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) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 621 | 626 | theorem val_neg_one (n : ℕ) : (-1 : ZMod n.succ).val = n := by |
dsimp [val, Fin.coe_neg]
cases n
· simp [Nat.mod_one]
· dsimp [ZMod, ZMod.cast]
rw [Fin.coe_neg_one]
|
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 438 | 440 | theorem isBigO_one_exp_comp {f : α → ℝ} :
((fun _ => 1 : α → ℝ) =O[l] fun x => exp (f x)) ↔ IsBoundedUnder (· ≥ ·) l f := by |
simp only [← exp_zero, isBigO_exp_comp_exp_comp, Pi.sub_def, zero_sub, isBoundedUnder_le_neg]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instan... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 421 | 447 | theorem exists_upperSemicontinuous_le_integral_le (f : α → ℝ≥0)
(fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
∃ g : α → ℝ≥0,
(∀ x, g x ≤ f x) ∧
UpperSemicontinuous g ∧
Integrable (fun x => (g x : ℝ)) μ ∧ (∫ x, (f x : ℝ) ∂μ) - ε ≤ ∫ x, ↑(g x) ∂μ := by |
lift ε to ℝ≥0 using εpos.le
rw [NNReal.coe_pos, ← ENNReal.coe_pos] at εpos
have If : (∫⁻ x, f x ∂μ) < ∞ := hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral
rcases exists_upperSemicontinuous_le_lintegral_le f If.ne εpos.ne' with ⟨g, gf, gcont, gint⟩
have Ig : (∫⁻ x, g x ∂μ) < ∞ := by
refine lt_of_l... |
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 593 | 594 | theorem inv_pow_lt_inv_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) : (a ^ n)⁻¹ < (a ^ m)⁻¹ := by |
convert one_div_pow_lt_one_div_pow_of_lt a1 mn using 1 <;> simp
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Locus of une... | Mathlib/Data/Finsupp/NeLocus.lean | 101 | 106 | theorem zipWith_neLocus_eq_right [DecidableEq M] [Zero M] [DecidableEq P] [Zero P] [Zero N]
{F : M → N → P} (F0 : F 0 0 = 0) (f₁ f₂ : α →₀ M) (g : α →₀ N)
(hF : ∀ g, Function.Injective fun f => F f g) :
(zipWith F F0 f₁ g).neLocus (zipWith F F0 f₂ g) = f₁.neLocus f₂ := by |
ext
simpa only [mem_neLocus] using (hF _).ne_iff
|
/-
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 | 150 | 150 | theorem fourier_one {x : AddCircle T} : fourier 1 x = toCircle x := by | rw [fourier_apply, one_zsmul]
|
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Scott Morrison
-/
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 174 | 174 | theorem one_sub_nonneg (x : Icc (0 : β) 1) : 0 ≤ 1 - (x : β) := by | simpa using x.2.2
|
/-
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 | 290 | 293 | theorem u_int_pos : 0 < ∫ x : E, u x ∂μ := by |
refine (integral_pos_iff_support_of_nonneg u_nonneg ?_).mpr ?_
· exact (u_continuous E).integrable_of_hasCompactSupport (u_compact_support E)
· rw [u_support]; exact measure_ball_pos _ _ zero_lt_one
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"... | Mathlib/FieldTheory/RatFunc/Defs.lean | 130 | 136 | theorem liftOn_condition_of_liftOn'_condition {P : Sort v} {f : K[X] → K[X] → P}
(H : ∀ {p q a} (hq : q ≠ 0) (_ha : a ≠ 0), f (a * p) (a * q) = f p q) ⦃p q p' q' : K[X]⦄
(hq : q ≠ 0) (hq' : q' ≠ 0) (h : q' * p = q * p') : f p q = f p' q' :=
calc
f p q = f (q' * p) (q' * q) := (H hq hq').symm
_ = f (q ... | rw [h, mul_comm q']
_ = f p' q' := H hq' hq
|
/-
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
-/
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib... | Mathlib/Topology/UrysohnsLemma.lean | 460 | 468 | theorem exists_continuous_nonneg_pos [RegularSpace X] [LocallyCompactSpace X] (x : X) :
∃ f : C(X, ℝ), HasCompactSupport f ∧ 0 ≤ (f : X → ℝ) ∧ f x ≠ 0 := by |
rcases exists_compact_mem_nhds x with ⟨k, hk, k_mem⟩
rcases exists_continuous_one_zero_of_isCompact hk isClosed_empty (disjoint_empty k)
with ⟨f, fk, -, f_comp, hf⟩
refine ⟨f, f_comp, fun x ↦ (hf x).1, ?_⟩
have := fk (mem_of_mem_nhds k_mem)
simp only [ContinuousMap.coe_mk, Pi.one_apply] at this
simp [t... |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | Mathlib/Probability/Distributions/Uniform.lean | 247 | 247 | theorem uniformOfFinset_apply_of_not_mem (ha : a ∉ s) : uniformOfFinset s hs a = 0 := by | simp [ha]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 48 | 52 | theorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) := by |
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, Vector.append, Append.append]
|
/-
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 | 231 | 233 | theorem logb_pos_iff (hx : 0 < x) : 0 < logb b x ↔ 1 < x := by |
rw [← @logb_one b]
rw [logb_lt_logb_iff hb zero_lt_one hx]
|
/-
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.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_im... | Mathlib/RingTheory/Coprime/Lemmas.lean | 295 | 297 | theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by |
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
|
/-
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.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 65 | 66 | theorem isCaratheodory_compl : IsCaratheodory m s₁ → IsCaratheodory m s₁ᶜ := by |
simp [IsCaratheodory, diff_eq, add_comm]
|
/-
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 | 706 | 709 | theorem det_smul_inv_vecMul_eq_cramer_transpose (A : Matrix n n α) (b : n → α) (h : IsUnit A.det) :
A.det • b ᵥ* A⁻¹ = cramer Aᵀ b := by |
rw [← A⁻¹.transpose_transpose, vecMul_transpose, transpose_nonsing_inv, ← det_transpose,
Aᵀ.det_smul_inv_mulVec_eq_cramer _ (isUnit_det_transpose A h)]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 90 | 95 | theorem SameCycle.cycleOf_eq (h : SameCycle f x y) : cycleOf f x = cycleOf f y := by |
ext z
rw [Equiv.Perm.cycleOf_apply]
split_ifs with hz
· exact (h.symm.trans hz).cycleOf_apply.symm
· exact (cycleOf_apply_of_not_sameCycle (mt h.trans hz)).symm
|
/-
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.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 126 | 135 | theorem gradedComm_tmul_of_zero (a : ⨁ i, 𝒜 i) (b : ℬ 0) :
gradedComm R 𝒜 ℬ (a ⊗ₜ lof R _ ℬ 0 b) = lof R _ ℬ _ b ⊗ₜ a := by |
suffices
(gradedComm R 𝒜 ℬ).toLinearMap ∘ₗ
(TensorProduct.mk R (⨁ i, 𝒜 i) (⨁ i, ℬ i)).flip (lof R _ ℬ 0 b) =
TensorProduct.mk R _ _ (lof R _ ℬ 0 b) from
DFunLike.congr_fun this a
ext i a
dsimp
rw [gradedComm_of_tmul_of, zero_mul, uzpow_zero, one_smul]
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Extensions of Groups
This file defines the HNN extension of a group `G`, `HNN... | Mathlib/GroupTheory/HNNExtension.lean | 458 | 480 | theorem unitsSMul_one_group_smul (g : A) (w : NormalWord d) :
unitsSMul φ 1 ((g : G) • w) = (φ g : G) • (unitsSMul φ 1 w) := by |
unfold unitsSMul
have : Cancels 1 ((g : G) • w) ↔ Cancels 1 w := by
simp [Cancels, Subgroup.mul_mem_cancel_left]
by_cases hcan : Cancels 1 w
· simp [unitsSMulWithCancel, dif_pos (this.2 hcan), dif_pos hcan]
cases w using consRecOn
· simp [Cancels] at hcan
· simp only [smul_cons, consRecOn_cons,... |
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 225 | 236 | theorem smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioc]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_lt_mul_left hr).mpr a_h_left_left
· exact (mul_le_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(lt_div_iff' hr).mpr... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 164 | 166 | theorem filter_product_right (q : β → Prop) [DecidablePred q] :
((s ×ˢ t).filter fun x : α × β => q x.2) = s ×ˢ t.filter q := by |
simpa using filter_product (fun _ : α => true) q
|
/-
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 | 442 | 443 | theorem diff_subset_closure_iff : s \ t ⊆ closure t ↔ s ⊆ closure t := by |
rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
|
/-
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.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Rin... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 200 | 214 | theorem norm_fract_le [HasSolidNorm K] (m : E) : ‖fract b m‖ ≤ ∑ i, ‖b i‖ := by |
classical
calc
‖fract b m‖ = ‖∑ i, b.repr (fract b m) i • b i‖ := by rw [b.sum_repr]
_ = ‖∑ i, Int.fract (b.repr m i) • b i‖ := by simp_rw [repr_fract_apply]
_ ≤ ∑ i, ‖Int.fract (b.repr m i) • b i‖ := norm_sum_le _ _
_ = ∑ i, ‖Int.fract (b.repr m i)‖ * ‖b i‖ := by simp_rw [norm_smul]
_ ≤ ∑ i, ‖... |
/-
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 | 414 | 419 | theorem wbtw_lineMap_iff [NoZeroSMulDivisors R V] {x y : P} {r : R} :
Wbtw R x (lineMap x y r) y ↔ x = y ∨ r ∈ Set.Icc (0 : R) 1 := by |
by_cases hxy : x = y
· rw [hxy, lineMap_same_apply]
simp
rw [or_iff_right hxy, Wbtw, affineSegment, (lineMap_injective R hxy).mem_set_image]
|
/-
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.Computability.Language
import Mathlib.Tactic.AdaptationNote
#align_import computability.regular_expressions from "leanprover-community/mathlib"@"369525b73f2... | Mathlib/Computability/RegularExpressions.lean | 231 | 242 | theorem char_rmatch_iff (a : α) (x : List α) : rmatch (char a) x ↔ x = [a] := by |
cases' x with _ x
· exact of_decide_eq_true rfl
cases' x with head tail
· rw [rmatch, deriv]
split_ifs
· tauto
· simp [List.singleton_inj]; tauto
· rw [rmatch, rmatch, deriv]
split_ifs with h
· simp only [deriv_one, zero_rmatch, cons.injEq, and_false]
· simp only [deriv_zero, zero_rma... |
/-
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.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 50 | 50 | theorem star_ι (m : M) : star (ι Q m) = -ι Q m := by | rw [star_def, involute_ι, map_neg, reverse_ι]
|
/-
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.Geometry.RingedSpace.Basic
import Mathlib.Geometry.RingedSpace.Stalks
#align_import algebraic_geometry.locally_ringed_space from "leanprover-community... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean | 302 | 315 | theorem preimage_basicOpen {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) {U : Opens Y}
(s : Y.presheaf.obj (op U)) :
(Opens.map f.1.base).obj (Y.toRingedSpace.basicOpen s) =
@RingedSpace.basicOpen X.toRingedSpace ((Opens.map f.1.base).obj U) (f.1.c.app _ s) := by |
ext x
constructor
· rintro ⟨⟨y, hyU⟩, hy : IsUnit _, rfl : y = _⟩
erw [RingedSpace.mem_basicOpen _ _ ⟨x, show x ∈ (Opens.map f.1.base).obj U from hyU⟩,
← PresheafedSpace.stalkMap_germ_apply]
exact (PresheafedSpace.stalkMap f.1 _).isUnit_map hy
· rintro ⟨y, hy : IsUnit _, rfl⟩
erw [RingedSpace... |
/-
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 | 117 | 123 | theorem fourier_coe_apply {n : ℤ} {x : ℝ} :
fourier n (x : AddCircle T) = Complex.exp (2 * π * Complex.I * n * x / T) := by |
rw [fourier_apply, ← QuotientAddGroup.mk_zsmul, toCircle, Function.Periodic.lift_coe,
expMapCircle_apply, Complex.ofReal_mul, Complex.ofReal_div, Complex.ofReal_mul, zsmul_eq_mul,
Complex.ofReal_mul, Complex.ofReal_intCast]
norm_num
congr 1; ring
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 453 | 455 | theorem inner_rotation_pi_div_two_left_smul (x : V) (r : ℝ) :
⟪o.rotation (π / 2 : ℝ) x, r • x⟫ = 0 := by |
rw [inner_smul_right, inner_rotation_pi_div_two_left, mul_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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 757 | 762 | theorem dist_div_cos_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₁ p₂ / Real.Angle.cos (∡ p₃ p₁ p₂) = dist p₁ p₃ := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.cos_coe, dist_comm p₁ p₃,
dist_div_cos_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inr (left_ne_of_oang... |
/-
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 | 297 | 298 | theorem map_subtype_le (p' : Submodule R p) : map p.subtype p' ≤ p := by |
simpa using (map_le_range : map p.subtype p' ≤ range p.subtype)
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 480 | 511 | theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
(N : ℝ) * exp (-4 * √(log N)) < rothNumberNat N := by |
let n := nValue N
have hn : 0 < (n : ℝ) := cast_pos.2 (nValue_pos <| hN.trans' <| by norm_num1)
have hd : 0 < dValue N := dValue_pos (hN.trans' <| by norm_num1)
have hN₀ : 0 < (N : ℝ) := cast_pos.2 (hN.trans' <| by norm_num1)
have hn₂ : 2 < n := three_le_nValue <| hN.trans' <| by norm_num1
have : (2 * dVal... |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 573 | 577 | theorem map_const_principal_coprod_map_id_principal {α β ι : Type*} (a : α) (b : β) (i : ι) :
(map (fun _ => b) (𝓟 {a})).coprod (map id (𝓟 {i})) =
𝓟 ((({b} : Set β) ×ˢ univ) ∪ (univ ×ˢ ({i} : Set ι))) := by |
simp only [map_principal, Filter.coprod, comap_principal, sup_principal, image_singleton,
image_id, prod_univ, univ_prod, id]
|
/-
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.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 60 | 61 | theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π := by |
rw [← not_exists, not_iff_not, sin_eq_zero_iff]
|
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_impor... | Mathlib/RingTheory/ChainOfDivisors.lean | 99 | 108 | theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
(hq : q ≠ 0) : Irreducible (c 1) := by |
cases' n with n; · contradiction
refine (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨?_, fun b hb => ?_⟩
· exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
cases i
· exact (Associates.isUnit_iff_eq_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.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 657 | 667 | theorem mapDomain_injOn (S : Set α) {f : α → β} (hf : Set.InjOn f S) :
Set.InjOn (mapDomain f : (α →₀ M) → β →₀ M) { w | (w.support : Set α) ⊆ S } := by |
intro v₁ hv₁ v₂ hv₂ eq
ext a
classical
by_cases h : a ∈ v₁.support ∪ v₂.support
· rw [← mapDomain_apply' S _ hv₁ hf _, ← mapDomain_apply' S _ hv₂ hf _, eq] <;>
· apply Set.union_subset hv₁ hv₂
exact mod_cast h
· simp only [not_or, mem_union, not_not, mem_support_iff] at h
simp... |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 198 | 201 | theorem ker_eq_bot_iff_ker_flip_eq_bot (H : B.IsRefl) :
LinearMap.ker B = ⊥ ↔ LinearMap.ker B.flip = ⊥ := by |
refine ⟨ker_flip_eq_bot H, fun h ↦ ?_⟩
exact (congr_arg _ B.flip_flip.symm).trans (ker_flip_eq_bot (flip_isRefl_iff.mpr H) h)
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 131 | 132 | theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by |
simp only [One.one, OfNat.ofNat, RatFunc.one]
|
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bra... | Mathlib/GroupTheory/Commutator.lean | 65 | 66 | theorem map_commutatorElement : (f ⁅g₁, g₂⁆ : G') = ⁅f g₁, f g₂⁆ := by |
simp_rw [commutatorElement_def, map_mul f, map_inv f]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over fro... | Mathlib/RingTheory/Ideal/Over.lean | 361 | 382 | theorem exists_ideal_over_prime_of_isIntegral_of_isDomain [Algebra.IsIntegral R S] (P : Ideal R)
[IsPrime P] (hP : RingHom.ker (algebraMap R S) ≤ P) :
∃ Q : Ideal S, IsPrime Q ∧ Q.comap (algebraMap R S) = P := by |
have hP0 : (0 : S) ∉ Algebra.algebraMapSubmonoid S P.primeCompl := by
rintro ⟨x, ⟨hx, x0⟩⟩
exact absurd (hP x0) hx
let Rₚ := Localization P.primeCompl
let Sₚ := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)
letI : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) :=
I... |
/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathl... | Mathlib/Combinatorics/SetFamily/LYM.lean | 65 | 87 | theorem card_mul_le_card_shadow_mul (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
𝒜.card * r ≤ (∂ 𝒜).card * (Fintype.card α - r + 1) := by |
let i : DecidableRel ((· ⊆ ·) : Finset α → Finset α → Prop) := fun _ _ => Classical.dec _
refine card_mul_le_card_mul' (· ⊆ ·) (fun s hs => ?_) (fun s hs => ?_)
· rw [← h𝒜 hs, ← card_image_of_injOn s.erase_injOn]
refine card_le_card ?_
simp_rw [image_subset_iff, mem_bipartiteBelow]
exact fun a ha =>... |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
#align_import algebra.lie.universal_enveloping fr... | Mathlib/Algebra/Lie/UniversalEnveloping.lean | 153 | 155 | theorem lift_ι_apply' (x : L) :
lift R f ((UniversalEnvelopingAlgebra.mkAlgHom R L) (ιₜ x)) = f x := by |
simpa using lift_ι_apply R f x
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Ma... | Mathlib/CategoryTheory/Limits/Final.lean | 807 | 827 | theorem IsFilteredOrEmpty.of_final (F : C ⥤ D) [Final F] [IsFilteredOrEmpty C] :
IsFilteredOrEmpty D where
cocone_objs X Y := ⟨F.obj (IsFiltered.max (Final.lift F X) (Final.lift F Y)),
Final.homToLift F X ≫ F.map (IsFiltered.leftToMax _ _),
⟨Final.homToLift F Y ≫ F.map (IsFiltered.rightToMax _ _), trivial... |
let P : StructuredArrow X F → Prop := fun h => ∃ (Z : C) (q₁ : h.right ⟶ Z)
(q₂ : Final.lift F Y ⟶ Z), h.hom ≫ F.map q₁ = f ≫ Final.homToLift F Y ≫ F.map q₂
rsuffices ⟨Z, q₁, q₂, h⟩ : Nonempty (P (StructuredArrow.mk (g ≫ Final.homToLift F Y)))
· refine ⟨F.obj (IsFiltered.coeq q₁ q₂),
Final.ho... |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import prob... | Mathlib/Probability/Martingale/BorelCantelli.lean | 263 | 267 | theorem Martingale.ae_not_tendsto_atTop_atTop [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ¬Tendsto (fun n => f n ω) atTop atTop := by |
filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with ω hω htop using
unbounded_of_tendsto_atTop htop (hω.2 <| bddBelow_range_of_tendsto_atTop_atTop htop)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 337 | 340 | theorem compAlongComposition_nnnorm {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
(p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) :
‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊ := by |
rw [← NNReal.coe_le_coe]; push_cast; exact q.compAlongComposition_norm p c
|
/-
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.Order.Archimedean
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Order.... | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 167 | 170 | theorem tprod_le_one (h : ∀ i, f i ≤ 1) : ∏' i, f i ≤ 1 := by |
by_cases hf : Multipliable f
· exact hf.hasProd.le_one h
· rw [tprod_eq_one_of_not_multipliable hf]
|
/-
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 | 936 | 938 | theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) :
(0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) := by |
simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support]
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 537 | 539 | theorem infsep_pos_iff_nontrivial_of_finite [Finite s] : 0 < s.infsep ↔ s.Nontrivial := by |
rw [infsep_pos, einfsep_lt_top_iff, and_iff_right_iff_imp]
exact fun _ => einfsep_pos_of_finite
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
/-!
# Power ... | Mathlib/RingTheory/PowerBasis.lean | 470 | 476 | theorem minpolyGen_map (pb : PowerBasis A S) (e : S ≃ₐ[A] S') :
(pb.map e).minpolyGen = pb.minpolyGen := by |
dsimp only [minpolyGen, map_dim]
-- Turn `Fin (pb.map e).dim` into `Fin pb.dim`
simp only [LinearEquiv.trans_apply, map_basis, Basis.map_repr, map_gen,
AlgEquiv.toLinearEquiv_apply, e.toLinearEquiv_symm, AlgEquiv.map_pow,
AlgEquiv.symm_apply_apply, sub_right_inj]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Ta... | Mathlib/Logic/Relation.lean | 607 | 610 | theorem ReflTransGen.lift' {p : β → β → Prop} {a b : α} (f : α → β)
(h : ∀ a b, r a b → ReflTransGen p (f a) (f b))
(hab : ReflTransGen r a b) : ReflTransGen p (f a) (f b) := by |
simpa [reflTransGen_idem] using hab.lift f h
|
/-
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.Data.Multiset.Sum
import Mathlib.Data.Finset.Card
#align_import data.finset.sum from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900... | Mathlib/Data/Finset/Sum.lean | 54 | 56 | theorem disjoint_map_inl_map_inr : Disjoint (s.map Embedding.inl) (t.map Embedding.inr) := by |
simp_rw [disjoint_left, mem_map]
rintro x ⟨a, _, rfl⟩ ⟨b, _, ⟨⟩⟩
|
/-
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 | 2,024 | 2,025 | theorem insert_diff_singleton {a : α} {s : Set α} : insert a (s \ {a}) = insert a s := by |
simp [insert_eq, union_diff_self, -union_singleton, -singleton_union]
|
/-
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
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 289 | 293 | theorem toBlock_diagonal_disjoint (d : m → α) {p q : m → Prop} (hpq : Disjoint p q) :
Matrix.toBlock (diagonal d) p q = 0 := by |
ext ⟨i, hi⟩ ⟨j, hj⟩
have : i ≠ j := fun heq => hpq.le_bot i ⟨hi, heq.symm ▸ hj⟩
simp [diagonal_apply_ne d this]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 296 | 303 | theorem add_one_mul_T_eq_poly_in_U (n : ℤ) :
((n : R[X]) + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n) := by |
have h₁ := congr_arg derivative <| T_eq_X_mul_T_sub_pol_U R n
simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,
T_derivative_eq_U, C_eq_natCast] at h₁
have h₂ := T_eq_U_sub_X_mul_U R (n + 1)
linear_combination (norm := (push_cast; ring_nf))
h₁ + (n + 2) * h₂
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 666 | 677 | theorem isSheafFor_iso {P' : Cᵒᵖ ⥤ Type w} (i : P ≅ P') : IsSheafFor P R → IsSheafFor P' R := by |
intro h x hx
let x' := x.compPresheafMap i.inv
have : x'.Compatible := FamilyOfElements.Compatible.compPresheafMap i.inv hx
obtain ⟨t, ht1, ht2⟩ := h x' this
use i.hom.app _ t
fconstructor
· convert FamilyOfElements.IsAmalgamation.compPresheafMap i.hom ht1
simp [x']
· intro y hy
rw [show y = (i... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 472 | 478 | theorem symm_eq_on_of_inter_eq_of_eqOn {e' : PartialEquiv α β} (h : e.IsImage s t)
(hs : e.source ∩ s = e'.source ∩ s) (heq : EqOn e e' (e.source ∩ s)) :
EqOn e.symm e'.symm (e.target ∩ t) := by |
rw [← h.image_eq]
rintro y ⟨x, hx, rfl⟩
have hx' := hx; rw [hs] at hx'
rw [e.left_inv hx.1, heq hx, e'.left_inv hx'.1]
|
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Jordan.Basic
import Mathlib.Algebra.Module.Defs
#align_import algebra.symmetrized from "leanprover-community/mathlib"@"933547832736be6... | Mathlib/Algebra/Symmetrized.lean | 329 | 330 | theorem sym_mul_self [Semiring α] [Invertible (2 : α)] (a : α) : sym (a * a) = sym a * sym a := by |
rw [sym_mul_sym, ← two_mul, invOf_mul_self_assoc]
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Shapes.Types
#alig... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 271 | 298 | theorem objSupIsoProdEqLocus_inv_eq_iff {X : TopCat.{u}} (F : X.Sheaf CommRingCat.{u})
{U V W UW VW : Opens X} (e : W ≤ U ⊔ V) (x) (y : F.1.obj (op W))
(h₁ : UW = U ⊓ W) (h₂ : VW = V ⊓ W) :
F.1.map (homOfLE e).op ((F.objSupIsoProdEqLocus U V).inv x) = y ↔
F.1.map (homOfLE (h₁ ▸ inf_le_left : UW ≤ U)).op... |
subst h₁ h₂
constructor
· rintro rfl
rw [← TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, ← TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd]
-- `simp` doesn't see through the type equality of objects in `CommRingCat`, so use `rw` #8386
repeat rw [← comp_apply]
simp only [← Functor.map_comp, ← op_comp, Cate... |
/-
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.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanp... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 219 | 221 | theorem gcd_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).gcd = GCDMonoid.gcd a s.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_cons]
simp
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 289 | 292 | theorem HasDerivWithinAt.const_mul (c : 𝔸) (hd : HasDerivWithinAt d d' s x) :
HasDerivWithinAt (fun y => c * d y) (c * d') s x := by |
convert (hasDerivWithinAt_const x s c).mul hd using 1
rw [zero_mul, zero_add]
|
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.PDeriv
/-!... | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | 206 | 209 | theorem polynomial_relation (P : Fin 3 → R) : 3 * eval P W.polynomial =
P x * eval P W.polynomialX + P y * eval P W.polynomialY + P z * eval P W.polynomialZ := by |
rw [eval_polynomial, eval_polynomialX, eval_polynomialY, eval_polynomialZ]
ring1
|
/-
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 | 256 | 262 | theorem intervalIntegral_add_eq_of_pos (hf : Periodic f T) (hT : 0 < T) (t s : ℝ) :
∫ x in t..t + T, f x = ∫ x in s..s + T, f x := by |
simp only [integral_of_le, hT.le, le_add_iff_nonneg_right]
haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) ℝ volume :=
⟨fun c s _ => measure_preimage_add _ _ _⟩
apply IsAddFundamentalDomain.setIntegral_eq (G := AddSubgroup.zmultiples T)
exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomai... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,105 | 1,107 | theorem Tendsto.atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) :
Tendsto (fun x => f x / r) l atTop := by |
simpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr)
|
/-
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, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof fr... | Mathlib/Analysis/ODE/PicardLindelof.lean | 361 | 369 | theorem exists_solution :
∃ f : ℝ → E, f v.t₀ = v.x₀ ∧ ∀ t ∈ Icc v.tMin v.tMax,
HasDerivWithinAt f (v t (f t)) (Icc v.tMin v.tMax) t := by |
rcases v.exists_fixed with ⟨f, hf⟩
refine ⟨f ∘ v.proj, ?_, fun t ht => ?_⟩
· simp only [(· ∘ ·), proj_coe, f.map_t₀]
· simp only [(· ∘ ·), v.proj_of_mem ht]
lift t to Icc v.tMin v.tMax using ht
simpa only [hf, v.proj_coe] using f.hasDerivWithinAt_next t
|
/-
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, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENa... | Mathlib/RingTheory/Multiplicity.lean | 464 | 471 | theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by |
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
|
/-
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.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 247 | 270 | theorem toJordanDecomposition_eq_of_eq_add_withDensity {f : α → ℝ} (hf : Measurable f)
(hfi : Integrable f μ) (htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
s.toJordanDecomposition =
@JordanDecomposition.mk α _
(t.toJordanDecomposition.posPart + μ.withDensity fun x => EN... |
haveI := isFiniteMeasure_withDensity_ofReal hfi.2
haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2
refine toJordanDecomposition_eq ?_
simp_rw [JordanDecomposition.toSignedMeasure, hadd]
ext i hi
rw [VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi,
toSignedMeasure_apply_measurable hi... |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 228 | 229 | theorem bit1_add' [One M] (a b : M) : bit1 (a + b) = bit1 a + bit0 b := by |
rw [add_comm, bit1_add, add_comm]
|
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 168 | 170 | theorem dimH_mono {s t : Set X} (h : s ⊆ t) : dimH s ≤ dimH t := by |
borelize X
exact dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top <| top_unique <| hd ▸ measure_mono h
|
/-
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.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 226 | 231 | theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by |
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq
use n₀
intro n hn
convert hn₀ (n - n₀)
rw [add_comm, tsub_add_cancel_of_le hn]
|
/-
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 | 320 | 322 | theorem _root_.RingHom.map_det (f : R →+* S) (M : Matrix n n R) :
f M.det = Matrix.det (f.mapMatrix M) := by |
simp [Matrix.det_apply', map_sum f, map_prod f]
|
/-
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.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 297 | 305 | theorem integrable_cexp_neg_mul_sq_norm_add (hb : 0 < b.re) (c : ℂ) (w : V) :
Integrable (fun (v : V) ↦ cexp (-b * ‖v‖^2 + c * ⟪w, v⟫)) := by |
let e := (stdOrthonormalBasis ℝ V).repr.symm
rw [← e.measurePreserving.integrable_comp_emb e.toHomeomorph.measurableEmbedding]
convert integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace
hb c (e.symm w) with v
simp only [neg_mul, Function.comp_apply, LinearIsometryEquiv.norm_map,
LinearIsometryEquiv.... |
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 53 | 62 | theorem cpow_eq_nhds' {p : ℂ × ℂ} (hp_fst : p.fst ≠ 0) :
(fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) := by |
suffices ∀ᶠ x : ℂ × ℂ in 𝓝 p, x.1 ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [cpow_def_of_ne_zero hx]
refine IsOpen.eventually_mem ?_ hp_fst
change IsOpen { x : ℂ × ℂ | x.1 = 0 }ᶜ
rw [isOpen_compl_iff]
exact isClosed_eq continuous_fst continuous_const
|
/-
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.Data.Vector.Basic
import Mathlib.Data.List.Zip
#align_import data.vector.zip from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f67... | Mathlib/Data/Vector/Zip.lean | 33 | 36 | theorem zipWith_get (x : Vector α n) (y : Vector β n) (i) :
(Vector.zipWith f x y).get i = f (x.get i) (y.get i) := by |
dsimp only [Vector.zipWith, Vector.get]
simp only [List.get_zipWith, Fin.cast]
|
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Nick Kuhn
-/
import Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves
/-!
# Description of the covering sieves of the coherent topology
This file characterises the coveri... | Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean | 29 | 36 | theorem coherentTopology.mem_sieves_of_hasEffectiveEpiFamily (S : Sieve X) :
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
EffectiveEpiFamily Y π ∧ (∀ a : α, (S.arrows) (π a)) ) →
(S ∈ GrothendieckTopology.sieves (coherentTopology C) X) := by |
intro ⟨α, _, Y, π, hπ⟩
apply (coherentCoverage C).mem_toGrothendieck_sieves_of_superset (R := Presieve.ofArrows Y π)
· exact fun _ _ h ↦ by cases h; exact hπ.2 _
· exact ⟨_, inferInstance, Y, π, rfl, hπ.1⟩
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen
-/
import Mathlib.Logic.Relation
import Mathlib.Order.GaloisConnection
#align_import data.setoid.basic from "leanprover-community/mathlib"@"bbe... | Mathlib/Data/Setoid/Basic.lean | 194 | 196 | theorem eq_top_iff {s : Setoid α} : s = (⊤ : Setoid α) ↔ ∀ x y : α, s.Rel x y := by |
rw [_root_.eq_top_iff, Setoid.le_def, Setoid.top_def]
simp only [Pi.top_apply, Prop.top_eq_true, forall_true_left]
|
/-
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, Sander Dahmen, Scott Morrison
-/
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dimension from "leanprover-community/mathl... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 235 | 238 | theorem rank_eq_of_equiv_equiv (i : R ≃+* R') (j : S ≃+* S')
(hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)) :
Module.rank R S = Module.rank R' S' := by |
simpa only [lift_id] using lift_rank_eq_of_equiv_equiv i j hc
|
/-
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.Data.Finset.Sum
import Mathlib.Logic.Embedding.Set
#align_import data.fintype.sum from "leanprover-community/mathlib"... | Mathlib/Data/Fintype/Sum.lean | 105 | 115 | theorem Set.MapsTo.exists_equiv_extend_of_card_eq [Fintype α] {t : Finset β}
(hαt : Fintype.card α = t.card) {s : Set α} {f : α → β} (hfst : s.MapsTo f t)
(hfs : Set.InjOn f s) : ∃ g : α ≃ t, ∀ i ∈ s, (g i : β) = f i := by |
classical
let s' : Finset α := s.toFinset
have hfst' : s'.image f ⊆ t := by simpa [s', ← Finset.coe_subset] using hfst
have hfs' : Set.InjOn f s' := by simpa [s'] using hfs
obtain ⟨g, hg⟩ := Finset.exists_equiv_extend_of_card_eq hαt hfst' hfs'
refine ⟨g, fun i hi => ?_⟩
apply hg
simpa [s'... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@... | Mathlib/Analysis/Calculus/Dslope.lean | 72 | 74 | theorem dslope_sub_smul_of_ne (f : 𝕜 → E) (h : b ≠ a) :
dslope (fun x => (x - a) • f x) a b = f b := by |
rw [dslope_of_ne _ h, slope_sub_smul _ h.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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 291 | 293 | theorem boundedBy_zero : boundedBy (0 : Set α → ℝ≥0∞) = 0 := by |
rw [← coe_bot, eq_bot_iff]
apply boundedBy_le
|
/-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
/-!
# Eisenstein Series
##... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean | 92 | 100 | theorem eisSummand_SL2_apply (k : ℤ) (i : (Fin 2 → ℤ)) (A : SL(2, ℤ)) (z : ℍ) :
eisSummand k i (A • z) = (z.denom A) ^ k * eisSummand k (i ᵥ* A) z := by |
simp only [eisSummand, specialLinearGroup_apply, algebraMap_int_eq, eq_intCast, ofReal_intCast,
one_div, vecMul, vec2_dotProduct, Int.cast_add, Int.cast_mul]
have h (a b c d u v : ℂ) (hc : c * z + d ≠ 0) : ((u * ((a * z + b) / (c * z + d)) + v) ^ k)⁻¹ =
(c * z + d) ^ k * (((u * a + v * c) * z + (u * b + ... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 398 | 400 | theorem lift_comp_lift_apply_eq_self [PerfectRing L p] (x : L) :
lift j i p (lift i j p x) = x := by |
rw [lift_comp_lift_apply, lift_self_apply]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 359 | 361 | theorem isUnit (h : IsPrimitiveRoot ζ k) (h0 : 0 < k) : IsUnit ζ := by |
apply isUnit_of_mul_eq_one ζ (ζ ^ (k - 1))
rw [← pow_succ', tsub_add_cancel_of_le h0.nat_succ_le, h.pow_eq_one]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_t... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 425 | 429 | theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) :
f.measure (Iic x) = ofReal (f x - l) := by |
refine tendsto_nhds_unique (tendsto_measure_Ioc_atBot _ _) ?_
simp_rw [measure_Ioc]
exact ENNReal.tendsto_ofReal (Tendsto.const_sub _ hf)
|
/-
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 | 158 | 168 | theorem content_eq_zero_iff {p : R[X]} : content p = 0 ↔ p = 0 := by |
rw [content, Finset.gcd_eq_zero_iff]
constructor <;> intro h
· ext n
by_cases h0 : n ∈ p.support
· rw [h n h0, coeff_zero]
· rw [mem_support_iff] at h0
push_neg at h0
simp [h0]
· intro x
simp [h]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 236 | 242 | theorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) :
∃ y ∈ s, x ≤ y := by |
by_cases hx : IsTop x
· exact ⟨x, htop x hx, le_rfl⟩
· simp only [IsTop, not_forall, not_le] at hx
rcases hs.exists_mem_open isOpen_Ioi hx with ⟨y, hys, hy : x < y⟩
exact ⟨y, hys, hy.le⟩
|
/-
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.Interval.Finset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith
... | Mathlib/Algebra/BigOperators/Intervals.lean | 103 | 106 | theorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α]
(f : α → M) (a b c : α) : (∏ x ∈ Ico a b, f (c + x)) = ∏ x ∈ Ico (a + c) (b + c), f x := by |
convert prod_Ico_add' f a b c using 2
rw [add_comm]
|
/-
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.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 405 | 408 | theorem neg_sub' {X Y : C} (a b : X ⟶ Y) : -(a - b) = -a + b := by |
rw [neg_def, neg_def]
conv_lhs => rw [← sub_self (0 : X ⟶ Y)]
rw [sub_sub_sub, add_def, neg_def]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,559 | 1,562 | theorem map_inf_eq (H K : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
map f (H ⊓ K) = map f H ⊓ map f K := by |
rw [← SetLike.coe_set_eq]
simp [Set.image_inter hf]
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.Norm
#align_import linear_algebra.free_module.norm from "leanprover-community/mathlib"@"90b0d53... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 30 | 50 | theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i) := by |
have hI := span_singleton_eq_bot.not.2 hf
let b' := ringBasis b (span {f}) hI
classical
rw [← Matrix.det_diagonal, ← LinearMap.det_toLin b']
let e :=
(b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans
((LinearEquiv.coord S S f hf).restrictScalars R)
refine (LinearMap.associated_det_of_e... |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 167 | 172 | theorem Ideal.mem_prime_of_mul_mem_pow [IsDedekindDomain S] {P : Ideal S} [P_prime : P.IsPrime]
(hP : P ≠ ⊥) {i : ℕ} {a b : S} (a_not_mem : a ∉ P ^ (i + 1)) (ab_mem : a * b ∈ P ^ (i + 1)) :
b ∈ P := by |
simp only [← Ideal.span_singleton_le_iff_mem, ← Ideal.dvd_iff_le, pow_succ, ←
Ideal.span_singleton_mul_span_singleton] at a_not_mem ab_mem ⊢
exact (prime_pow_succ_dvd_mul (Ideal.prime_of_isPrime hP P_prime) ab_mem).resolve_left a_not_mem
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 271 | 273 | theorem EuclideanSpace.single_apply (i : ι) (a : 𝕜) (j : ι) :
(EuclideanSpace.single i a) j = ite (j = i) a 0 := by |
rw [EuclideanSpace.single, WithLp.equiv_symm_pi_apply, ← Pi.single_apply i a j]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 394 | 394 | theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by | rw [← inv_lt_inv_iff, inv_inv]
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 521 | 523 | theorem norm_map_iff_adjoint_comp_self (u : H →L[𝕜] K) :
(∀ x : H, ‖u x‖ = ‖x‖) ↔ adjoint u ∘L u = 1 := by |
rw [LinearMap.norm_map_iff_inner_map_map u, u.inner_map_map_iff_adjoint_comp_self]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.q... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 302 | 306 | theorem snd_comp_quotientInfEquivQuotientProd (I J : Ideal R) (coprime : IsCoprime I J) :
(RingHom.snd _ _).comp
(quotientInfEquivQuotientProd I J coprime : R ⧸ I ⊓ J →+* (R ⧸ I) × R ⧸ J) =
Ideal.Quotient.factor (I ⊓ J) J inf_le_right := by |
apply Quotient.ringHom_ext; ext; rfl
|
/-
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 | 556 | 563 | theorem Wbtw.sameRay_vsub {x y z : P} (h : Wbtw R x y z) : SameRay R (y -ᵥ x) (z -ᵥ y) := by |
rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩
simp_rw [lineMap_apply]
rcases ht0.lt_or_eq with (ht0' | rfl); swap; · simp
rcases ht1.lt_or_eq with (ht1' | rfl); swap; · simp
refine Or.inr (Or.inr ⟨1 - t, t, sub_pos.2 ht1', ht0', ?_⟩)
simp only [vadd_vsub, smul_smul, vsub_vadd_eq_vsub_sub, smul_sub, ← sub_smul]
ring... |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/... | Mathlib/Analysis/SumIntegralComparisons.lean | 98 | 123 | theorem AntitoneOn.sum_le_integral (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∑ i ∈ Finset.range a, f (x₀ + (i + 1 : ℕ))) ≤ ∫ x in x₀..x₀ + a, f x := by |
have hint : ∀ k : ℕ, k < a → IntervalIntegrable f volume (x₀ + k) (x₀ + (k + 1 : ℕ)) := by
intro k hk
refine (hf.mono ?_).intervalIntegrable
rw [uIcc_of_le]
· apply Icc_subset_Icc
· simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]
· simp only [add_le_add_iff_left, Nat.cast_le, Nat.suc... |
/-
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.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 141 | 144 | theorem map_cast_heq {α β} (h : α = β) (s : Finset α) :
HEq (s.map (Equiv.cast h).toEmbedding) s := by |
subst h
simp
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 431 | 435 | theorem degree_lt_card_verts [DecidableRel G.Adj] (v : V) : G.degree v < Fintype.card V := by |
classical
apply Finset.card_lt_card
rw [Finset.ssubset_iff]
exact ⟨v, by simp, Finset.subset_univ _⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
/-!
# Lemmas on infini... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 585 | 587 | theorem tprod_dite_left (P : Prop) [Decidable P] (x : β → P → α) :
∏' b : β, (if h : P then x b h else 1) = if h : P then ∏' b : β, x b h else 1 := by |
by_cases hP : P <;> simp [hP]
|
/-
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.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 404 | 408 | theorem isRoot_of_eval₂_map_eq_zero (hf : Function.Injective f) {r : R} :
eval₂ f (f r) p = 0 → p.IsRoot r := by |
intro h
apply hf
rw [← eval₂_hom, h, f.map_zero]
|
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