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) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Order.Interval.Finset.Basic
#align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd... | Mathlib/Data/Int/Interval.lean | 141 | 142 | theorem card_Ioc_of_le (h : a ≤ b) : ((Ioc a b).card : ℤ) = b - a := by |
rw [card_Ioc, toNat_sub_of_le h]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 331 | 332 | theorem polar_smul_left_of_tower (a : S) (x y : M) : polar Q (a • x) y = a • polar Q x y := by |
rw [← IsScalarTower.algebraMap_smul R a x, polar_smul_left, Algebra.smul_def]
|
/-
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.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 145 | 160 | theorem hasFDerivAt_stereoInvFunAux (v : E) :
HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) 0 := by |
have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : E →L[ℝ] ℝ) 0 := by
convert (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt
simp
have h₁ : HasFDerivAt (fun w : E => (‖w‖ ^ 2 + 4)⁻¹) (0 : E →L[ℝ] ℝ) 0 := by
convert (hasFDerivAt_inv _).comp _ (h₀.add (hasFDerivAt_const 4 0)) <;> simp
have h₂ : HasFDer... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 2,007 | 2,015 | theorem sumAddHom_comm {ι₁ ι₂ : Sort _} {β₁ : ι₁ → Type*} {β₂ : ι₂ → Type*} {γ : Type*}
[DecidableEq ι₁] [DecidableEq ι₂] [∀ i, AddZeroClass (β₁ i)] [∀ i, AddZeroClass (β₂ i)]
[AddCommMonoid γ] (f₁ : Π₀ i, β₁ i) (f₂ : Π₀ i, β₂ i) (h : ∀ i j, β₁ i →+ β₂ j →+ γ) :
sumAddHom (fun i₂ => sumAddHom (fun i₁ => h i... |
obtain ⟨⟨f₁, s₁, h₁⟩, ⟨f₂, s₂, h₂⟩⟩ := f₁, f₂
simp only [sumAddHom, AddMonoidHom.finset_sum_apply, Quotient.liftOn_mk, AddMonoidHom.coe_mk,
AddMonoidHom.flip_apply, Trunc.lift, toFun_eq_coe, ZeroHom.coe_mk, coe_mk']
exact Finset.sum_comm
|
/-
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.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 131 | 141 | theorem targetAffineLocally_respectsIso {P : AffineTargetMorphismProperty}
(hP : P.toProperty.RespectsIso) : (targetAffineLocally P).RespectsIso := by |
constructor
· introv H U
rw [morphismRestrict_comp, affine_cancel_left_isIso hP]
exact H U
· introv H
rintro ⟨U, hU : IsAffineOpen U⟩; dsimp
haveI : IsAffine _ := hU.map_isIso e.hom
rw [morphismRestrict_comp, affine_cancel_right_isIso hP]
exact H ⟨(Opens.map e.hom.val.base).obj U, hU.map_... |
/-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_imp... | Mathlib/Analysis/Convex/Slope.lean | 315 | 320 | theorem StrictConvexOn.secant_strict_mono_aux3 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
(hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f z - f x) / (z - x) < (f z - f y) / (z - y) := by |
have hyz' : 0 < z - y := by linarith
have hxz' : 0 < z - x := by linarith
rw [div_lt_div_iff hxz' hyz']
linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]
|
/-
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 | 547 | 551 | theorem coe_update [DecidableEq α] : (f.update a b : α → M) = Function.update f a b := by |
delta update Function.update
ext
dsimp
split_ifs <;> 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.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 65 | 73 | theorem Monic.map [Semiring S] (f : R →+* S) (hp : Monic p) : Monic (p.map f) := by |
unfold Monic
nontriviality
have : f p.leadingCoeff ≠ 0 := by
rw [show _ = _ from hp, f.map_one]
exact one_ne_zero
rw [Polynomial.leadingCoeff, coeff_map]
suffices p.coeff (p.map f).natDegree = 1 by simp [this]
rwa [natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f this)]
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 269 | 269 | theorem not_isRightDescent_one (i : B) : ¬cs.IsRightDescent 1 i := by | simp [IsRightDescent]
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,597 | 1,605 | theorem map_span (s : Set P₁) : (affineSpan k s).map f = affineSpan k (f '' s) := by |
rcases s.eq_empty_or_nonempty with (rfl | ⟨p, hp⟩);
· rw [image_empty, span_empty, span_empty, map_bot]
-- Porting note: I don't know exactly why this `simp` was broken.
apply ext_of_direction_eq
· simp [direction_affineSpan]
· exact
⟨f p, mem_image_of_mem f (subset_affineSpan k _ hp),
subs... |
/-
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.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congr... | Mathlib/GroupTheory/Coprod/Basic.lean | 512 | 513 | theorem fst_comp_toProd : (MonoidHom.fst M N).comp toProd = fst := by |
rw [← fst_prod_snd, MonoidHom.fst_comp_prod]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 142 | 146 | theorem preservesColimitsIso_inv_comp_desc (t : Cocone F) :
(preservesColimitIso G F).inv ≫ G.map (colimit.desc _ t) =
colimit.desc _ (G.mapCocone t) := by |
ext
simp [← G.map_comp]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RingHomProperties
import Mathlib.RingTheory.IntegralClosure
#align_import ring_theory.ring_hom.integral from "leanprover-community/mathlib"@"a7c0... | Mathlib/RingTheory/RingHom/Integral.lean | 24 | 25 | theorem isIntegral_stableUnderComposition : StableUnderComposition fun f => f.IsIntegral := by |
introv R hf hg; exact hf.trans _ _ hg
|
/-
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
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstance... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 343 | 347 | theorem subsingleton_iff : Subsingleton (Submodule R M) ↔ Subsingleton M :=
have h : Subsingleton (Submodule R M) ↔ Subsingleton (AddSubmonoid M) := by |
rw [← subsingleton_iff_bot_eq_top, ← subsingleton_iff_bot_eq_top, ← toAddSubmonoid_eq,
bot_toAddSubmonoid, top_toAddSubmonoid]
h.trans AddSubmonoid.subsingleton_iff
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 653 | 660 | theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞}
(hf : ∀ b ∈ s, AEMeasurable (f b) μ) :
∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ := by |
induction' s using Finset.induction_on with a s has ih
· simp
· simp only [Finset.sum_insert has]
rw [Finset.forall_mem_insert] at hf
rw [lintegral_add_left' hf.1, ih hf.2]
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,521 | 1,523 | theorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a := by |
rw [← succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb
exact (cantor a).trans_le (power_le_power_right hb)
|
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.D... | Mathlib/CategoryTheory/CofilteredSystem.lean | 222 | 230 | theorem eventualRange_eq_iff {f : i ⟶ j} :
F.eventualRange j = range (F.map f) ↔
∀ ⦃k⦄ (g : k ⟶ i), range (F.map f) ⊆ range (F.map <| g ≫ f) := by |
rw [subset_antisymm_iff, eventualRange, and_iff_right (iInter₂_subset _ _), subset_iInter₂_iff]
refine ⟨fun h k g => h _ _, fun h j' f' => ?_⟩
obtain ⟨k, g, g', he⟩ := cospan f f'
refine (h g).trans ?_
rw [he, F.map_comp]
apply range_comp_subset_range
|
/-
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.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 172 | 175 | theorem pullbackDiagonalMapIso_inv_snd_fst :
(pullbackDiagonalMapIso f i i₁ i₂).inv ≫ pullback.snd ≫ pullback.fst = pullback.fst := by |
delta pullbackDiagonalMapIso
simp
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathl... | Mathlib/Analysis/Matrix.lean | 613 | 615 | theorem frobenius_norm_row (v : m → α) : ‖row v‖ = ‖(WithLp.equiv 2 _).symm v‖ := by |
rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]
simp only [row_apply, Real.rpow_two, WithLp.equiv_symm_pi_apply]
|
/-
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 | 1,396 | 1,405 | theorem setToFun_smul [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]
(hT : DominatedFinMeasAdditive μ T C) (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜)
(f : α → E) : setToFun μ T hT (c • f) = c • setToFun μ T hT f := by |
by_cases hf : Integrable f μ
· rw [setToFun_eq hT hf, setToFun_eq hT, Integrable.toL1_smul',
L1.setToL1_smul hT h_smul c _]
· by_cases hr : c = 0
· rw [hr]; simp
· have hf' : ¬Integrable (c • f) μ := by rwa [integrable_smul_iff hr f]
rw [setToFun_undef hT hf, setToFun_undef hT hf', smul_zero]... |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Equivalence
#align_import algebraic_topology.dold_kan.compatibility from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 86 | 88 | theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF := by |
ext Y
simp [equivalence₁, equivalence₀]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 287 | 289 | theorem span_single_image (s : Set M) (a : α) :
Submodule.span R (single a '' s) = (Submodule.span R s).map (lsingle a : M →ₗ[R] α →₀ M) := by |
rw [← span_image]; rfl
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 556 | 557 | theorem right_inv_eq_left_inv (h : A * B = 1) (g : C * A = 1) : B = C := by |
rw [← inv_eq_right_inv h, ← inv_eq_left_inv g]
|
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprov... | Mathlib/Algebra/CharP/MixedCharZero.lean | 161 | 169 | theorem of_algebraRat [Algebra ℚ R] : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) := by |
intro I hI
constructor
intro a b h_ab
contrapose! hI
-- `↑a - ↑b` is a unit contained in `I`, which contradicts `I ≠ ⊤`.
refine I.eq_top_of_isUnit_mem ?_ (IsUnit.map (algebraMap ℚ R) (IsUnit.mk0 (a - b : ℚ) ?_))
· simpa only [← Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]
simpa on... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 186 | 190 | theorem bodd_mul (m n : ℤ) : bodd (m * n) = (bodd m && bodd n) := by |
cases' m with m m <;> cases' n with n n <;>
simp only [ofNat_eq_coe, ofNat_mul_negSucc, negSucc_mul_ofNat, ofNat_mul_ofNat,
negSucc_mul_negSucc] <;>
simp only [negSucc_coe, bodd_neg, bodd_coe, ← Nat.bodd_mul]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 541 | 547 | theorem hasBasis_iInf_of_directed' {ι : Type*} {ι' : ι → Sort _} [Nonempty ι] {l : ι → Filter α}
(s : ∀ i, ι' i → Set α) (p : ∀ i, ι' i → Prop) (hl : ∀ i, (l i).HasBasis (p i) (s i))
(h : Directed (· ≥ ·) l) :
(⨅ i, l i).HasBasis (fun ii' : Σi, ι' i => p ii'.1 ii'.2) fun ii' => s ii'.1 ii'.2 := by |
refine ⟨fun t => ?_⟩
rw [mem_iInf_of_directed h, Sigma.exists]
exact exists_congr fun i => (hl i).mem_iff
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "lean... | Mathlib/Algebra/CharZero/Lemmas.lean | 132 | 133 | theorem nat_mul_inj' {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) (w : n ≠ 0) : a = b := by |
simpa [w] using nat_mul_inj h
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 128 | 151 | theorem log_le_of_logEmbedding_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
(w : InfinitePlace K) : |Real.log (w x)| ≤ (Fintype.card (InfinitePlace K)) * r := by |
have tool : ∀ x : ℝ, 0 ≤ x → x ≤ mult w * x := fun x hx => by
nth_rw 1 [← one_mul x]
refine mul_le_mul ?_ le_rfl hx ?_
all_goals { rw [mult]; split_ifs <;> norm_num }
by_cases hw : w = w₀
· have hyp := congr_arg (‖·‖) (sum_logEmbedding_component x).symm
replace hyp := (le_of_eq hyp).trans (norm_s... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 296 | 297 | theorem nonsing_inv_mul_cancel_right (B : Matrix m n α) (h : IsUnit A.det) : B * A⁻¹ * A = B := by |
simp [Matrix.mul_assoc, nonsing_inv_mul A h]
|
/-
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.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 853 | 854 | theorem Finset.univ_map_embedding {α : Type*} [Fintype α] (e : α ↪ α) : univ.map e = univ := by |
rw [← e.equiv_of_fintype_self_embedding_to_embedding, univ_map_equiv_to_embedding]
|
/-
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 | 166 | 170 | theorem projRestricts_comp_projRestrict (h : ∀ i, J i → K i) :
ProjRestricts C h ∘ ProjRestrict C K = ProjRestrict C J := by |
ext x i
simp only [π, Proj, Function.comp_apply, ProjRestricts_coe, ProjRestrict_coe]
aesop
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 254 | 256 | theorem degreeOf_lt_iff {n : σ} {f : MvPolynomial σ R} {d : ℕ} (h : 0 < d) :
degreeOf n f < d ↔ ∀ m : σ →₀ ℕ, m ∈ f.support → m n < d := by |
rwa [degreeOf_eq_sup, Finset.sup_lt_iff]
|
/-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 193 | 204 | theorem norm_torusIntegral_le_of_norm_le_const {C : ℝ} (hf : ∀ θ, ‖f (torusMap c R θ)‖ ≤ C) :
‖∯ x in T(c, R), f x‖ ≤ ((2 * π) ^ (n : ℕ) * ∏ i, |R i|) * C :=
calc
‖∯ x in T(c, R), f x‖ ≤ (∏ i, |R i|) * C * (volume (Icc (0 : ℝⁿ) fun _ => 2 * π)).toReal :=
norm_setIntegral_le_of_norm_le_const' measure_Icc... | simp [norm_smul]
_ ≤ (∏ i : Fin n, |R i|) * C := mul_le_mul_of_nonneg_left (hf _) <| by positivity
_ = ((2 * π) ^ (n : ℕ) * ∏ i, |R i|) * C := by
simp only [Pi.zero_def, Real.volume_Icc_pi_toReal fun _ => Real.two_pi_pos.le, sub_zero,
Fin.prod_const, mul_assoc, mul_comm ((2 * π) ^ (n : ℕ))]... |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 160 | 161 | theorem sub_coe_pi_eq_add_coe_pi (θ : Angle) : θ - π = θ + π := by |
rw [sub_eq_add_neg, neg_coe_pi]
|
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 327 | 327 | theorem neg_mul_neg (a b : α) : -a * -b = a * b := by | simp
|
/-
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 | 644 | 657 | theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
| [], h => absurd rfl h
| List.cons a l, _ =>
have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
(a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by |
intro l'
induction' l' with a₁ l₁ ih
· intros
rw [corec_eq]
rfl
· intros
rw [corec_eq, Stream'.cycle_g_cons, ih a₁]
rfl
gen l a
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# The... | Mathlib/CategoryTheory/Comma/Arrow.lean | 272 | 274 | theorem square_to_iso_invert (i : Arrow T) {X Y : T} (p : X ≅ Y) (sq : i ⟶ Arrow.mk p.hom) :
i.hom ≫ sq.right ≫ p.inv = sq.left := by |
simpa only [Category.assoc] using (Iso.comp_inv_eq p).mpr (Arrow.w_mk_right sq).symm
|
/-
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.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 465 | 481 | theorem ext_of_generateFrom_of_cover {S T : Set (Set α)} (h_gen : ‹_› = generateFrom S)
(hc : T.Countable) (h_inter : IsPiSystem S) (hU : ⋃₀ T = univ) (htop : ∀ t ∈ T, μ t ≠ ∞)
(ST_eq : ∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)) (T_eq : ∀ t ∈ T, μ t = ν t) : μ = ν := by |
refine ext_of_sUnion_eq_univ hc hU fun t ht => ?_
ext1 u hu
simp only [restrict_apply hu]
refine induction_on_inter h_gen h_inter ?_ (ST_eq t ht) ?_ ?_ hu
· simp only [Set.empty_inter, measure_empty]
· intro v hv hvt
have := T_eq t ht
rw [Set.inter_comm] at hvt ⊢
rwa [← measure_inter_add_diff t... |
/-
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 | 306 | 312 | theorem setToSimpleFunc_eq_sum_filter {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F')
(f : α →ₛ F) :
setToSimpleFunc T f = ∑ x ∈ f.range.filter fun x => x ≠ 0, (T (f ⁻¹' {x})) x := by |
symm
refine sum_filter_of_ne fun x _ => mt fun hx0 => ?_
rw [hx0]
exact ContinuousLinearMap.map_zero _
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Order.Group.WithTop
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.V... | Mathlib/RingTheory/HahnSeries/Summable.lean | 518 | 521 | theorem one_sub_self_mul_hsum_powers : (1 - x) * (powers x hx).hsum = 1 := by |
rw [← hsum_smul, sub_smul 1 x (powers x hx), one_smul, hsum_sub, ←
hsum_embDomain (x • powers x hx) ⟨Nat.succ, Nat.succ_injective⟩, embDomain_succ_smul_powers]
simp
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Cont... | Mathlib/Data/Real/Cardinality.lean | 219 | 221 | theorem not_countable_real : ¬(Set.univ : Set ℝ).Countable := by |
rw [← le_aleph0_iff_set_countable, not_le, mk_univ_real]
apply cantor
|
/-
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
-/
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.O... | Mathlib/Data/Rat/Cast/Defs.lean | 120 | 121 | theorem cast_natCast (n : ℕ) : ((n : ℚ) : α) = n := by |
rw [← Int.cast_natCast, cast_intCast, Int.cast_natCast]
|
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.... | Mathlib/Data/ENNReal/Basic.lean | 703 | 704 | theorem iUnion_Ioo_coe_nat : ⋃ n : ℕ, Ioo a n = Ioi a \ {∞} := by |
simp only [← Ioi_inter_Iio, ← inter_iUnion, iUnion_Iio_coe_nat, diff_eq]
|
/-
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.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
/-!
# Cycles of a li... | Mathlib/Data/List/Cycle.lean | 930 | 930 | theorem Chain.nil (r : α → α → Prop) : Cycle.Chain r (@nil α) := by | trivial
|
/-
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 | 804 | 806 | theorem mk_d_2_1 : (mk X₀ X₁ X₂ d₀ d₁ s succ).d 2 1 = d₁ := by |
change ite (2 = 1 + 1) (𝟙 X₂ ≫ d₁) 0 = d₁
rw [if_pos rfl, Category.id_comp]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "lean... | Mathlib/GroupTheory/Abelianization.lean | 299 | 310 | theorem image_commutatorSet_closureCommutatorRepresentatives :
(closureCommutatorRepresentatives G).subtype ''
commutatorSet (closureCommutatorRepresentatives G) =
commutatorSet G := by |
apply Set.Subset.antisymm
· rintro - ⟨-, ⟨g₁, g₂, rfl⟩, rfl⟩
exact ⟨g₁, g₂, rfl⟩
· exact fun g hg =>
⟨_,
⟨⟨_, subset_closure (Or.inl ⟨_, ⟨⟨g, hg⟩, rfl⟩, rfl⟩)⟩,
⟨_, subset_closure (Or.inr ⟨_, ⟨⟨g, hg⟩, rfl⟩, rfl⟩)⟩, rfl⟩,
hg.choose_spec.choose_spec⟩
|
/-
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.Algebra.Ring.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import number_theo... | Mathlib/NumberTheory/SumFourSquares.lean | 78 | 96 | theorem exists_sq_add_sq_add_one_eq_mul (p : ℕ) [hp : Fact p.Prime] :
∃ (a b k : ℕ), 0 < k ∧ k < p ∧ a ^ 2 + b ^ 2 + 1 = k * p := by |
rcases hp.1.eq_two_or_odd' with (rfl | hodd)
· use 1, 0, 1; simp
rcases Nat.sq_add_sq_zmodEq p (-1) with ⟨a, b, ha, hb, hab⟩
rcases Int.modEq_iff_dvd.1 hab.symm with ⟨k, hk⟩
rw [sub_neg_eq_add, mul_comm] at hk
have hk₀ : 0 < k := by
refine pos_of_mul_pos_left ?_ (Nat.cast_nonneg p)
rw [← hk]
po... |
/-
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 | 194 | 198 | theorem Continuous.matrix_det [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
{A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).det := by |
simp_rw [Matrix.det_apply]
refine continuous_finset_sum _ fun l _ => Continuous.const_smul ?_ _
exact continuous_finset_prod _ fun l _ => hA.matrix_elem _ _
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 255 | 256 | theorem coeff_zero_C (a : R) : coeff R 0 (C R a) = a := by |
rw [coeff_C, if_pos rfl]
|
/-
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.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,217 | 2,218 | theorem map_apply_eq_iff_map_symm_apply_eq (e : α ≃ᵐ β) : μ.map e = ν ↔ ν.map e.symm = μ := by |
rw [← (map_measurableEquiv_injective e).eq_iff, map_map_symm, eq_comm]
|
/-
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 | 98 | 102 | theorem natDegree_mul_C_le (f : R[X]) (a : R) : (f * C a).natDegree ≤ f.natDegree :=
calc
(f * C a).natDegree ≤ f.natDegree + (C a).natDegree := natDegree_mul_le
_ = f.natDegree + 0 := by | rw [natDegree_C a]
_ = f.natDegree := add_zero _
|
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 117 | 125 | theorem Monic.mul (hp : Monic p) (hq : Monic q) : Monic (p * q) :=
letI := Classical.decEq R
if h0 : (0 : R) = 1 then
haveI := subsingleton_of_zero_eq_one h0
Subsingleton.elim _ _
else by
have : p.leadingCoeff * q.leadingCoeff ≠ 0 := by |
simp [Monic.def.1 hp, Monic.def.1 hq, Ne.symm h0]
rw [Monic.def, leadingCoeff_mul' this, Monic.def.1 hp, Monic.def.1 hq, one_mul]
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
/-!
# Fractional ideal norms
This file defines the absolute ideal norm of a frac... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 106 | 128 | theorem abs_det_basis_change [NoZeroDivisors K] {ι : Type*} [Fintype ι]
[DecidableEq ι] (b : Basis ι ℤ R) (I : FractionalIdeal R⁰ K) (bI : Basis ι ℤ I) :
|(b.localizationLocalization ℚ ℤ⁰ K).det ((↑) ∘ bI)| = absNorm I := by |
have := IsFractionRing.nontrivial R K
let b₀ : Basis ι ℚ K := b.localizationLocalization ℚ ℤ⁰ K
let bI.num : Basis ι ℤ I.num := bI.map
((equivNum (nonZeroDivisors.coe_ne_zero _)).restrictScalars ℤ)
rw [absNorm_eq, ← Ideal.natAbs_det_basis_change b I.num bI.num, Int.cast_natAbs, Int.cast_abs,
Int.cast... |
/-
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 | 1,326 | 1,328 | theorem clusterPt_lift'_closure_iff {F : Filter X} :
ClusterPt x (F.lift' closure) ↔ ClusterPt x F := by |
simp [clusterPt_iff_lift'_closure, lift'_lift'_assoc (monotone_closure X) (monotone_closure X)]
|
/-
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.Analysis.Calculus.MeanValue
#align_import analysis.calculus.extend_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a... | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 145 | 175 | theorem has_deriv_at_interval_right_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ}
{f : ℝ → E} (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a)
(hs : s ∈ 𝓝[<] a) (f_lim' : Tendsto (fun x => deriv f x) (𝓝[<] a) (𝓝 e)) :
HasDerivWithinAt f e (Iic a) a := by |
/- This is a specialization of `has_fderiv_at_boundary_of_differentiable`. To be in the setting of
this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
call `t = (b, a)`. Then, we check all the assumptions of this theorem and we apply it. -/
obtain ⟨b, ba, sab⟩ : ∃... |
/-
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 | 366 | 372 | theorem scanl_head : (scanl f b v).head = b := by |
cases n
· have : v = nil := by simp only [Nat.zero_eq, eq_iff_true_of_subsingleton]
simp only [this, scanl_nil, head_cons]
· rw [← cons_head_tail v]
simp only [← get_zero, get_eq_get, toList_scanl, toList_cons, List.scanl, Fin.val_zero,
List.get]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Matrix.CharP
#align_import l... | Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean | 61 | 62 | theorem ZMod.trace_pow_card {p : ℕ} [Fact p.Prime] (M : Matrix n n (ZMod p)) :
trace (M ^ p) = trace M ^ p := by | have h := FiniteField.trace_pow_card M; rwa [ZMod.card] at h
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 671 | 719 | theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n n R) :
(Matrix.fromBlocks A B 0 D).det = A.det * D.det := by |
classical
simp_rw [det_apply']
convert Eq.symm <|
sum_subset (β := R) (subset_univ ((sumCongrHom m n).range : Set (Perm (Sum m n))).toFinset) ?_
· simp_rw [sum_mul_sum, ← sum_product', univ_product_univ]
refine sum_nbij (fun σ ↦ σ.fst.sumCongr σ.snd) ?_ ?_ ?_ ?_
· intro σ₁₂ _
si... |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 144 | 160 | theorem dropWhile_eq_self_iff : dropWhile p l = l ↔ ∀ hl : 0 < l.length, ¬p (l.get ⟨0, hl⟩) := by |
cases' l with hd tl
· simp only [dropWhile, true_iff]
intro h
by_contra
rwa [length_nil, lt_self_iff_false] at h
· rw [dropWhile]
refine ⟨fun h => ?_, fun h => ?_⟩
· intro _ H
rw [get] at H
refine (cons_ne_self hd tl) (Sublist.antisymm ?_ (sublist_cons _ _))
rw [← h]
s... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Set... | Mathlib/GroupTheory/GroupAction/Basic.lean | 790 | 805 | theorem le_stabilizer_iff_smul_le (s : Set α) (H : Subgroup G) :
H ≤ stabilizer G s ↔ ∀ g ∈ H, g • s ⊆ s := by |
constructor
· intro hyp g hg
apply Eq.subset
rw [← mem_stabilizer_iff]
exact hyp hg
· intro hyp g hg
rw [mem_stabilizer_iff]
apply subset_antisymm (hyp g hg)
intro x hx
use g⁻¹ • x
constructor
· apply hyp g⁻¹ (inv_mem hg)
simp only [Set.smul_mem_smul_set_iff, hx]
· s... |
/-
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.MeasureTheory.Measure.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 148 | 150 | theorem measureUnivNNReal_pos [IsFiniteMeasure μ] (hμ : μ ≠ 0) : 0 < measureUnivNNReal μ := by |
contrapose! hμ
simpa [measureUnivNNReal_eq_zero, Nat.le_zero] using hμ
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 637 | 640 | theorem mul_eq_max' {a b : Cardinal} (h : ℵ₀ ≤ a * b) : a * b = max a b := by |
rcases aleph0_le_mul_iff.mp h with ⟨ha, hb, ha' | hb'⟩
· exact mul_eq_max_of_aleph0_le_left ha' hb
· exact mul_eq_max_of_aleph0_le_right ha hb'
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 208 | 211 | theorem χ₈'_eq_χ₄_mul_χ₈ (a : ZMod 8) : χ₈' a = χ₄ (cast a) * χ₈ a := by |
-- Porting note: was `decide!`
fin_cases a
all_goals decide
|
/-
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 | 669 | 673 | theorem orderOf_dvd_iff_zpow_eq_one : (orderOf x : ℤ) ∣ i ↔ x ^ i = 1 := by |
rcases Int.eq_nat_or_neg i with ⟨i, rfl | rfl⟩
· rw [Int.natCast_dvd_natCast, orderOf_dvd_iff_pow_eq_one, zpow_natCast]
· rw [dvd_neg, Int.natCast_dvd_natCast, zpow_neg, inv_eq_one, zpow_natCast,
orderOf_dvd_iff_pow_eq_one]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 243 | 244 | theorem sum_apply {α : Type*} (f : α → MultilinearMap R M₁ M₂) (m : ∀ i, M₁ i) {s : Finset α} :
(∑ a ∈ s, f a) m = ∑ a ∈ s, f a m := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.... | Mathlib/Topology/UniformSpace/Separation.lean | 155 | 157 | theorem t0Space_iff_uniformity' :
T0Space α ↔ Pairwise fun x y ↦ ∃ r ∈ 𝓤 α, (x, y) ∉ r := by |
simp [t0Space_iff_not_inseparable, inseparable_iff_ker_uniformity]
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 754 | 756 | theorem finprod_mem_div_distrib [DivisionCommMonoid G] (f g : α → G) (hs : s.Finite) :
∏ᶠ i ∈ s, f i / g i = (∏ᶠ i ∈ s, f i) / ∏ᶠ i ∈ s, g i := by |
simp only [div_eq_mul_inv, finprod_mem_mul_distrib hs, finprod_mem_inv_distrib g hs]
|
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
... | Mathlib/Data/Nat/Dist.lean | 81 | 82 | theorem dist_add_add_left (k n m : ℕ) : dist (k + n) (k + m) = dist n m := by |
rw [add_comm k n, add_comm k m]; apply dist_add_add_right
|
/-
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 | 2,189 | 2,191 | theorem isBigO_nat_atTop_iff {f : ℕ → E''} {g : ℕ → F''} (h : ∀ x, g x = 0 → f x = 0) :
f =O[atTop] g ↔ ∃ C, ∀ x, ‖f x‖ ≤ C * ‖g x‖ := by |
rw [← Nat.cofinite_eq_atTop, isBigO_cofinite_iff h]
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algebra.bilinear_form.tensor_product from "leanprover-community/mat... | Mathlib/LinearAlgebra/BilinearForm/TensorProduct.lean | 131 | 135 | theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ := M₁) (M₂ := M₂)).toLinearMap = tensorDistrib R R := by |
ext B₁ B₂ : 3
ext
exact mul_comm _ _
|
/-
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 | 1,360 | 1,363 | theorem mem_factorSet_some {p : Associates α} {hp : Irreducible p}
{l : Multiset { a : Associates α // Irreducible a }} :
p ∈ (l : FactorSet α) ↔ Subtype.mk p hp ∈ l := by |
dsimp only [Membership.mem]; dsimp only [FactorSetMem]; split_ifs; rfl
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 395 | 396 | theorem cancel_counit_right {X Y : D} (f f' : X ⟶ e.functor.obj (e.inverse.obj Y)) :
f ≫ e.counit.app Y = f' ≫ e.counit.app Y ↔ f = f' := by | simp only [cancel_mono]
|
/-
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 | 179 | 182 | theorem sum_smul_vsub_eq_weightedVSubOfPoint_sub (w : ι → k) (p₁ p₂ : ι → P) (b : P) :
(∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i)) =
s.weightedVSubOfPoint p₁ b w - s.weightedVSubOfPoint p₂ b w := by |
simp_rw [weightedVSubOfPoint_apply, ← sum_sub_distrib, ← smul_sub, vsub_sub_vsub_cancel_right]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.Mea... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 387 | 393 | theorem lpMeasSubgroupToLpTrim_left_inv (hm : m ≤ m0) :
Function.LeftInverse (lpTrimToLpMeasSubgroup F p μ hm) (lpMeasSubgroupToLpTrim F p μ hm) := by |
intro f
ext1
ext1
rw [← lpMeasSubgroup_coe]
exact (lpTrimToLpMeasSubgroup_ae_eq hm _).trans (lpMeasSubgroupToLpTrim_ae_eq hm _)
|
/-
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, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 485 | 485 | theorem integral_zero : (∫ _ in a..b, (0 : E) ∂μ) = 0 := by | simp [intervalIntegral]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 123 | 124 | theorem eq_loopyOn_or_rkPos (M : Matroid α) : M = loopyOn M.E ∨ RkPos M := by |
rw [← empty_base_iff, rkPos_iff_empty_not_base]; apply em
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_spac... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 108 | 114 | theorem Metric.closedBall_infDist_compl_subset_closure {x : F} {s : Set F} (hx : x ∈ s) :
closedBall x (infDist x sᶜ) ⊆ closure s := by |
rcases eq_or_ne (infDist x sᶜ) 0 with h₀ | h₀
· rw [h₀, closedBall_zero']
exact closure_mono (singleton_subset_iff.2 hx)
· rw [← closure_ball x h₀]
exact closure_mono ball_infDist_compl_subset
|
/-
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 | 164 | 174 | theorem starConvex_iff_forall_pos (hx : x ∈ s) : StarConvex 𝕜 x s ↔
∀ ⦃y⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s := by |
refine ⟨fun h y hy a b ha hb hab => h hy ha.le hb.le hab, ?_⟩
intro h y hy a b ha hb hab
obtain rfl | ha := ha.eq_or_lt
· rw [zero_add] at hab
rwa [hab, one_smul, zero_smul, zero_add]
obtain rfl | hb := hb.eq_or_lt
· rw [add_zero] at hab
rwa [hab, one_smul, zero_smul, add_zero]
exact h hy ha hb h... |
/-
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.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 147 | 149 | theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s := by |
ext x
exact (Submodule.gi R R).gc s x.asIdeal
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.monad fro... | Mathlib/Algebra/MvPolynomial/Monad.lean | 190 | 191 | theorem join₂_map (f : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) :
join₂ (map f φ) = bind₂ f φ := by | simp only [join₂, bind₂, eval₂Hom_map_hom, RingHom.id_comp]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Init.Order.LinearOrder
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.Spread
import Mathlib.Tactic.Conv... | Mathlib/Order/Basic.lean | 1,038 | 1,038 | theorem const_le_const : const β a ≤ const β b ↔ a ≤ b := by | simp [Pi.le_def]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 180 | 182 | theorem Set.Subsingleton.maximals_eq (h : s.Subsingleton) : maximals r s = s := by |
rcases h.eq_empty_or_singleton with (rfl | ⟨x, rfl⟩)
exacts [minimals_empty _, maximals_singleton _ _]
|
/-
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 | 74 | 77 | theorem Ico_mul_Icc_subset' (a b c d : α) : Ico a b * Icc c d ⊆ Ico (a * c) (b * d) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_le_mul' hya hzc, mul_lt_mul_of_lt_of_le hyb hzd⟩
|
/-
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, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Gr... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 457 | 463 | theorem setIntegral_eq_tsum (h : IsFundamentalDomain G s μ) {f : α → E} {t : Set α}
(hf : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∑' g : G, ∫ x in t ∩ g • s, f x ∂μ :=
calc
∫ x in t, f x ∂μ = ∑' g : G, ∫ x in g • s, f x ∂μ.restrict t :=
h.integral_eq_tsum_of_ac restrict_le_self.absolutelyContinuous f h... |
simp only [h.restrict_restrict, measure_smul, inter_comm]
|
/-
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 | 418 | 419 | theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by |
rwa [mem_ball, dist_self]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathl... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 104 | 106 | theorem padicNorm_p_lt_one (hp : 1 < p) : padicNorm p p < 1 := by |
rw [padicNorm_p hp, inv_lt_one_iff]
exact mod_cast Or.inr hp
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... | Mathlib/RingTheory/LocalProperties.lean | 307 | 324 | theorem localization_isReduced : LocalizationPreserves fun R hR => IsReduced R := by |
introv R _ _
constructor
rintro x ⟨_ | n, e⟩
· simpa using congr_arg (· * x) e
obtain ⟨⟨y, m⟩, hx⟩ := IsLocalization.surj M x
dsimp only at hx
let hx' := congr_arg (· ^ n.succ) hx
simp only [mul_pow, e, zero_mul, ← RingHom.map_pow] at hx'
rw [← (algebraMap R S).map_zero] at hx'
obtain ⟨m', hm'⟩ := ... |
/-
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.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 117 | 117 | theorem circleMap_mem_sphere' (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∈ sphere c |R| := by | simp
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.GroupTheory.Subsemigroup.Center
#align_import group_theory.submonoid.center from "leanprover-community/mat... | Mathlib/GroupTheory/Submonoid/Center.lean | 79 | 81 | theorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g := by |
rw [← Semigroup.mem_center_iff]
exact Iff.rfl
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 590 | 607 | theorem lmapDomain_supported (f : α → α') (s : Set α) :
(supported M R s).map (lmapDomain M R f) = supported M R (f '' s) := by |
classical
cases isEmpty_or_nonempty α
· simp [s.eq_empty_of_isEmpty]
refine
le_antisymm
(map_le_iff_le_comap.2 <|
le_trans (supported_mono <| Set.subset_preimage_image _ _)
(supported_comap_lmapDomain M R _ _))
?_
intro l hl
refine ⟨(lmapDomain M R (Function.invFunOn f s) ... |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathl... | Mathlib/Control/Monad/Cont.lean | 249 | 250 | theorem ReaderT.goto_mkLabel {α ρ β} (x : Label α m β) (i : α) :
goto (ReaderT.mkLabel ρ x) i = monadLift (goto x i) := by | cases x; rfl
|
/-
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.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 518 | 521 | theorem measure_biUnion_eq_iSup {s : ι → Set α} {t : Set ι} (ht : t.Countable)
(hd : DirectedOn ((· ⊆ ·) on s) t) : μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i) := by |
haveI := ht.toEncodable
rw [biUnion_eq_iUnion, measure_iUnion_eq_iSup hd.directed_val, ← iSup_subtype'']
|
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variable {α : Type*}
namespace List
... | Mathlib/Data/List/Iterate.lean | 25 | 26 | theorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f a n = [] ↔ n = 0 := by |
rw [← length_eq_zero, length_iterate]
|
/-
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 | 492 | 493 | theorem inter_nonempty_iff_exists_left : (s ∩ t).Nonempty ↔ ∃ x ∈ s, x ∈ t := by |
simp_rw [inter_nonempty]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 426 | 427 | theorem coeff_mul_C (n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (a : R) :
coeff R n (φ * C σ R a) = coeff R n φ * a := by | simpa using coeff_add_mul_monomial n 0 φ a
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 185 | 205 | theorem isNormal_iff_strictMono_and_continuous (f : Ordinal.{u} → Ordinal.{u}) :
IsNormal f ↔ StrictMono f ∧ Continuous f := by |
refine ⟨fun h => ⟨h.strictMono, ?_⟩, ?_⟩
· rw [continuous_def]
intro s hs
rw [isOpen_iff] at *
intro o ho ho'
rcases hs _ ho (h.isLimit ho') with ⟨a, ha, has⟩
rw [← IsNormal.bsup_eq.{u, u} h ho', lt_bsup] at ha
rcases ha with ⟨b, hb, hab⟩
exact
⟨b, hb, fun c hc =>
Set.mem_... |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
/-!
# Sober spaces
A quasi... | Mathlib/Topology/Sober.lean | 218 | 240 | theorem quasiSober_of_open_cover (S : Set (Set α)) (hS : ∀ s : S, IsOpen (s : Set α))
[hS' : ∀ s : S, QuasiSober s] (hS'' : ⋃₀ S = ⊤) : QuasiSober α := by |
rw [quasiSober_iff]
intro t h h'
obtain ⟨x, hx⟩ := h.1
obtain ⟨U, hU, hU'⟩ : x ∈ ⋃₀ S := by
rw [hS'']
trivial
haveI : QuasiSober U := hS' ⟨U, hU⟩
have H : IsPreirreducible ((↑) ⁻¹' t : Set U) :=
h.2.preimage (hS ⟨U, hU⟩).openEmbedding_subtype_val
replace H : IsIrreducible ((↑) ⁻¹' t : Set U) ... |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 124 | 126 | theorem roots_of_cyclotomic (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] :
(cyclotomic' n R).roots = (primitiveRoots n R).val := by |
rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R)
|
/-
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.MeasureTheory.Measure.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 727 | 731 | theorem forall_measure_inter_spanningSets_eq_zero [MeasurableSpace α] {μ : Measure α}
[SigmaFinite μ] (s : Set α) : (∀ n, μ (s ∩ spanningSets μ n) = 0) ↔ μ s = 0 := by |
nth_rw 2 [show s = ⋃ n, s ∩ spanningSets μ n by
rw [← inter_iUnion, iUnion_spanningSets, inter_univ] ]
rw [measure_iUnion_null_iff]
|
/-
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.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Po... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 272 | 274 | theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ι → G} {s : Finset ι}
(hf : ∀ i, 0 ≤ f i) : |∑ i ∈ s, f i| = ∑ i ∈ s, f i := by |
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
|
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