure, zero_lt_top]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,016 | 1,031 | theorem snorm_indicator_ge_of_bdd_below (hp : p ≠ 0) (hp' : p ≠ ∞) {f : α → F} (C : ℝ≥0) {s : Set α}
(hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → C ≤ ‖s.indicator f x‖₊) :
C • μ s ^ (1 / p.toReal) ≤ snorm (s.indicator f) p μ := by |
rw [ENNReal.smul_def, smul_eq_mul, snorm_eq_lintegral_rpow_nnnorm hp hp',
ENNReal.le_rpow_one_div_iff (ENNReal.toReal_pos hp hp'),
ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg, ← ENNReal.rpow_mul,
one_div_mul_cancel (ENNReal.toReal_pos hp hp').ne.symm, ENNReal.rpow_one, ← set_lintegral_const,
... |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 145 | 151 | theorem dZero_comp_eq : dZero A ∘ₗ (zeroCochainsLequiv A) =
oneCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 0 1 := by |
ext x y
show A.ρ y (x default) - x default = _ + ({0} : Finset _).sum _
simp_rw [Fin.coe_fin_one, zero_add, pow_one, neg_smul, one_smul,
Finset.sum_singleton, sub_eq_add_neg]
rcongr i <;> exact Fin.elim0 i
|
/-
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.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 65 | 71 | theorem ghostComponent_verschiebungFun (x : 𝕎 R) (n : ℕ) :
ghostComponent (n + 1) (verschiebungFun x) = p * ghostComponent n x := by |
simp only [ghostComponent_apply, aeval_wittPolynomial]
rw [Finset.sum_range_succ', verschiebungFun_coeff, if_pos rfl,
zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero, add_zero, Finset.mul_sum, Finset.sum_congr rfl]
rintro i -
simp only [pow_succ', verschiebungFun_coeff_succ, Nat.succ_sub_succ_eq_sub, mul_a... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 185 | 190 | theorem IsAlgebraic.of_pow {r : A} {n : ℕ} (hn : 0 < n) (ht : IsAlgebraic R (r ^ n)) :
IsAlgebraic R r := by |
obtain ⟨p, p_nonzero, hp⟩ := ht
refine ⟨Polynomial.expand _ n p, ?_, ?_⟩
· rwa [Polynomial.expand_ne_zero hn]
· rwa [Polynomial.expand_aeval n p r]
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-comm... | Mathlib/Data/Nat/Log.lean | 326 | 331 | theorem clog_mono_right (b : ℕ) {n m : ℕ} (h : n ≤ m) : clog b n ≤ clog b m := by |
rcases le_or_lt b 1 with hb | hb
· rw [clog_of_left_le_one hb]
exact zero_le _
· rw [← le_pow_iff_clog_le hb]
exact h.trans (le_pow_clog hb _)
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 341 | 343 | theorem interpolate_one (hvs : Set.InjOn v s) (hs : s.Nonempty) : interpolate s v 1 = 1 := by |
simp_rw [interpolate_apply, Pi.one_apply, map_one, one_mul]
exact sum_basis hvs hs
|
/-
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 | 505 | 508 | theorem LinearIndependent.mono {t s : Set M} (h : t ⊆ s) :
LinearIndependent R (fun x => x : s → M) → LinearIndependent R (fun x => x : t → M) := by |
simp only [linearIndependent_subtype_disjoint]
exact Disjoint.mono_left (Finsupp.supported_mono h)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 240 | 244 | theorem tendsto_approxBounded_ae {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m m0 : MeasurableSpace α} {μ : Measure α} (hf : StronglyMeasurable[m] f) {c : ℝ}
(hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
∀ᵐ x ∂μ, Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) := by |
filter_upwards [hf_bound] with x hfx using tendsto_approxBounded_of_norm_le hf hfx
|
/-
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 | 162 | 163 | theorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) := by |
cases x; rw [Zsqrtd.norm, normSq]; simp
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 248 | 251 | theorem updateColumn_transpose [DecidableEq m] : updateColumn Mᵀ i b = (updateRow M i b)ᵀ := by |
ext
rw [transpose_apply, updateRow_apply, updateColumn_apply]
rfl
|
/-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | Mathlib/Topology/MetricSpace/Contracting.lean | 338 | 342 | theorem tendsto_iterate_fixedPoint (x) :
Tendsto (fun n ↦ f^[n] x) atTop (𝓝 <| fixedPoint f hf) := by |
convert tendsto_iterate_efixedPoint hf (edist_ne_top x _)
refine (fixedPoint_unique _ ?_).symm
apply efixedPoint_isFixedPt
|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheor... | Mathlib/Algebra/Module/Torsion.lean | 311 | 313 | theorem torsionBySet_univ : torsionBySet R M Set.univ = ⊥ := by |
rw [eq_bot_iff, ← torsionBy_one, ← torsionBySet_singleton_eq]
exact torsionBySet_le_torsionBySet_of_subset fun _ _ => trivial
|
/-
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.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Ring.Aut
import Mathlib.Algebra.Ring.CompTypeclasses
import ... | Mathlib/Algebra/Star/Basic.lean | 150 | 150 | theorem star_star_mul (x y : R) : star (star x * y) = star y * x := by | rw [star_mul, star_star]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 508 | 510 | theorem prod_properDivisors_prime_pow {α : Type*} [CommMonoid α] {k p : ℕ} {f : ℕ → α}
(h : p.Prime) : (∏ x ∈ (p ^ k).properDivisors, f x) = ∏ x ∈ range k, f (p ^ x) := by |
simp [h, properDivisors_prime_pow]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e31... | Mathlib/Data/QPF/Multivariate/Basic.lean | 258 | 264 | theorem suppPreservation_iff_isUniform : q.SuppPreservation ↔ q.IsUniform := by |
constructor
· intro h α a a' f f' h' i
rw [← MvPFunctor.supp_eq, ← MvPFunctor.supp_eq, ← h, h', h]
· rintro h α ⟨a, f⟩
ext
rwa [supp_eq_of_isUniform, MvPFunctor.supp_eq]
|
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 699 | 717 | theorem sbtw_iff_left_ne_and_right_mem_image_Ioi {x y z : P} :
Sbtw R x y z ↔ x ≠ y ∧ z ∈ lineMap x y '' Set.Ioi (1 : R) := by |
refine ⟨fun h => ⟨h.left_ne, ?_⟩, fun h => ?_⟩
· obtain ⟨r, ⟨hr, rfl⟩⟩ := h.wbtw.right_mem_image_Ici_of_left_ne h.left_ne
rw [Set.mem_Ici] at hr
rcases hr.lt_or_eq with (hrlt | rfl)
· exact Set.mem_image_of_mem _ hrlt
· exfalso
simp at h
· rcases h with ⟨hne, r, hr, rfl⟩
rw [Set.mem_Ioi... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,336 | 1,337 | theorem setToFun_zero (hT : DominatedFinMeasAdditive μ T C) : setToFun μ T hT (0 : α → E) = 0 := by |
erw [setToFun_eq hT (integrable_zero _ _ _), Integrable.toL1_zero, ContinuousLinearMap.map_zero]
|
/-
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.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 135 | 136 | theorem linearMap_vsub (f : P1 →ᵃ[k] P2) (p1 p2 : P1) : f.linear (p1 -ᵥ p2) = f p1 -ᵥ f p2 := by |
conv_rhs => rw [← vsub_vadd p1 p2, map_vadd, vadd_vsub]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 108 | 108 | theorem laverage_zero : ⨍⁻ _x, (0 : ℝ≥0∞) ∂μ = 0 := by | rw [laverage, lintegral_zero]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 253 | 255 | theorem reverse_reverse {v : Vector α n} : v.reverse.reverse = v := by |
cases v
simp [Vector.reverse]
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1d... | Mathlib/LinearAlgebra/Orientation.lean | 203 | 207 | theorem orientation_isEmpty [IsEmpty ι] (b : Basis ι R M) :
b.orientation = positiveOrientation := by |
rw [Basis.orientation]
congr
exact b.det_isEmpty
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.fold from "leanprover-community/mathlib"@"446eb51ce0a90f8385f260d2b5... | Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean | 121 | 122 | theorem foldl_one (f : M →ₗ[R] N →ₗ[R] N) (hf) (n : N) : foldl Q f hf n 1 = n := by |
rw [← foldr_reverse, reverse.map_one, foldr_one]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 104 | 104 | theorem pairwise_singleton (R) (a : α) : Pairwise R [a] := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 111 | 117 | theorem natDegree_C_mul_eq_of_mul_eq_one {ai : R} (au : ai * a = 1) :
(C a * p).natDegree = p.natDegree :=
le_antisymm (natDegree_C_mul_le a p)
(calc
p.natDegree = (1 * p).natDegree := by | nth_rw 1 [← one_mul p]
_ = (C ai * (C a * p)).natDegree := by rw [← C_1, ← au, RingHom.map_mul, ← mul_assoc]
_ ≤ (C a * p).natDegree := natDegree_C_mul_le ai (C a * p))
|
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
#align_import topology.homotopy.contractible from "leanprover-community/mathlib"... | Mathlib/Topology/Homotopy/Contractible.lean | 68 | 81 | theorem contractible_iff_id_nullhomotopic (Y : Type*) [TopologicalSpace Y] :
ContractibleSpace Y ↔ (ContinuousMap.id Y).Nullhomotopic := by |
constructor
· intro
apply id_nullhomotopic
rintro ⟨p, h⟩
refine
{ hequiv_unit' :=
⟨{ toFun := ContinuousMap.const _ ()
invFun := ContinuousMap.const _ p
left_inv := ?_
right_inv := ?_ }⟩ }
· exact h.symm
· convert Homotopic.refl (ContinuousMap.id Unit)
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 213 | 240 | theorem u_exists :
∃ u : E → ℝ,
ContDiff ℝ ⊤ u ∧ (∀ x, u x ∈ Icc (0 : ℝ) 1) ∧ support u = ball 0 1 ∧ ∀ x, u (-x) = u x := by |
have A : IsOpen (ball (0 : E) 1) := isOpen_ball
obtain ⟨f, f_support, f_smooth, f_range⟩ :
∃ f : E → ℝ, f.support = ball (0 : E) 1 ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1 :=
A.exists_smooth_support_eq
have B : ∀ x, f x ∈ Icc (0 : ℝ) 1 := fun x => f_range (mem_range_self x)
refine ⟨fun x => (f x ... |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Logic.Equiv.Functor
import Mathlib.RingTheory.FreeRing
#a... | Mathlib/RingTheory/FreeCommRing.lean | 377 | 390 | theorem coe_eq : ((↑) : FreeRing α → FreeCommRing α) =
@Functor.map FreeAbelianGroup _ _ _ fun l : List α => (l : Multiset α) := by |
funext x
erw [castFreeCommRing, toFreeCommRing, FreeRing.lift, Equiv.coe_trans, Function.comp,
FreeAbelianGroup.liftMonoid_coe (FreeMonoid.lift FreeCommRing.of)]
dsimp [Functor.map]
rw [← AddMonoidHom.coe_coe]
apply FreeAbelianGroup.lift.unique; intro L
erw [FreeAbelianGroup.lift.of, Function.comp]
e... |
/-
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.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 382 | 384 | theorem liftPropWithinAt_of_liftPropAt_of_mem_nhds (h : LiftPropAt P g x) (hs : s ∈ 𝓝 x) :
LiftPropWithinAt P g s x := by |
rwa [← univ_inter s, hG.liftPropWithinAt_inter hs]
|
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