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) 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 | 781 | 785 | theorem wbtw_or_wbtw_smul_vadd_of_nonpos (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0) (hr₂ : r₂ ≤ 0) :
Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) ∨ Wbtw R x (r₂ • v +ᵥ x) (r₁ • v +ᵥ x) := by |
rcases le_total r₁ r₂ with (h | h)
· exact Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_le x v hr₂ h)
· exact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonpos_of_le x v hr₁ h)
|
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 965 | 969 | theorem binomial_eq_binomial {k l m n : ℕ} {u v : R} (hu : u ≠ 0) (hv : v ≠ 0) :
C u * X ^ k + C v * X ^ l = C u * X ^ m + C v * X ^ n ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n := by |
simp_rw [C_mul_X_pow_eq_monomial, ← toFinsupp_inj, toFinsupp_add, toFinsupp_monomial]
exact Finsupp.single_add_single_eq_single_add_single hu hv
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 664 | 693 | theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
(hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by |
obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
refine ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => ?_⟩
have yr' : ‖y‖ < r' := by
rw [ball_zero_eq] at hy
exact hy
have h... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 436 | 442 | theorem sin_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
(h0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂) : Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃ := by |
rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [← @vsub_ne_zero V, @ne_comm _ p₃, ← @vsub_ne_zero V _ _ _ p₂, or_comm] at h0
rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,
... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 800 | 803 | theorem coplanar_of_finrank_eq_two (s : Set P) (h : finrank k V = 2) : Coplanar k s := by |
have : FiniteDimensional k V := .of_finrank_eq_succ h
rw [coplanar_iff_finrank_le_two, ← h]
exact Submodule.finrank_le _
|
/-
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, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | Mathlib/Algebra/Divisibility/Units.lean | 228 | 231 | theorem IsRelPrime.mul_dvd_of_right_isPrimal (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z)
(hy : IsPrimal y) : x * y ∣ z := by |
obtain ⟨w, rfl⟩ := H1
exact mul_dvd_mul_left x (H.symm.dvd_of_dvd_mul_left_of_isPrimal H2 hy)
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 668 | 669 | theorem mem_closure_singleton {x y : A} : y ∈ closure ({x} : Set A) ↔ ∃ n : ℕ, n • x = y := by |
rw [closure_singleton_eq, AddMonoidHom.mem_mrange]; rfl
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 599 | 600 | theorem diam_preimage (s : Set β) : Metric.diam (h ⁻¹' s) = Metric.diam s := by |
rw [← image_symm, diam_image]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,360 | 1,377 | theorem coeff_map (p : MvPolynomial σ R) : ∀ m : σ →₀ ℕ, coeff m (map f p) = f (coeff m p) := by |
classical
apply MvPolynomial.induction_on p <;> clear p
· intro r m
rw [map_C]
simp only [coeff_C]
split_ifs
· rfl
rw [f.map_zero]
· intro p q hp hq m
simp only [hp, hq, (map f).map_add, coeff_add]
rw [f.map_add]
· intro p i hp m
simp only [hp, (map f).map_mul, map_X]
simp... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Ma... | Mathlib/RingTheory/WittVector/Frobenius.lean | 293 | 306 | theorem coeff_frobenius_charP (x : 𝕎 R) (n : ℕ) : coeff (frobenius x) n = x.coeff n ^ p := by |
rw [coeff_frobenius]
letI : Algebra (ZMod p) R := ZMod.algebra _ _
-- outline of the calculation, proofs follow below
calc
aeval (fun k => x.coeff k) (frobeniusPoly p n) =
aeval (fun k => x.coeff k)
(MvPolynomial.map (Int.castRingHom (ZMod p)) (frobeniusPoly p n)) := ?_
_ = aeval (fun... |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 573 | 581 | theorem corec_def {X} (f : X → F X) (x₀ : X) : M.corec f x₀ = M.mk (F.map (M.corec f) (f x₀)) := by |
dsimp only [M.corec, M.mk]
congr with n
cases' n with n
· dsimp only [sCorec, Approx.sMk]
· dsimp only [sCorec, Approx.sMk]
cases f x₀
dsimp only [PFunctor.map]
congr
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 346 | 351 | theorem eval_interpolate_at_node (hvs : Set.InjOn v s) (hi : i ∈ s) :
eval (v i) (interpolate s v r) = r i := by |
rw [interpolate_apply, eval_finset_sum, ← add_sum_erase _ _ hi]
simp_rw [eval_mul, eval_C, eval_basis_self hvs hi, mul_one, add_right_eq_self]
refine sum_eq_zero fun j H => ?_
rw [eval_basis_of_ne (mem_erase.mp H).1 hi, mul_zero]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 542 | 548 | theorem setIntegral_indicatorConstLp [CompleteSpace E]
{p : ℝ≥0∞} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμt : μ t ≠ ∞) (e : E) :
∫ x in s, indicatorConstLp p ht hμt e x ∂μ = (μ (t ∩ s)).toReal • e :=
calc
∫ x in s, indicatorConstLp p ht hμt e x ∂μ = ∫ x in s, t.indicator (fun _ => e) x ∂μ := by |
rw [setIntegral_congr_ae hs (indicatorConstLp_coeFn.mono fun x hx _ => hx)]
_ = (μ (t ∩ s)).toReal • e := by rw [integral_indicator_const _ ht, Measure.restrict_apply ht]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 768 | 775 | theorem _root_.Set.PairwiseDisjoint.exists_mem_filter {ι : Type*} {l : ι → Filter α} {t : Set ι}
(hd : t.PairwiseDisjoint l) (ht : t.Finite) :
∃ s : ι → Set α, (∀ i, s i ∈ l i) ∧ t.PairwiseDisjoint s := by |
haveI := ht.to_subtype
rcases (hd.subtype _ _).exists_mem_filter_of_disjoint with ⟨s, hsl, hsd⟩
lift s to (i : t) → {s // s ∈ l i} using hsl
rcases @Subtype.exists_pi_extension ι (fun i => { s // s ∈ l i }) _ _ s with ⟨s, rfl⟩
exact ⟨fun i => s i, fun i => (s i).2, hsd.set_of_subtype _ _⟩
|
/-
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 | 1,379 | 1,388 | theorem unique' {p q : Multiset (Associates α)} :
(∀ a ∈ p, Irreducible a) → (∀ a ∈ q, Irreducible a) → p.prod = q.prod → p = q := by |
apply Multiset.induction_on_multiset_quot p
apply Multiset.induction_on_multiset_quot q
intro s t hs ht eq
refine Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique ?_ ?_ ?_)
· exact fun a ha => irreducible_mk.1 <| hs _ <| Multiset.mem_map_of_mem _ ha
· exact fun a ha => irreducible... |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.... | Mathlib/Topology/Algebra/Order/Field.lean | 117 | 119 | theorem Filter.Tendsto.neg_mul_atBot {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop := by |
simpa only [mul_comm] using hg.atBot_mul_neg hC hf
|
/-
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
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 76 | 84 | theorem mapAccumr₂_mapAccumr_right (f₁ : α → γ → σ₁ → σ₁ × ζ) (f₂ : β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ xs (mapAccumr f₂ ys s₂).snd s₁)
= let m := (mapAccumr₂ (fun x y s =>
let r₂ := f₂ y s.snd
let r₁ := f₁ x r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂))
(m.... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
/-
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.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 729 | 734 | theorem nmul_le_iff₃' :
a ⨳ (b ⨳ c) ≤ d ↔
∀ a' < a, ∀ b' < b, ∀ c' < c,
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') <
d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') := by |
rw [← not_iff_not]; simp [lt_nmul_iff₃']
|
/-
Copyright (c) 2020 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.Data.Part
import Mathlib.Order.Chain
import Mathlib.Order.Hom.Order
import Mathlib.Algebra.Order.Ring.Nat
#align_import o... | Mathlib/Order/OmegaCompletePartialOrder.lean | 716 | 722 | theorem seq_continuous' {β γ : Type v} (f : α → Part (β → γ)) (g : α → Part β) (hf : Continuous' f)
(hg : Continuous' g) : Continuous' fun x => f x <*> g x := by |
simp only [seq_eq_bind_map]
apply bind_continuous' _ _ hf
apply Pi.OmegaCompletePartialOrder.flip₂_continuous'
intro
apply map_continuous' _ _ hg
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 435 | 436 | theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by |
rw [← mul_right_inj' (two_ne_zero' ℂ), mul_sub, two_cosh, two_sinh, add_sub_sub_cancel, two_mul]
|
/-
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 | 436 | 443 | theorem norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle y (y - x)) = ‖y - x‖ := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.r... |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 129 | 129 | theorem bot_symmDiff : ⊥ ∆ a = a := by | rw [symmDiff_comm, symmDiff_bot]
|
/-
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.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 573 | 617 | theorem altitude_replace_orthocenter_eq_affineSpan {t₁ t₂ : Triangle ℝ P}
{i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3} (hi₁₂ : i₁ ≠ i₂) (hi₁₃ : i₁ ≠ i₃) (hi₂₃ : i₂ ≠ i₃) (hj₁₂ : j₁ ≠ j₂)
(hj₁₃ : j₁ ≠ j₃) (hj₂₃ : j₂ ≠ j₃) (h₁ : t₂.points j₁ = t₁.orthocenter)
(h₂ : t₂.points j₂ = t₁.points i₂) (h₃ : t₂.points j₃ = t₁.points i... |
symm
rw [← h₂, t₂.affineSpan_pair_eq_altitude_iff]
rw [h₂]
use t₁.independent.injective.ne hi₁₂
have he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points) := by
refine ext_of_direction_eq ?_
⟨t₁.points i₃, mem_affineSpan ℝ ⟨j₃, h₃⟩, mem_affineSpan ℝ (Set.mem_range_self _)⟩
... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 594 | 608 | theorem tensor_right_unitality (X₁ X₂ : C) :
(ρ_ (X₁ ⊗ X₂)).hom =
((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv) ≫
tensor_μ C (X₁, X₂) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ X₁).hom ⊗ (ρ_ X₂).hom) := by |
dsimp only [tensor_μ]
have :
((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv) ≫
(α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).inv) =
(α_ X₁ X₂ (𝟙_ C)).hom ≫ (X₁ ◁ (ρ_ X₂).inv ▷ 𝟙_ C) := by
coherence
slice_rhs 1 3 => rw [this]
clear this
slice_rhs 2 3 => rw [← MonoidalCategory.whiskerLe... |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community... | Mathlib/Probability/Process/Adapted.lean | 188 | 198 | theorem progMeasurable_of_tendsto' {γ} [MeasurableSpace ι] [PseudoMetrizableSpace β]
(fltr : Filter γ) [fltr.NeBot] [fltr.IsCountablyGenerated] {U : γ → ι → Ω → β}
(h : ∀ l, ProgMeasurable f (U l)) (h_tendsto : Tendsto U fltr (𝓝 u)) : ProgMeasurable f u := by |
intro i
apply @stronglyMeasurable_of_tendsto (Set.Iic i × Ω) β γ
(MeasurableSpace.prod _ (f i)) _ _ fltr _ _ _ _ fun l => h l i
rw [tendsto_pi_nhds] at h_tendsto ⊢
intro x
specialize h_tendsto x.fst
rw [tendsto_nhds] at h_tendsto ⊢
exact fun s hs h_mem => h_tendsto {g | g x.snd ∈ s} (hs.preimage (con... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,201 | 1,206 | theorem hasFTaylorSeriesUpToOn_pi' :
HasFTaylorSeriesUpToOn n Φ P' s ↔
∀ i, HasFTaylorSeriesUpToOn n (fun x => Φ x i)
(fun x m => (@ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ i).compContinuousMultilinearMap
(P' x m)) s := by |
convert hasFTaylorSeriesUpToOn_pi (𝕜 := 𝕜) (φ := fun i x ↦ Φ x i); ext; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, 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 | 517 | 520 | theorem restrict_sum_of_countable [Countable ι] (μ : ι → Measure α) (s : Set α) :
(sum μ).restrict s = sum fun i => (μ i).restrict s := by |
ext t ht
simp_rw [sum_apply _ ht, restrict_apply ht, sum_apply_of_countable]
|
/-
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 | 336 | 340 | theorem snd_comp_quotientMulEquivQuotientProd (I J : Ideal R) (coprime : IsCoprime I J) :
(RingHom.snd _ _).comp
(quotientMulEquivQuotientProd I J coprime : R ⧸ I * J →+* (R ⧸ I) × R ⧸ J) =
Ideal.Quotient.factor (I * J) J mul_le_left := by |
apply Quotient.ringHom_ext; ext; rfl
|
/-
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.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compac... | Mathlib/Topology/PartitionOfUnity.lean | 491 | 505 | theorem sum_toPOUFun_eq (x : X) : ∑ᶠ i, f.toPOUFun i x = 1 - ∏ᶠ i, (1 - f i x) := by |
set s := (f.point_finite x).toFinset
have hs : (s : Set ι) = { i | f i x ≠ 0 } := Finite.coe_toFinset _
have A : (support fun i => toPOUFun f i x) ⊆ s := by
rw [hs]
exact fun i hi => f.support_toPOUFun_subset i hi
have B : (mulSupport fun i => 1 - f i x) ⊆ s := by
rw [hs, mulSupport_one_sub]
ex... |
/-
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 | 336 | 338 | theorem rank_range_of_surjective (f : M →ₗ[R] M') (h : Surjective f) :
Module.rank R (LinearMap.range f) = Module.rank R M' := by |
rw [LinearMap.range_eq_top.2 h, rank_top]
|
/-
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.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 520 | 532 | theorem comp_le_of_antitoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t)
(φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s := by |
refine iSup_le ?_
rintro ⟨n, u, hu, ut⟩
rw [← Finset.sum_range_reflect]
refine (Finset.sum_congr rfl fun x hx => ?_).trans_le <| le_iSup_of_le
⟨n, fun i => φ (u <| n - i), fun x y xy => hφ (ut _) (ut _) (hu <| Nat.sub_le_sub_left xy n),
fun i => φst (ut _)⟩
le_rfl
rw [Finset.mem_range] at hx
... |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 274 | 278 | theorem submodule_map_mul_reverse (p q : Submodule R (CliffordAlgebra Q)) :
(p * q).map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
q.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) *
p.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) := by |
simp_rw [reverse, Submodule.map_comp, Submodule.map_mul, Submodule.map_unop_mul]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform... | Mathlib/Topology/UniformSpace/Cauchy.lean | 326 | 332 | theorem Filter.HasBasis.cauchySeq_iff {γ} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop}
{s : γ → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → (u m, u n) ∈ s i := by |
rw [cauchySeq_iff_tendsto, ← prod_atTop_atTop_eq]
refine (atTop_basis.prod_self.tendsto_iff h).trans ?_
simp only [exists_prop, true_and_iff, MapsTo, preimage, subset_def, Prod.forall, mem_prod_eq,
mem_setOf_eq, mem_Ici, and_imp, Prod.map, ge_iff_le, @forall_swap (_ ≤ _) β]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import rin... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 301 | 307 | theorem wittStructureInt_prop (Φ : MvPolynomial idx ℤ) (n) :
bind₁ (wittStructureInt p Φ) (wittPolynomial p ℤ n) =
bind₁ (fun i => rename (Prod.mk i) (W_ ℤ n)) Φ := by |
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
have := wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n
simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,
AlgHom.coe_toRingHom, map_wittStructureInt]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 448 | 450 | theorem iInf_coe_toSubmodule {ι} (p : ι → LieSubmodule R L M) :
(↑(⨅ i, p i) : Submodule R M) = ⨅ i, (p i : Submodule R M) := by |
rw [iInf, sInf_coe_toSubmodule]; ext; simp
|
/-
Copyright (c) 2019 Zhouhang Zhou. 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.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 818 | 819 | theorem integral_eq (f : α → E) (hf : Integrable f μ) : ∫ a, f a ∂μ = L1.integral (hf.toL1 f) := by |
simp [integral, hE, hf]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
#align_import category_theor... | Mathlib/CategoryTheory/Limits/IsLimit.lean | 836 | 838 | theorem comp_coconePointsIsoOfNatIso_inv {F G : J ⥤ C} {s : Cocone F} {t : Cocone G}
(P : IsColimit s) (Q : IsColimit t) (w : F ≅ G) (j : J) :
t.ι.app j ≫ (coconePointsIsoOfNatIso P Q w).inv = w.inv.app j ≫ s.ι.app j := by | simp
|
/-
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.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compac... | Mathlib/Topology/PartitionOfUnity.lean | 482 | 488 | theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι)
(ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) :
f.toPOUFun i x = f i x * ∏ j ∈ t.filter fun j => WellOrderingRel j i, (1 - f j x) := by |
refine congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun {j} hj => ?_)
rw [Ne, sub_eq_self] at hj
rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp]
exact flip (ht j) hj
|
/-
Copyright (c) 2017 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker, Jon Eugster
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Logic.Uni... | Mathlib/Algebra/Group/Units.lean | 754 | 755 | theorem isUnit_iff_exists_inv' [CommMonoid M] {a : M} : IsUnit a ↔ ∃ b, b * a = 1 := by |
simp [isUnit_iff_exists_inv, mul_comm]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 1,160 | 1,162 | theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
{x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by |
simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]; rfl
|
/-
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
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 1,022 | 1,029 | theorem mk_dvd_mk {a b : α} : Associates.mk a ∣ Associates.mk b ↔ a ∣ b := by |
simp only [dvd_def, mk_surjective.exists, mk_mul_mk, mk_eq_mk_iff_associated,
Associated.comm (x := b)]
constructor
· rintro ⟨x, u, rfl⟩
exact ⟨_, mul_assoc ..⟩
· rintro ⟨c, rfl⟩
use c
|
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb... | Mathlib/Data/Real/EReal.lean | 808 | 812 | theorem toReal_add {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :
toReal (x + y) = toReal x + toReal y := by |
lift x to ℝ using ⟨hx, h'x⟩
lift y to ℝ using ⟨hy, h'y⟩
rfl
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 1,208 | 1,208 | theorem nnabs_coe (x : ℝ≥0) : nnabs x = x := by | simp
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 127 | 131 | theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by |
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-communit... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 311 | 328 | theorem Gamma_eq_GammaAux (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) : Gamma s = GammaAux n s := by |
have u : ∀ k : ℕ, GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s := by
intro k; induction' k with k hk
· simp [Gamma]
· rw [← hk, ← add_assoc]
refine (GammaAux_recurrence2 s (⌊1 - s.re⌋₊ + k) ?_).symm
rw [Nat.cast_add]
have i0 := Nat.sub_one_lt_floor (1 - s.re)
simp only [sub_sub_cancel_l... |
/-
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.Tower
import Mathlib.RingTheory.Algebraic
import Mathlib.FieldTheory.Minpoly.Basic
#align_import field_theory.intermediate_field from "leanprove... | Mathlib/FieldTheory/IntermediateField.lean | 599 | 603 | theorem toSubalgebra_injective :
Function.Injective (IntermediateField.toSubalgebra : IntermediateField K L → _) := by |
intro S S' h
ext
rw [← mem_toSubalgebra, ← mem_toSubalgebra, h]
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
#align_import category_theory.limits.preserves... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean | 142 | 147 | theorem preservesZeroMorphisms_of_map_zero_object (i : F.obj 0 ≅ 0) : PreservesZeroMorphisms F where
map_zero X Y :=
calc
F.map (0 : X ⟶ Y) = F.map (0 : X ⟶ 0) ≫ F.map 0 := by | rw [← Functor.map_comp, comp_zero]
_ = F.map 0 ≫ (i.hom ≫ i.inv) ≫ F.map 0 := by rw [Iso.hom_inv_id, Category.id_comp]
_ = 0 := by simp only [zero_of_to_zero i.hom, zero_comp, comp_zero]
|
/-
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.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.eq... | Mathlib/Order/Partition/Equipartition.lean | 38 | 42 | theorem isEquipartition_iff_card_parts_eq_average :
P.IsEquipartition ↔
∀ a : Finset α,
a ∈ P.parts → a.card = s.card / P.parts.card ∨ a.card = s.card / P.parts.card + 1 := by |
simp_rw [IsEquipartition, Finset.equitableOn_iff, P.sum_card_parts]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 352 | 365 | theorem kreplace_self {a : α} {b : β a} {l : List (Sigma β)} (nd : NodupKeys l)
(h : Sigma.mk a b ∈ l) : kreplace a b l = l := by |
refine (lookmap_congr ?_).trans (lookmap_id' (Option.guard fun (s : Sigma β) => a = s.1) ?_ _)
· rintro ⟨a', b'⟩ h'
dsimp [Option.guard]
split_ifs
· subst a'
simp [nd.eq_of_mk_mem h h']
· rfl
· rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
dsimp [Option.guard]
split_ifs
· simp
· rintro ⟨⟩
|
/-
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 | 519 | 522 | theorem TransGen.lift' {p : β → β → Prop} {a b : α} (f : α → β)
(h : ∀ a b, r a b → TransGen p (f a) (f b)) (hab : TransGen r a b) :
TransGen p (f a) (f b) := by |
simpa [transGen_idem] using hab.lift f h
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 346 | 348 | theorem ContinuousLinearEquiv.comp_contDiffAt_iff (e : F ≃L[𝕜] G) :
ContDiffAt 𝕜 n (e ∘ f) x ↔ ContDiffAt 𝕜 n f x := by |
simp only [← contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 3,025 | 3,030 | theorem takeWhile_eq_nil_iff : takeWhile p l = [] ↔ ∀ hl : 0 < l.length, ¬p (l.nthLe 0 hl) := by |
induction' l with x xs IH
· simp only [takeWhile_nil, Bool.not_eq_true, true_iff]
intro h
simp at h
· by_cases hp : p x <;> simp [hp, takeWhile_cons, IH, nthLe_cons]
|
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.LightProfinite.Basic
import Mathlib.Topology.Category.Profinite.Limits
/-!
# Explicit limits and colimits
This file collects so... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 128 | 131 | theorem pullback_snd_eq :
LightProfinite.pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
|
/-
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.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import ring_theory.roots_of_unity.complex from "leanprover-c... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 53 | 55 | theorem isPrimitiveRoot_exp (n : ℕ) (h0 : n ≠ 0) : IsPrimitiveRoot (exp (2 * π * I / n)) n := by |
simpa only [Nat.cast_one, one_div] using
isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 698 | 699 | theorem leadingCoeff_eq_zero_iff_deg_eq_bot : leadingCoeff p = 0 ↔ degree p = ⊥ := by |
rw [leadingCoeff_eq_zero, degree_eq_bot]
|
/-
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
-/
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be4... | Mathlib/Order/Filter/Pi.lean | 165 | 166 | theorem pi_inf_principal_univ_pi_neBot :
NeBot (pi f ⊓ 𝓟 (Set.pi univ s)) ↔ ∀ i, NeBot (f i ⊓ 𝓟 (s i)) := by | simp [neBot_iff]
|
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.Measur... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 335 | 344 | theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by |
apply ae_of_forall_measure_lt_top_ae_restrict
intro t t_meas t_lt_top
apply ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter (hf_int_finite t t_meas t_lt_top)
intro s s_meas _
exact
hf_zero _ (s_meas.inter t_meas)
(lt_of_le_of_lt (measure_mono (Set.inter_subset_right)) t_lt_top)
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
/-!
# zip & u... | Mathlib/Data/List/Zip.lean | 254 | 256 | theorem reverse_revzip (l : List α) : reverse l.revzip = revzip l.reverse := by |
rw [← zip_unzip (revzip l).reverse]
simp [unzip_eq_map, revzip, map_reverse, map_fst_zip, map_snd_zip]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 670 | 670 | theorem smul_coe : x • (y : ℍ[R,c₁,c₂]) = ↑(x * y) := by | rw [coe_mul, coe_mul_eq_smul]
|
/-
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.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 110 | 114 | theorem t'_fst_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
simp only [t', Category.assoc, pullbackSymmetry_hom_comp_fst_assoc,
pullbackRightPullbackFstIso_inv_snd_fst_assoc, pullback.lift_fst_assoc, t_fst_fst,
pullbackRightPullbackFstIso_hom_fst_assoc]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 608 | 612 | theorem index_center_le_pow [Finite (commutatorSet G)] [Group.FG G] :
(center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G := by |
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]
exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Dual
/-!
# Matroid Restriction
Given `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`
are the independ... | Mathlib/Data/Matroid/Restrict.lean | 299 | 300 | theorem StrictRestriction.ne (h : N <r M) : N ≠ M := by |
rintro rfl; rw [← ofMatroid_lt_iff] at h; simp at h
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
# Basic... | Mathlib/Topology/Bornology/Basic.lean | 143 | 144 | theorem isBounded_compl_iff : IsBounded sᶜ ↔ IsCobounded s := by |
rw [isBounded_def, isCobounded_def, compl_compl]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Hom.Defs
#align_import data.matrix.dmatrix from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Depende... | Mathlib/Data/Matrix/DMatrix.lean | 143 | 146 | theorem map_add [∀ i j, AddMonoid (α i j)] {β : m → n → Type w} [∀ i j, AddMonoid (β i j)]
(f : ∀ ⦃i j⦄, α i j →+ β i j) (M N : DMatrix m n α) :
((M + N).map fun i j => @f i j) = (M.map fun i j => @f i j) + N.map fun i j => @f i j := by |
ext; simp
|
/-
Copyright (c) 2019 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.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,394 | 1,398 | theorem coprodComparison_inv_natural (f : A ⟶ A') (g : B ⟶ B') [IsIso (coprodComparison F A B)]
[IsIso (coprodComparison F A' B')] :
inv (coprodComparison F A B) ≫ coprod.map (F.map f) (F.map g) =
F.map (coprod.map f g) ≫ inv (coprodComparison F A' B') := by |
rw [IsIso.eq_comp_inv, Category.assoc, IsIso.inv_comp_eq, coprodComparison_natural]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ModularLattice
import Mathlib.Order.WellFounded
import Mathlib.Tactic.Nontriviality
#align_import order.atoms fr... | Mathlib/Order/Atoms.lean | 828 | 841 | theorem isAtom_iff [OrderBot α] [IsAtomic α] [OrderBot β] {l : α → β} {u : β → α}
(gi : GaloisInsertion l u) (hbot : u ⊥ = ⊥) (h_atom : ∀ a, IsAtom a → u (l a) = a) (a : α) :
IsAtom (l a) ↔ IsAtom a := by |
refine ⟨fun hla => ?_, fun ha => gi.isAtom_of_u_bot hbot ((h_atom a ha).symm ▸ ha)⟩
obtain ⟨a', ha', hab'⟩ :=
(eq_bot_or_exists_atom_le (u (l a))).resolve_left (hbot ▸ fun h => hla.1 (gi.u_injective h))
have :=
(hla.le_iff.mp <| (gi.l_u_eq (l a) ▸ gi.gc.monotone_l hab' : l a' ≤ l a)).resolve_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.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 994 | 996 | theorem measurable_uniqueElim [Unique δ] [∀ i, MeasurableSpace (π i)] :
Measurable (uniqueElim : π (default : δ) → ∀ i, π i) := by |
simp_rw [measurable_pi_iff, Unique.forall_iff, uniqueElim_default]; exact measurable_id
|
/-
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
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 276 | 279 | theorem mapAccumr₂_eq_map₂_of_constant_state (f : α → β → σ → σ × γ) (s : σ)
(h : ∀ a b, (f a b s).fst = s) :
mapAccumr₂ f xs ys s = (s, (map₂ (fun x y => (f x y s).snd) xs ys)) := by |
induction xs, ys using revInductionOn₂ <;> simp_all
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import M... | Mathlib/CategoryTheory/Filtered/Basic.lean | 806 | 822 | theorem cone_nonempty (F : J ⥤ C) : Nonempty (Cone F) := by |
classical
let O := Finset.univ.image F.obj
let H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y) :=
Finset.univ.biUnion fun X : J =>
Finset.univ.biUnion fun Y : J =>
Finset.univ.image fun f : X ⟶ Y => ⟨F.obj X, F.obj Y, by simp [O], by simp [O], F.map f⟩
obtain ⟨Z, f, w⟩ := inf_exists ... |
/-
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 | 127 | 129 | theorem fourier_coe_apply' {n : ℤ} {x : ℝ} :
toCircle (n • (x : AddCircle T) :) = Complex.exp (2 * π * Complex.I * n * x / T) := by |
rw [← fourier_apply]; exact fourier_coe_apply
|
/-
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.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.GroupTheory.Congruence.Basic
#align_import... | Mathlib/RingTheory/Congruence.lean | 564 | 565 | theorem sup_def {c d : RingCon R} : c ⊔ d = ringConGen (⇑c ⊔ ⇑d) := by |
rw [sup_eq_ringConGen]; rfl
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11... | Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean | 63 | 72 | theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppredB := by |
obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.b = ppredB :=
exists_conts_b_of_denom nth_denom_eq
obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.b = predB :=
exists_conts_b_of_denom succ_nth_denom_eq
rw [denom_eq_co... |
/-
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
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 263 | 264 | theorem card_image_le [DecidableEq β] : (s.image f).card ≤ s.card := by |
simpa only [card_map] using (s.1.map f).toFinset_card_le
|
/-
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.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#a... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 832 | 839 | theorem isCycle_iff_exists_isCycleOn :
f.IsCycle ↔ ∃ s : Set α, s.Nontrivial ∧ f.IsCycleOn s ∧ ∀ ⦃x⦄, ¬IsFixedPt f x → x ∈ s := by |
refine ⟨fun hf => ⟨{ x | f x ≠ x }, ?_, hf.isCycleOn, fun _ => id⟩, ?_⟩
· obtain ⟨a, ha⟩ := hf
exact ⟨f a, f.injective.ne ha.1, a, ha.1, ha.1⟩
· rintro ⟨s, hs, hf, hsf⟩
obtain ⟨a, ha⟩ := hs.nonempty
exact ⟨a, hf.apply_ne hs ha, fun b hb => hf.2 ha <| hsf hb⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 394 | 394 | theorem arccos_neg_one : arccos (-1) = π := by | simp [arccos, add_halves]
|
/-
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.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Sites.EqualizerShe... | Mathlib/CategoryTheory/Sites/Preserves.lean | 97 | 109 | theorem piComparison_fac :
have : HasCoproduct X := ⟨⟨c, hc⟩⟩
piComparison F (fun x ↦ op (X x)) = F.map (opCoproductIsoProduct' hc (productIsProduct _)).inv ≫
Equalizer.Presieve.Arrows.forkMap F X c.inj := by |
have : HasCoproduct X := ⟨⟨c, hc⟩⟩
dsimp only [Equalizer.Presieve.Arrows.forkMap]
have h : Pi.lift (fun i ↦ F.map (c.inj i).op) =
F.map (Pi.lift (fun i ↦ (c.inj i).op)) ≫ piComparison F _ := by simp
rw [h, ← Category.assoc, ← Functor.map_comp]
have hh : Pi.lift (fun i ↦ (c.inj i).op) = (productIsProduc... |
/-
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 | 315 | 356 | theorem map_setToSimpleFunc (T : Set α → F →L[ℝ] F') (h_add : FinMeasAdditive μ T) {f : α →ₛ G}
(hf : Integrable f μ) {g : G → F} (hg : g 0 = 0) :
(f.map g).setToSimpleFunc T = ∑ x ∈ f.range, T (f ⁻¹' {x}) (g x) := by |
have T_empty : T ∅ = 0 := h_add.map_empty_eq_zero
have hfp : ∀ x ∈ f.range, x ≠ 0 → μ (f ⁻¹' {x}) ≠ ∞ := fun x _ hx0 =>
(measure_preimage_lt_top_of_integrable f hf hx0).ne
simp only [setToSimpleFunc, range_map]
refine Finset.sum_image' _ fun b hb => ?_
rcases mem_range.1 hb with ⟨a, rfl⟩
by_cases h0 : ... |
/-
Copyright (c) 2019 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.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 66 | 69 | theorem hasDerivWithinAt_iff_tendsto_slope :
HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s \ {x}] x) (𝓝 f') := by |
simp only [HasDerivWithinAt, nhdsWithin, diff_eq, ← inf_assoc, inf_principal.symm]
exact hasDerivAtFilter_iff_tendsto_slope
|
/-
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 | 347 | 354 | theorem integral_cexp_neg_mul_sq_norm_add
(hb : 0 < b.re) (c : ℂ) (w : V) :
∫ v : V, cexp (- b * ‖v‖^2 + c * ⟪w, v⟫) =
(π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (c ^ 2 * ‖w‖^2 / (4 * b)) := by |
let e := (stdOrthonormalBasis ℝ V).repr.symm
rw [← e.measurePreserving.integral_comp e.toHomeomorph.measurableEmbedding]
convert integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
hb c (e.symm w) <;> simp [LinearIsometryEquiv.inner_map_eq_flip]
|
/-
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.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 388 | 398 | theorem IsPreconnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β}
(hs : IsPreconnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f)
(hsf : s ⊆ range f) : IsPreconnected (f ⁻¹' s) :=
isPreconnected_closed_iff.2 fun u v hu hv hsuv hsu hsv => by
replace hsf : f '' (f ⁻¹' s)... |
refine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_
· simpa only [hsf, image_union] using image_subset f hsuv
· simpa only [image_preimage_inter] using hsu.image f
· simpa only [image_preimage_inter] using hsv.image f
· exact ⟨a, has, hau, hinj.mem_set_image... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 588 | 596 | theorem isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i)) (U : Set X) :
IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by |
constructor
· exact fun ⟨h₁, h₂⟩ ↦ eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open _ hb U h₁ h₂
· rintro ⟨s, hs, rfl⟩
constructor
· exact hs.isCompact_biUnion fun i _ => hb' i
· exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _)
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Eric Wieser
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.Tacti... | Mathlib/Algebra/Group/Pi/Basic.lean | 404 | 410 | theorem apply_mulSingle₂ (f' : ∀ i, f i → g i → h i) (hf' : ∀ i, f' i 1 1 = 1) (i : I)
(x : f i) (y : g i) (j : I) :
f' j (mulSingle i x j) (mulSingle i y j) = mulSingle i (f' i x y) j := by |
by_cases h : j = i
· subst h
simp only [mulSingle_eq_same]
· simp only [mulSingle_eq_of_ne h, hf']
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathl... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 136 | 138 | theorem dist_center_eq_dist_center_of_mem_sphere {p₁ p₂ : P} {s : Sphere P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) : dist p₁ s.center = dist p₂ s.center := by |
rw [mem_sphere.1 hp₁, mem_sphere.1 hp₂]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,512 | 1,518 | theorem Iic_union_Icc (h : min c d ≤ b) : Iic b ∪ Icc c d = Iic (max b d) := by |
rcases le_or_lt c d with hcd | hcd <;> simp [hcd] at h
· exact Iic_union_Icc' h
· cases' h with h h
· have hdb : d ≤ b := hcd.le.trans h
simp [*]
· 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 | 795 | 800 | theorem IsOrthoᵢ.not_isOrtho_basis_self_of_separatingRight [Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁} {v : Basis n R M} (h : B.IsOrthoᵢ v) (hB : B.SeparatingRight)
(i : n) : ¬B.IsOrtho (v i) (v i) := by |
rw [isOrthoᵢ_flip] at h
rw [isOrtho_flip]
exact h.not_isOrtho_basis_self_of_separatingLeft (flip_separatingLeft.mpr hB) i
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 334 | 336 | theorem midpoint_mem_segment [Invertible (2 : 𝕜)] (x y : E) : midpoint 𝕜 x y ∈ [x -[𝕜] y] := by |
rw [segment_eq_image_lineMap]
exact ⟨⅟ 2, ⟨invOf_nonneg.mpr zero_le_two, invOf_le_one one_le_two⟩, rfl⟩
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canoni... | Mathlib/Algebra/Order/Sub/Canonical.lean | 333 | 334 | theorem tsub_eq_zero_iff_le : a - b = 0 ↔ a ≤ b := by |
rw [← nonpos_iff_eq_zero, tsub_le_iff_left, add_zero]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/math... | Mathlib/Probability/Kernel/CondDistrib.lean | 171 | 174 | theorem _root_.MeasureTheory.Integrable.norm_integral_condDistrib_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
Integrable (fun x => ‖∫ y, f (x, y) ∂condDistrib Y X μ x‖) (μ.map X) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.norm_integral_condKernel
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 57 | 69 | theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*} [LinearOrderedCommRing β] {f : α → β}
[DecidablePred fun x => f x ≤ 0] {s : Finset α} (h0 : Even (s.filter fun x => f x ≤ 0).card) :
0 ≤ ∏ x ∈ s, f x :=
calc
0 ≤ ∏ x ∈ s, (if f x ≤ 0 then (-1 : β) else 1) * f x :=
Finset.prod_nonneg fun x ... |
rw [Finset.prod_mul_distrib, Finset.prod_ite, Finset.prod_const_one, mul_one,
Finset.prod_const, neg_one_pow_eq_pow_mod_two, Nat.even_iff.1 h0, pow_zero, one_mul]
|
/-
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 | 1,483 | 1,486 | theorem tendsto_atTop_embedding [Preorder β] [Preorder γ] {f : α → β} {e : β → γ} {l : Filter α}
(hm : ∀ b₁ b₂, e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ c, ∃ b, c ≤ e b) :
Tendsto (e ∘ f) l atTop ↔ Tendsto f l atTop := by |
rw [← comap_embedding_atTop hm hu, tendsto_comap_iff]
|
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417... | Mathlib/Algebra/Polynomial/Lifts.lean | 69 | 70 | theorem lifts_iff_ringHom_rangeS (p : S[X]) : p ∈ lifts f ↔ p ∈ (mapRingHom f).rangeS := by |
simp only [coe_mapRingHom, lifts, Set.mem_range, RingHom.mem_rangeS]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 871 | 875 | theorem dotProduct_single (x : α) (i : m) : v ⬝ᵥ Pi.single i x = v i * x := by |
-- Porting note: (implicit arg) added `(f := fun _ => α)`
have : ∀ j ≠ i, v j * Pi.single (f := fun _ => α) i x j = 0 := fun j hij => by
simp [Pi.single_eq_of_ne hij]
convert Finset.sum_eq_single i (fun j _ => this j) _ using 1 <;> simp
|
/-
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.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.r... | Mathlib/Data/ENNReal/Operations.lean | 130 | 130 | theorem not_lt_zero : ¬a < 0 := by | simp
|
/-
Copyright (c) 2020. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Massot
-/
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FailIfNoProgress
import Mathlib.Algebra.Group.Commutator
#align_import tactic.group from "leanprover-community/mat... | Mathlib/Tactic/Group.lean | 43 | 44 | theorem zpow_trick_one {G : Type*} [Group G] (a b : G) (m : ℤ) :
a * b * b ^ m = a * b ^ (m + 1) := by | rw [mul_assoc, mul_self_zpow]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 1,468 | 1,473 | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) := by |
simp only [expNear, mul_zero, add_zero]
convert exp_bound (n := m) h ?_ using 1
· field_simp [mul_comm]
· omega
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 643 | 657 | theorem nodupKeys_dedupKeys (l : List (Sigma β)) : NodupKeys (dedupKeys l) := by |
dsimp [dedupKeys]
generalize hl : nil = l'
have : NodupKeys l' := by
rw [← hl]
apply nodup_nil
clear hl
induction' l with x xs l_ih
· apply this
· cases x
simp only [foldr_cons, kinsert_def, nodupKeys_cons, ne_eq, not_true]
constructor
· simp only [keys_kerase]
apply l_ih.not_me... |
/-
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 | 476 | 477 | theorem det_nonsing_inv_mul_det (h : IsUnit A.det) : A⁻¹.det * A.det = 1 := by |
rw [← det_mul, A.nonsing_inv_mul h, det_one]
|
/-
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.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "l... | Mathlib/Order/SuccPred/Basic.lean | 890 | 891 | theorem Ico_pred_right_eq_insert (h : a ≤ b) : Ioc (pred a) b = insert a (Ioc a b) := by |
simp_rw [← Ioi_inter_Iic, Ioi_pred_eq_insert, insert_inter_of_mem (mem_Iic.2 h)]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 901 | 909 | theorem Orthonormal.orthonormal_of_forall_eq_or_eq_neg {v w : ι → E} (hv : Orthonormal 𝕜 v)
(hw : ∀ i, w i = v i ∨ w i = -v i) : Orthonormal 𝕜 w := by |
classical
rw [orthonormal_iff_ite] at *
intro i j
cases' hw i with hi hi <;> cases' hw j with hj hj <;>
replace hv := hv i j <;> split_ifs at hv ⊢ with h <;>
simpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true,
neg_eq_zero] using hv
|
/-
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.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf... | Mathlib/Data/Nat/WithBot.lean | 35 | 40 | theorem add_eq_one_iff {n m : WithBot ℕ} : n + m = 1 ↔ n = 0 ∧ m = 1 ∨ n = 1 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_one_iff
|
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