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) 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 | 1,535 | 1,536 | theorem real_inner_smul_self_right (x : F) (r : ℝ) : ⟪x, r • x⟫_ℝ = r * (‖x‖ * ‖x‖) := by |
rw [inner_smul_right, ← real_inner_self_eq_norm_mul_norm]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 259 | 262 | theorem inj_right {w : σ → M} (hφ : φ ≠ 0) (hm : IsWeightedHomogeneous w φ m)
(hn : IsWeightedHomogeneous w φ n) : m = n := by |
obtain ⟨d, hd⟩ : ∃ d, coeff d φ ≠ 0 := exists_coeff_ne_zero hφ
rw [← hm hd, ← hn hd]
|
/-
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 | 484 | 486 | theorem Ico_succ_right_eq_insert_of_not_isMax (h₁ : a ≤ b) (h₂ : ¬IsMax b) :
Ico a (succ b) = insert b (Ico a b) := by |
simp_rw [← Iio_inter_Ici, Iio_succ_eq_insert_of_not_isMax h₂, insert_inter_of_mem (mem_Ici.2 h₁)]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathli... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 195 | 198 | theorem zeta_eq_tsum_one_div_nat_cpow {s : ℂ} (hs : 1 < re s) :
riemannZeta s = ∑' n : ℕ, 1 / (n : ℂ) ^ s := by |
simpa only [QuotientAddGroup.mk_zero, cosZeta_zero, mul_zero, zero_mul, Real.cos_zero,
ofReal_one] using (hasSum_nat_cosZeta 0 hs).tsum_eq.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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 1,292 | 1,295 | theorem mem_closure_iff_comap_neBot :
x ∈ closure s ↔ NeBot (comap ((↑) : s → X) (𝓝 x)) := by |
simp_rw [mem_closure_iff_nhds, comap_neBot_iff, Set.inter_nonempty_iff_exists_right,
SetCoe.exists, exists_prop]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-co... | Mathlib/Order/Interval/Set/ProjIcc.lean | 300 | 308 | theorem IccExtend_eq_self (f : α → β) (ha : ∀ x < a, f x = f a) (hb : ∀ x, b < x → f x = f b) :
IccExtend h (f ∘ (↑)) = f := by |
ext x
cases' lt_or_le x a with hxa hax
· simp [IccExtend_of_le_left _ _ hxa.le, ha x hxa]
· rcases le_or_lt x b with hxb | hbx
· lift x to Icc a b using ⟨hax, hxb⟩
rw [IccExtend_val, comp_apply]
· simp [IccExtend_of_right_le _ _ hbx.le, hb x hbx]
|
/-
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.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 102 | 114 | theorem contLinear_eq_zero_iff_exists_const (f : P →ᴬ[R] Q) :
f.contLinear = 0 ↔ ∃ q, f = const R P q := by |
have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun ... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,016 | 1,019 | theorem continuousOn_extend : ContinuousOn (f.extend I) (f.extend I).source := by |
refine I.continuous.comp_continuousOn ?_
rw [extend_source]
exact f.continuousOn
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 878 | 884 | theorem snormEssSup_le_nnreal_smul_snormEssSup_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ≥0}
(h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) : snormEssSup f μ ≤ c • snormEssSup g μ :=
calc
essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ ≤ essSup (fun x => (↑(c * ‖g x‖₊) : ℝ≥0∞)) μ :=
essSup_mono_ae <| h.mono fun x hx => ENNRea... | simp_rw [ENNReal.coe_mul]
_ = c • essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ := ENNReal.essSup_const_mul
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,900 | 1,912 | theorem exists_eq_add_of_lt {n : ℕ} {a b : Fin (n + 1)} (h : a < b) :
∃ k < b, k + 1 ≤ b ∧ b = a + k + 1 := by |
cases n
· cases' a with a ha
cases' b with b hb
simp only [Nat.zero_eq, Nat.zero_add, Nat.lt_one_iff] at ha hb
simp [ha, hb] at h
obtain ⟨k, hk⟩ : ∃ k : ℕ, (b : ℕ) = a + k + 1 := Nat.exists_eq_add_of_lt h
have hkb : k < b := by omega
refine ⟨⟨k, hkb.trans b.is_lt⟩, hkb, ?_, ?_⟩
· rw [Fin.le_iff... |
/-
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.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 164 | 170 | theorem comap_uniformity_le (hf : AntilipschitzWith K f) : (𝓤 β).comap (Prod.map f f) ≤ 𝓤 α := by |
refine ((uniformity_basis_edist.comap _).le_basis_iff uniformity_basis_edist).2 fun ε h₀ => ?_
refine ⟨(↑K)⁻¹ * ε, ENNReal.mul_pos (ENNReal.inv_ne_zero.2 ENNReal.coe_ne_top) h₀.ne', ?_⟩
refine fun x hx => (hf x.1 x.2).trans_lt ?_
rw [mul_comm, ← div_eq_mul_inv] at hx
rw [mul_comm]
exact ENNReal.mul_lt_of_l... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 83 | 90 | theorem det_comm' [DecidableEq m] [DecidableEq n] {M : Matrix n m A} {N : Matrix m n A}
{M' : Matrix m n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) : det (M * N) = det (N * M) := by |
nontriviality A
-- Although `m` and `n` are different a priori, we will show they have the same cardinality.
-- This turns the problem into one for square matrices, which is easy.
let e := indexEquivOfInv hMM' hM'M
rw [← det_submatrix_equiv_self e, ← submatrix_mul_equiv _ _ _ (Equiv.refl n) _, det_comm,
... |
/-
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 Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 774 | 776 | theorem replaceF_cons_of_some {l : List α} (p) (h : p a = some a') :
(a :: l).replaceF p = a' :: l := by |
simp [replaceF_cons, h]
|
/-
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.Inverse
import Mathlib.Analysis.SpecialFunctions... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 30 | 49 | theorem deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x := by |
cases' h₁.lt_or_lt with h₁ h₁
· have : 1 - x ^ 2 < 0 := by nlinarith [h₁]
rw [sqrt_eq_zero'.2 this.le, div_zero]
have : arcsin =ᶠ[𝓝 x] fun _ => -(π / 2) :=
(gt_mem_nhds h₁).mono fun y hy => arcsin_of_le_neg_one hy.le
exact ⟨(hasStrictDerivAt_const _ _).congr_of_eventuallyEq this.symm,
cont... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 1,972 | 1,976 | theorem mem_sup {α β} [DecidableEq β] {s : Finset α} {f : α → Finset β} {x : β} :
x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v := by |
change _ ↔ ∃ v ∈ s, x ∈ (f v).val
rw [← Multiset.mem_sup, ← Multiset.mem_toFinset, sup_toFinset]
simp_rw [val_toFinset]
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-comm... | Mathlib/Data/Nat/Log.lean | 284 | 286 | theorem clog_eq_one {b n : ℕ} (hn : 2 ≤ n) (h : n ≤ b) : clog b n = 1 := by |
rw [clog_of_two_le (hn.trans h) hn, clog_of_right_le_one]
rw [← Nat.lt_succ_iff, Nat.div_lt_iff_lt_mul] <;> omega
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,557 | 1,558 | theorem NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z := by |
simp only [NNReal.nndist_eq, max_eq_right, tsub_zero, zero_tsub, zero_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.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,500 | 2,505 | theorem prod_sum {α : Type*} {ι : Type*} [CommMonoid α] (f : ι → Multiset α) (s : Finset ι) :
(∑ x ∈ s, f x).prod = ∏ x ∈ s, (f x).prod := by |
classical
induction' s using Finset.induction_on with a t hat ih
· rw [Finset.sum_empty, Finset.prod_empty, Multiset.prod_zero]
· rw [Finset.sum_insert hat, Finset.prod_insert hat, Multiset.prod_add, ih]
|
/-
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 | 82 | 86 | theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ = f.lintegral μ := by |
rw [MeasureTheory.lintegral]
exact le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <| le_rfl)
(le_iSup₂_of_le f le_rfl le_rfl)
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
/-!
# The Turán graph
This file defines the Turán graph and proves some of its basic properties.
## Main declarations
* `S... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 54 | 62 | theorem turanGraph_eq_top : turanGraph n r = ⊤ ↔ r = 0 ∨ n ≤ r := by |
simp_rw [SimpleGraph.ext_iff, Function.funext_iff, turanGraph, top_adj, eq_iff_iff, not_iff_not]
refine ⟨fun h ↦ ?_, ?_⟩
· contrapose! h
use ⟨0, (Nat.pos_of_ne_zero h.1).trans h.2⟩, ⟨r, h.2⟩
simp [h.1.symm]
· rintro (rfl | h) a b
· simp [Fin.val_inj]
· rw [Nat.mod_eq_of_lt (a.2.trans_le h), Nat... |
/-
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 | 1,123 | 1,123 | theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) := by | rw [← neg_sub, cosh_sub_sinh]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat... | Mathlib/Data/Nat/Multiplicity.lean | 288 | 312 | theorem multiplicity_two_factorial_lt : ∀ {n : ℕ} (_ : n ≠ 0), multiplicity 2 n ! < n := by |
have h2 := prime_two.prime
refine binaryRec ?_ ?_
· exact fun h => False.elim <| h rfl
· intro b n ih h
by_cases hn : n = 0
· subst hn
simp only [ne_eq, bit_eq_zero, true_and, Bool.not_eq_false] at h
simp only [h, bit_true, bit1_zero, factorial, mul_one, Nat.isUnit_iff, cast_one]
rw [... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | Mathlib/Topology/Sheaves/Stalks.lean | 174 | 179 | theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom := by |
ext
simp only [stalkPushforward, germ, colim_map, ι_colimMap_assoc, whiskerRight_app]
erw [CategoryTheory.Functor.map_id]
simp [stalkFunctor]
|
/-
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 | 384 | 388 | theorem ite_le_natDegree_coeff (p : R[X]) (n : ℕ) (I : Decidable (n < 1 + natDegree p)) :
@ite _ (n < 1 + natDegree p) I (coeff p n) 0 = coeff p n := by |
split_ifs with h
· rfl
· exact (coeff_eq_zero_of_natDegree_lt (not_le.1 fun w => h (Nat.lt_one_add_iff.2 w))).symm
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 893 | 896 | theorem map_comap (μ : Measure β) : (comap f μ).map f = μ.restrict (range f) := by |
ext1 t ht
rw [hf.map_apply, comap_apply f hf.injective hf.measurableSet_image' _ (hf.measurable ht),
image_preimage_eq_inter_range, 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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 659 | 660 | theorem mul_eq_right {a b : Cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) (ha' : a ≠ 0) : a * b = b := by |
rw [mul_comm, mul_eq_left hb ha ha']
|
/-
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.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 272 | 274 | theorem swap_mem_interedges_iff {x : α × α} : x.swap ∈ interedges r s t ↔ x ∈ interedges r t s := by |
rw [mem_interedges_iff, mem_interedges_iff, hr.iff]
exact and_left_comm
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsup... | Mathlib/LinearAlgebra/DFinsupp.lean | 307 | 315 | theorem iSup_eq_range_dfinsupp_lsum (p : ι → Submodule R N) :
iSup p = LinearMap.range (DFinsupp.lsum ℕ (M := fun i ↦ ↥(p i)) fun i => (p i).subtype) := by |
apply le_antisymm
· apply iSup_le _
intro i y hy
simp only [LinearMap.mem_range, lsum_apply_apply]
exact ⟨DFinsupp.single i ⟨y, hy⟩, DFinsupp.sumAddHom_single _ _ _⟩
· rintro x ⟨v, rfl⟩
exact dfinsupp_sumAddHom_mem _ v _ fun i _ => (le_iSup p i : p i ≤ _) (v i).2
|
/-
Copyright (c) 2022 Hans Parshall. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hans Parshall
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Matrix
import Mathlib.Analysis.RCLike.Basic
import Mathlib.LinearAlgebra.UnitaryGroup
import M... | Mathlib/Analysis/NormedSpace/Star/Matrix.lean | 49 | 77 | theorem entry_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜)
(i j : n) : ‖U i j‖ ≤ 1 := by |
-- The norm squared of an entry is at most the L2 norm of its row.
have norm_sum : ‖U i j‖ ^ 2 ≤ ∑ x, ‖U i x‖ ^ 2 := by
apply Multiset.single_le_sum
· intro x h_x
rw [Multiset.mem_map] at h_x
cases' h_x with a h_a
rw [← h_a.2]
apply sq_nonneg
· rw [Multiset.mem_map]
use j
... |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 142 | 150 | theorem zmod_congr_of_sub_mem_span_aux (n : ℕ) (x : ℤ_[p]) (a b : ℤ)
(ha : x - a ∈ (Ideal.span {(p : ℤ_[p]) ^ n}))
(hb : x - b ∈ (Ideal.span {(p : ℤ_[p]) ^ n})) : (a : ZMod (p ^ n)) = b := by |
rw [Ideal.mem_span_singleton] at ha hb
rw [← sub_eq_zero, ← Int.cast_sub, ZMod.intCast_zmod_eq_zero_iff_dvd, Int.natCast_pow]
rw [← dvd_neg, neg_sub] at ha
have := dvd_add ha hb
rwa [sub_eq_add_neg, sub_eq_add_neg, add_assoc, neg_add_cancel_left, ← sub_eq_add_neg, ←
Int.cast_sub, pow_p_dvd_int_iff] at th... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Wieser
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.p... | Mathlib/Data/Real/Pointwise.lean | 37 | 46 | theorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRight ha').map_csInf' hs h).symm
· rw [Real.sInf_of_not_bddBelow (mt (b... |
/-
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 | 425 | 426 | theorem one_le_one_div (h1 : 0 < a) (h2 : a ≤ 1) : 1 ≤ 1 / a := by |
rwa [le_one_div (@zero_lt_one α _ _ _ _ _) h1, one_div_one]
|
/-
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 | 845 | 853 | theorem eq_pow_of_pow_eq_one {k : ℕ} {ζ ξ : R} (h : IsPrimitiveRoot ζ k) (hξ : ξ ^ k = 1)
(h0 : 0 < k) : ∃ i < k, ζ ^ i = ξ := by |
lift ζ to Rˣ using h.isUnit h0
lift ξ to Rˣ using isUnit_ofPowEqOne hξ h0.ne'
lift k to ℕ+ using h0
simp only [← Units.val_pow_eq_pow_val, ← Units.ext_iff]
rw [coe_units_iff] at h
apply h.eq_pow_of_mem_rootsOfUnity
rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, hξ, Units.val_one]
|
/-
Copyright (c) 2022 Matej Penciak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matej Penciak, Moritz Doll, Fabien Clery
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 189 | 198 | theorem inv_left_mul_aux (hA : A ∈ symplecticGroup l R) : -(J l R * Aᵀ * J l R * A) = 1 :=
calc
-(J l R * Aᵀ * J l R * A) = (-J l R) * (Aᵀ * J l R * A) := by |
simp only [Matrix.mul_assoc, Matrix.neg_mul]
_ = (-J l R) * J l R := by
rw [mem_iff'] at hA
rw [hA]
_ = (-1 : R) • (J l R * J l R) := by simp only [Matrix.neg_mul, neg_smul, one_smul]
_ = (-1 : R) • (-1 : Matrix _ _ _) := by rw [J_squared]
_ = 1 := by simp only [neg_smul_neg, one_smul... |
/-
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 | 161 | 165 | theorem fromBlocks_submatrix_sum_swap_left (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (f : p → Sum l m) :
(fromBlocks A B C D).submatrix Sum.swap f = (fromBlocks C D A B).submatrix id f := by |
ext i j
cases i <;> dsimp <;> cases f j <;> rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 111 | 111 | theorem log_neg_eq_log (x : ℝ) : log (-x) = log x := by | rw [← log_abs x, ← log_abs (-x), abs_neg]
|
/-
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 | 1,120 | 1,123 | theorem re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two (x y : E) :
re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2 := by |
rw [@norm_add_mul_self 𝕜]
ring
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-co... | Mathlib/Order/Interval/Set/ProjIcc.lean | 72 | 73 | theorem projIcc_of_le_left (hx : x ≤ a) : projIcc a b h x = ⟨a, left_mem_Icc.2 h⟩ := by |
simp [projIcc, hx, hx.trans h]
|
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Coalgebras
In this file we define... | Mathlib/RingTheory/Coalgebra/Basic.lean | 133 | 135 | theorem comul_comp_inr :
comul ∘ₗ inr R A B = TensorProduct.map (.inr R A B) (.inr R A B) ∘ₗ comul := by |
ext; simp
|
/-
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.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.quotient_nilpotent from "leanprover-community/mathl... | Mathlib/RingTheory/QuotientNilpotent.lean | 54 | 78 | theorem IsNilpotent.isUnit_quotient_mk_iff {R : Type*} [CommRing R] {I : Ideal R}
(hI : IsNilpotent I) {x : R} : IsUnit (Ideal.Quotient.mk I x) ↔ IsUnit x := by |
refine ⟨?_, fun h => h.map <| Ideal.Quotient.mk I⟩
revert x
apply Ideal.IsNilpotent.induction_on (R := R) (S := R) I hI <;> clear hI I
swap
· introv e h₁ h₂ h₃
apply h₁
apply h₂
exact
h₃.map
((DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_righ... |
/-
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, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 752 | 754 | theorem fst_comp_prodComm :
(LinearMap.fst R M₂ M).comp (prodComm R M M₂).toLinearMap = (LinearMap.snd R M M₂) := by |
ext <;> simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 488 | 493 | theorem ListBlank.cons_bind {Γ Γ'} [Inhabited Γ] [Inhabited Γ'] (a : Γ) (l : ListBlank Γ)
(f : Γ → List Γ') (hf) : (l.cons a).bind f hf = (l.bind f hf).append (f a) := by |
refine l.inductionOn fun l ↦ ?_
-- Porting note: Added `suffices` to get `simp` to work.
suffices ((mk l).cons a).bind f hf = ((mk l).bind f hf).append (f a) by exact this
simp only [ListBlank.append_mk, ListBlank.bind_mk, ListBlank.cons_mk, List.cons_bind]
|
/-
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.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 58 | 71 | theorem normBound_pos : 0 < normBound abv bS := by |
obtain ⟨i, j, k, hijk⟩ : ∃ i j k, Algebra.leftMulMatrix bS (bS i) j k ≠ 0 := by
by_contra! h
obtain ⟨i⟩ := bS.index_nonempty
apply bS.ne_zero i
apply
(injective_iff_map_eq_zero (Algebra.leftMulMatrix bS)).mp (Algebra.leftMulMatrix_injective bS)
ext j k
simp [h, DMatrix.zero_apply]
sim... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 2,168 | 2,173 | theorem exists_open_nhds_disjoint_closure [T25Space X] {x y : X} (h : x ≠ y) :
∃ u : Set X,
x ∈ u ∧ IsOpen u ∧ ∃ v : Set X, y ∈ v ∧ IsOpen v ∧ Disjoint (closure u) (closure v) := by |
simpa only [exists_prop, and_assoc] using
((nhds_basis_opens x).lift'_closure.disjoint_iff (nhds_basis_opens y).lift'_closure).1
(disjoint_lift'_closure_nhds.2 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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 219 | 223 | theorem Set.Finite.isClosed_biUnion {s : Set α} {f : α → Set X} (hs : s.Finite)
(h : ∀ i ∈ s, IsClosed (f i)) :
IsClosed (⋃ i ∈ s, f i) := by |
simp only [← isOpen_compl_iff, compl_iUnion] at *
exact hs.isOpen_biInter h
|
/-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Option.Basic
import Mathlib.SetTheory.Cardinal.Basic
#al... | Mathlib/Computability/Encoding.lean | 134 | 140 | theorem decode_encodePosNum : ∀ n, decodePosNum (encodePosNum n) = n := by |
intro n
induction' n with m hm m hm <;> unfold encodePosNum decodePosNum
· rfl
· rw [hm]
exact if_neg (encodePosNum_nonempty m)
· exact congr_arg PosNum.bit0 hm
|
/-
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.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 1,037 | 1,046 | theorem ncard_le_one_iff_eq (hs : s.Finite := by | toFinite_tac) :
s.ncard ≤ 1 ↔ s = ∅ ∨ ∃ a, s = {a} := by
obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
· exact iff_of_true (by simp) (Or.inl rfl)
rw [ncard_le_one_iff hs]
refine ⟨fun h ↦ Or.inr ⟨x, (singleton_subset_iff.mpr hx).antisymm' fun y hy ↦ h hy hx⟩, ?_⟩
rintro (rfl | ⟨a, rfl⟩)
· exact (not_me... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 303 | 309 | theorem PartiallyWellOrderedOn.image_of_monotone_on (hs : s.PartiallyWellOrderedOn r)
(hf : ∀ a₁ ∈ s, ∀ a₂ ∈ s, r a₁ a₂ → r' (f a₁) (f a₂)) : (f '' s).PartiallyWellOrderedOn r' := by |
intro g' hg'
choose g hgs heq using hg'
obtain rfl : f ∘ g = g' := funext heq
obtain ⟨m, n, hlt, hmn⟩ := hs g hgs
exact ⟨m, n, hlt, hf _ (hgs m) _ (hgs n) hmn⟩
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 226 | 229 | theorem dist_eq_add_dist_of_angle_eq_pi {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) :
dist p1 p3 = dist p1 p2 + dist p3 p2 := by |
rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, ← vsub_sub_vsub_cancel_right]
exact norm_sub_eq_add_norm_of_angle_eq_pi h
|
/-
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.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align... | Mathlib/Computability/Ackermann.lean | 89 | 92 | theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 := by |
induction' n with n IH
· rfl
· simpa [mul_succ]
|
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury G. Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepar... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 157 | 163 | theorem separating_of_generateFrom (S : Set (Set α))
[h : @SeparatesPoints α (generateFrom S)] :
∀ x y : α, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y := by |
letI := generateFrom S
intros x y hxy
rw [← forall_generateFrom_mem_iff_mem_iff] at hxy
exact separatesPoints_def $ fun _ hs ↦ (hxy _ hs).mp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,402 | 2,404 | theorem natCast_mod (m n : ℕ) : ((m % n : ℕ) : Ordinal) = m % n := by |
rw [← add_left_cancel, div_add_mod, ← natCast_div, ← natCast_mul, ← Nat.cast_add,
Nat.div_add_mod]
|
/-
Copyright (c) 2024 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
/-!
# The span of a s... | Mathlib/Analysis/NormedSpace/Span.lean | 36 | 39 | theorem toSpanSingleton_homothety (x : E) (c : 𝕜) :
‖LinearMap.toSpanSingleton 𝕜 E x c‖ = ‖x‖ * ‖c‖ := by |
rw [mul_comm]
exact norm_smul _ _
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 2,009 | 2,015 | theorem directedOn_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by |
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,134 | 1,134 | theorem algEquiv_mk (x y) : algEquiv M S (mk x y) = mk' S x y := by | rw [mk_eq_mk', algEquiv_mk']
|
/-
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 | 83 | 83 | theorem disjSum_eq_empty : s.disjSum t = ∅ ↔ s = ∅ ∧ t = ∅ := by | simp [ext_iff]
|
/-
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.Algebra.Group.Defs
import Mathlib.Control.Functor
#align_import control.applicative from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516ab... | Mathlib/Control/Applicative.lean | 40 | 63 | theorem Applicative.ext {F} :
∀ {A1 : Applicative F} {A2 : Applicative F} [@LawfulApplicative F A1] [@LawfulApplicative F A2],
(∀ {α : Type u} (x : α), @Pure.pure _ A1.toPure _ x = @Pure.pure _ A2.toPure _ x) →
(∀ {α β : Type u} (f : F (α → β)) (x : F α),
@Seq.seq _ A1.toSeq _ _ f (fun _ => x)... |
funext α x
apply H1
obtain rfl : @s1 = @s2 := by
funext α β f x
exact H2 f (x Unit.unit)
obtain ⟨seqLeft_eq1, seqRight_eq1, pure_seq1, -⟩ := L1
obtain ⟨seqLeft_eq2, seqRight_eq2, pure_seq2, -⟩ := L2
obtain rfl : F1 = F2 := by
apply Functor.ext
intros
exact (pur... |
/-
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.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62... | Mathlib/Algebra/Group/Support.lean | 175 | 177 | theorem mulSupport_const {c : M} (hc : c ≠ 1) : (mulSupport fun _ : α => c) = Set.univ := by |
ext x
simp [hc]
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 839 | 891 | theorem denseRange_coeSimpleFuncNonnegToLpNonneg [hp : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) :
DenseRange (coeSimpleFuncNonnegToLpNonneg p μ G) := fun g ↦ by
borelize G
rw [mem_closure_iff_seq_limit]
have hg_memℒp : Memℒp (g : α → G) p μ := Lp.memℒp (g : Lp G p μ)
have zero_mem : (0 : G) ∈ (range (g : α → G) ∪ ... |
simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff,
mem_setOf_eq, le_refl, and_self_iff]
have : SeparableSpace ((range (g : α → G) ∪ {0}) ∩ { y | 0 ≤ y } : Set G) := by
apply IsSeparable.separableSpace
apply IsSeparable.mono _ Set.inter_subset_left
exact
... |
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 218 | 223 | theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ)
(hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) :
⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) := by |
simpa only [union_compl_self, restrict_univ] using
laverage_union_mem_openSegment aedisjoint_compl_right hs.compl hs₀ hsc₀ (measure_ne_top _ _)
(measure_ne_top _ _)
|
/-
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.Topology.GDelta
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.Instances.EReal
import Mathlib.Analysis.Normed.G... | Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 303 | 306 | theorem IsCompact.closure_subset_measurableSet [R1Space γ] {K s : Set γ} (hK : IsCompact K)
(hs : MeasurableSet s) (hKs : K ⊆ s) : closure K ⊆ s := by |
rw [hK.closure_eq_biUnion_inseparable, iUnion₂_subset_iff]
exact fun x hx y hy ↦ (hy.mem_measurableSet_iff hs).1 (hKs hx)
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.inner_product_space.linear_pmap from "leanprover-comm... | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 140 | 147 | theorem adjointAux_inner (y : T.adjointDomain) (x : T.domain) :
⟪adjointAux hT y, x⟫ = ⟪(y : F), T x⟫ := by |
simp only [adjointAux, LinearMap.coe_mk, InnerProductSpace.toDual_symm_apply,
adjointDomainMkCLMExtend_apply]
-- Porting note(https://github.com/leanprover-community/mathlib4/issues/5026):
-- mathlib3 was finished here
simp only [AddHom.coe_mk, InnerProductSpace.toDual_symm_apply]
rw [adjointDomainMkCLME... |
/-
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.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.legendre_symbol.gauss_eisenstein_lemmas from "lea... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 105 | 115 | theorem gauss_lemma {p : ℕ} [h : Fact p.Prime] {a : ℤ} (hp : p ≠ 2) (ha0 : (a : ZMod p) ≠ 0) :
legendreSym p a = (-1) ^ ((Ico 1 (p / 2).succ).filter fun x : ℕ =>
p / 2 < (a * x : ZMod p).val).card := by |
replace hp : Odd p := h.out.odd_of_ne_two hp
have : (legendreSym p a : ZMod p) = (((-1) ^ ((Ico 1 (p / 2).succ).filter fun x : ℕ =>
p / 2 < (a * x : ZMod p).val).card : ℤ) : ZMod p) := by
rw [legendreSym.eq_pow, gauss_lemma_aux p ha0]
cases legendreSym.eq_one_or_neg_one p ha0 <;>
cases neg_one_pow_eq... |
/-
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 | 1,450 | 1,460 | theorem support_map_of_injective (p : MvPolynomial σ R) {f : R →+* S₁} (hf : Injective f) :
(map f p).support = p.support := by |
apply Finset.Subset.antisymm
· exact MvPolynomial.support_map_subset _ _
intro x hx
rw [mem_support_iff]
contrapose! hx
simp only [Classical.not_not, mem_support_iff]
replace hx : (map f p).coeff x = 0 := hx
rw [coeff_map, ← f.map_zero] at hx
exact hf hx
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 307 | 315 | theorem liftPropWithinAt_indep_chart [HasGroupoid M G] [HasGroupoid M' G']
(he : e ∈ G.maximalAtlas M) (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M')
(xf : g x ∈ f.source) :
LiftPropWithinAt P g s x ↔
ContinuousWithinAt g s x ∧ P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) := by |
simp only [liftPropWithinAt_iff']
exact and_congr_right <|
hG.liftPropWithinAt_indep_chart_aux (chart_mem_maximalAtlas _ _) (mem_chart_source _ _) he xe
(chart_mem_maximalAtlas _ _) (mem_chart_source _ _) hf xf
|
/-
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.Basic
import Mathlib.RingTheory.Noetherian
#align_import algebra.lie.subalgebra from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350d... | Mathlib/Algebra/Lie/Subalgebra.lean | 329 | 333 | theorem surjective_rangeRestrict : Function.Surjective f.rangeRestrict := by |
rintro ⟨y, hy⟩
erw [mem_range] at hy; obtain ⟨x, rfl⟩ := hy
use x
simp only [Subtype.mk_eq_mk, rangeRestrict_apply]
|
/-
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.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
#align_impo... | Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 172 | 174 | theorem measure_eq_iInf' (μ : Measure α) (s : Set α) :
μ s = ⨅ t : { t // s ⊆ t ∧ MeasurableSet t }, μ t := by |
simp_rw [iInf_subtype, iInf_and, ← measure_eq_iInf]
|
/-
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 Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 613 | 616 | theorem length_filterMap_le (f : α → Option β) (l : List α) :
(filterMap f l).length ≤ l.length := by |
rw [← length_map _ some, map_filterMap_some_eq_filter_map_is_some, ← length_map _ f]
apply length_filter_le
|
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Data.Fintype.Perm
import Mathlib.LinearAlgebra.Multilinear.Basis
#align_import linear_algebra.... | Mathlib/LinearAlgebra/Alternating/Basic.lean | 960 | 969 | theorem Basis.ext_alternating {f g : N₁ [⋀^ι]→ₗ[R'] N₂} (e : Basis ι₁ R' N₁)
(h : ∀ v : ι → ι₁, Function.Injective v → (f fun i => e (v i)) = g fun i => e (v i)) :
f = g := by |
classical
refine AlternatingMap.coe_multilinearMap_injective (Basis.ext_multilinear e fun v => ?_)
by_cases hi : Function.Injective v
· exact h v hi
· have : ¬Function.Injective fun i => e (v i) := hi.imp Function.Injective.of_comp
rw [coe_multilinearMap, coe_multilinearMap, f.map_eq_zero_of_no... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,405 | 1,407 | theorem preimage_Icc {a b : α} : ((↑) : ℤ → α) ⁻¹' Set.Icc a b = Set.Icc ⌈a⌉ ⌊b⌋ := by |
ext
simp [ceil_le, le_floor]
|
/-
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.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-communi... | Mathlib/MeasureTheory/Group/Prod.lean | 198 | 210 | theorem lintegral_lintegral_mul_inv [IsMulLeftInvariant ν] (f : G → G → ℝ≥0∞)
(hf : AEMeasurable (uncurry f) (μ.prod ν)) :
(∫⁻ x, ∫⁻ y, f (y * x) x⁻¹ ∂ν ∂μ) = ∫⁻ x, ∫⁻ y, f x y ∂ν ∂μ := by |
have h : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) :=
(measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv
have h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν) :=
hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv μ ν).quasiMeasurePreserving
simp_rw [lintegral_... |
/-
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.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/ODE/Gronwall.lean | 355 | 364 | theorem ODE_solution_unique_of_eventually
(hf : ∀ᶠ t in 𝓝 t₀, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t)
(hg : ∀ᶠ t in 𝓝 t₀, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t)
(heq : f t₀ = g t₀) : f =ᶠ[𝓝 t₀] g := by |
obtain ⟨ε, hε, h⟩ := eventually_nhds_iff_ball.mp (hf.and hg)
rw [Filter.eventuallyEq_iff_exists_mem]
refine ⟨ball t₀ ε, ball_mem_nhds _ hε, ?_⟩
simp_rw [Real.ball_eq_Ioo] at *
apply ODE_solution_unique_of_mem_Ioo hv (Real.ball_eq_Ioo t₀ ε ▸ mem_ball_self hε)
(fun _ ht ↦ (h _ ht).1) (fun _ ht ↦ (h _ ht).2... |
/-
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 | 121 | 121 | theorem symmDiff_self : a ∆ a = ⊥ := by | rw [symmDiff, sup_idem, sdiff_self]
|
/-
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 | 1,194 | 1,197 | theorem measurableSet_range_inl [MeasurableSpace α] :
MeasurableSet (range Sum.inl : Set (α ⊕ β)) := by |
rw [← image_univ]
exact MeasurableSet.univ.inl_image
|
/-
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.Sites.Coherent.RegularSheaves
/-!
# Description of the covering sieves of the regular topology
This file characterises the coverin... | Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean | 64 | 78 | theorem mem_sieves_iff_hasEffectiveEpi (S : Sieve X) :
(S ∈ (regularTopology C).sieves X) ↔
∃ (Y : C) (π : Y ⟶ X), EffectiveEpi π ∧ (S.arrows π) := by |
constructor
· intro h
induction' h with Y T hS Y Y R S _ _ a b
· rcases hS with ⟨Y', π, h'⟩
refine ⟨Y', π, h'.2, ?_⟩
rcases h' with ⟨rfl, _⟩
exact ⟨Y', 𝟙 Y', π, Presieve.ofArrows.mk (), (by simp)⟩
· exact ⟨Y, (𝟙 Y), inferInstance, by simp only [Sieve.top_apply, forall_const]⟩
· ... |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
... | Mathlib/NumberTheory/Pell.lean | 241 | 245 | theorem d_nonsquare_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : ¬IsSquare d := by |
have hp := a.prop
rintro ⟨b, rfl⟩
simp_rw [← sq, ← mul_pow, sq_sub_sq, Int.mul_eq_one_iff_eq_one_or_neg_one] at hp
rcases hp with (⟨hp₁, hp₂⟩ | ⟨hp₁, hp₂⟩) <;> omega
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanp... | Mathlib/RingTheory/Jacobson.lean | 303 | 355 | theorem isIntegral_isLocalization_polynomial_quotient
(P : Ideal R[X]) (pX : R[X]) (hpX : pX ∈ P) [Algebra (R ⧸ P.comap (C : R →+* R[X])) Rₘ]
[IsLocalization.Away (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff Rₘ]
[Algebra (R[X] ⧸ P) Sₘ] [IsLocalization ((Submonoid.powers (pX.map (Quotient.m... |
let P' : Ideal R := P.comap C
let M : Submonoid (R ⧸ P') :=
Submonoid.powers (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff
let M' : Submonoid (R[X] ⧸ P) :=
(Submonoid.powers (pX.map (Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff).map
(quotientMap P C le_rfl)
let φ : R ⧸ P... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 1,190 | 1,200 | theorem ker_pow_eq_ker_pow_finrank_of_le [FiniteDimensional K V] {f : End K V} {m : ℕ}
(hm : finrank K V ≤ m) : LinearMap.ker (f ^ m) = LinearMap.ker (f ^ finrank K V) := by |
obtain ⟨k, h_k_le, hk⟩ :
∃ k, k ≤ finrank K V ∧ LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ) :=
exists_ker_pow_eq_ker_pow_succ f
calc
LinearMap.ker (f ^ m) = LinearMap.ker (f ^ (k + (m - k))) := by
rw [add_tsub_cancel_of_le (h_k_le.trans hm)]
_ = LinearMap.ker (f ^ k) := by rw [ker_pow_... |
/-
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 | 83 | 87 | theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 70 | 80 | theorem nth_stream_fr_nonneg_lt_one {ifp_n : IntFractPair K}
(nth_stream_eq : IntFractPair.stream v n = some ifp_n) : 0 ≤ ifp_n.fr ∧ ifp_n.fr < 1 := by |
cases n with
| zero =>
have : IntFractPair.of v = ifp_n := by injection nth_stream_eq
rw [← this, IntFractPair.of]
exact ⟨fract_nonneg _, fract_lt_one _⟩
| succ =>
rcases succ_nth_stream_eq_some_iff.1 nth_stream_eq with ⟨_, _, _, ifp_of_eq_ifp_n⟩
rw [← ifp_of_eq_ifp_n, IntFractPair.of]
ex... |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 861 | 862 | theorem inv_last {i j} (w : NeWord G i j) : w.inv.last = w.head⁻¹ := by |
induction w <;> simp [inv, *]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb... | Mathlib/Algebra/CharP/Two.lean | 65 | 66 | theorem neg_eq (x : R) : -x = x := by |
rw [neg_eq_iff_add_eq_zero, ← two_smul R x, two_eq_zero, zero_smul]
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 797 | 799 | theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
(G'.neighborSet v).toFinset.card = G'.degree v := by |
rw [degree, Set.toFinset_card]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Chris Hughes, Michael Howes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Semiconj.Units
#align_import algebra.group.conj from "leanprover-community... | Mathlib/Algebra/Group/Conj.lean | 117 | 122 | theorem conj_zpow {i : ℤ} {a b : α} : (a * b * a⁻¹) ^ i = a * b ^ i * a⁻¹ := by |
induction' i
· change (a * b * a⁻¹) ^ (_ : ℤ) = a * b ^ (_ : ℤ) * a⁻¹
simp [zpow_natCast]
· simp only [zpow_negSucc, conj_pow, mul_inv_rev, inv_inv]
rw [mul_assoc]
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algeb... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 432 | 438 | theorem toTrivSqZeroExt_comp_map [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module Rᵐᵒᵖ N]
[IsCentralScalar R N] (f : M →ₗ[R] N) :
toTrivSqZeroExt.comp (map f) = (TrivSqZeroExt.map f).comp toTrivSqZeroExt := by |
apply hom_ext
apply LinearMap.ext
simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, Function.comp_apply,
AlgHom.toLinearMap_apply, map_apply_ι, toTrivSqZeroExt_ι, TrivSqZeroExt.map_inr, forall_const]
|
/-
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.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-c... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 850 | 864 | theorem eqOn_right_half_plane_of_superexponential_decay {g : ℂ → E}
(hfd : DiffContOnCl ℂ f {z | 0 < z.re}) (hgd : DiffContOnCl ℂ g {z | 0 < z.re})
(hfexp : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * abs z ^ c))
(hgexp : ∃ c < (2 : ℝ), ∃ B,
g =O[cobounded ℂ ⊓ 𝓟... |
suffices EqOn (f - g) 0 {z : ℂ | 0 ≤ z.re} by
simpa only [EqOn, Pi.sub_apply, Pi.zero_apply, sub_eq_zero] using this
refine eq_zero_on_right_half_plane_of_superexponential_decay (hfd.sub hgd) ?_ hre ?_
· exact isBigO_sub_exp_rpow hfexp hgexp
· rcases hfim with ⟨Cf, hCf⟩; rcases hgim with ⟨Cg, hCg⟩
exac... |
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.Bin... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 184 | 185 | theorem pullbackZeroZeroIso_hom_fst (X Y : C) [HasBinaryProduct X Y] :
(pullbackZeroZeroIso X Y).hom ≫ prod.fst = pullback.fst := by | simp [← Iso.eq_inv_comp]
|
/-
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 | 757 | 760 | theorem OrthonormalBasis.det_to_matrix_orthonormalBasis_real :
a.toBasis.det b = 1 ∨ a.toBasis.det b = -1 := by |
rw [← sq_eq_one_iff]
simpa [unitary, sq] using Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)
|
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth, Antoine Chambert-Loir
-/
import Mathlib.Data.Set.Pointwise.SMul
impor... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 58 | 60 | theorem image_smul_setₛₗ :
h '' (c • s) = σ c • h '' s := by |
simp only [← image_smul, image_image, map_smulₛₗ h]
|
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.Submodule.Localization
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.... | Mathlib/LinearAlgebra/Dimension/Localization.lean | 85 | 93 | theorem exists_set_linearIndependent_of_isDomain [IsDomain R] :
∃ s : Set M, #s = Module.rank R M ∧ LinearIndependent (ι := s) R Subtype.val := by |
obtain ⟨w, hw⟩ :=
IsLocalizedModule.linearIndependent_lift R⁰ (LocalizedModule.mkLinearMap R⁰ M) le_rfl
(Module.Free.chooseBasis (FractionRing R) (LocalizedModule R⁰ M)).linearIndependent
refine ⟨Set.range w, ?_, (linearIndependent_subtype_range hw.injective).mpr hw⟩
apply Cardinal.lift_injective.{max ... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 305 | 310 | theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
IsCompact s := by |
rcases eq_empty_or_nonempty s with (rfl | ⟨x, -⟩)
· exact isCompact_empty
· rcases hb.subset_closedBall x with ⟨r, hr⟩
exact (isCompact_closedBall x r).of_isClosed_subset hc hr
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,150 | 1,153 | theorem _root_.MeasurableEquiv.sigmaFinite_map {μ : Measure α} (f : α ≃ᵐ β) (h : SigmaFinite μ) :
SigmaFinite (μ.map f) := by |
refine SigmaFinite.of_map _ f.symm.measurable.aemeasurable ?_
rwa [map_map f.symm.measurable f.measurable, f.symm_comp_self, Measure.map_id]
|
/-
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, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
/-!
# The dominated convergence t... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 107 | 137 | theorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AEStronglyMeasurable (f i) μ)
(hf' : ∑' i, ∫⁻ a : α, ‖f i a‖₊ ∂μ ≠ ∞) :
∫ a : α, ∑' i, f i a ∂μ = ∑' i, ∫ a : α, f i a ∂μ := by |
by_cases hG : CompleteSpace G; swap
· simp [integral, hG]
have hf'' : ∀ i, AEMeasurable (fun x => (‖f i x‖₊ : ℝ≥0∞)) μ := fun i => (hf i).ennnorm
have hhh : ∀ᵐ a : α ∂μ, Summable fun n => (‖f n a‖₊ : ℝ) := by
rw [← lintegral_tsum hf''] at hf'
refine (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mon... |
/-
Copyright (c) 2023 Mantas Bakšys, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Rearrangement
i... | Mathlib/Algebra/Order/Chebyshev.lean | 70 | 72 | theorem AntivaryOn.card_smul_sum_le_sum_smul_sum (hfg : AntivaryOn f g s) :
(s.card • ∑ i ∈ s, f i • g i) ≤ (∑ i ∈ s, f i) • ∑ i ∈ s, g i := by |
exact hfg.dual_right.sum_smul_sum_le_card_smul_sum
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0... | Mathlib/Data/Finset/Option.lean | 135 | 138 | theorem eraseNone_inter [DecidableEq (Option α)] [DecidableEq α] (s t : Finset (Option α)) :
eraseNone (s ∩ t) = eraseNone s ∩ eraseNone t := by |
ext
simp
|
/-
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 | 462 | 498 | theorem union (f : α → E) {s t : Set α} {x : α} (hs : IsGreatest s x) (ht : IsLeast t x) :
eVariationOn f (s ∪ t) = eVariationOn f s + eVariationOn f t := by |
classical
apply le_antisymm _ (eVariationOn.add_le_union f fun a ha b hb => le_trans (hs.2 ha) (ht.2 hb))
apply iSup_le _
rintro ⟨n, ⟨u, hu, ust⟩⟩
obtain ⟨v, m, hv, vst, xv, huv⟩ : ∃ (v : ℕ → α) (m : ℕ),
Monotone v ∧ (∀ i, v i ∈ s ∪ t) ∧ x ∈ v '' Iio m ∧
(∑ i ∈ Finset.range n, edist (f (u (i + 1)))... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 436 | 440 | theorem getVert_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) (i : ℕ) :
(p.append q).getVert i = if i < p.length then p.getVert i else q.getVert (i - p.length) := by |
induction p generalizing i with
| nil => simp
| cons h p ih => cases i <;> simp [getVert, ih, Nat.succ_lt_succ_iff]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 1,064 | 1,071 | theorem hnot_hnot_sdiff_distrib (a b : α) : ￢￢(a \ b) = ￢￢a \ ￢￢b := by |
apply le_antisymm
· refine hnot_le_comm.1 ((hnot_anti sdiff_le_inf_hnot).trans' ?_)
rw [hnot_inf_distrib, hnot_le_iff_codisjoint_right, codisjoint_left_comm, ←
hnot_le_iff_codisjoint_right]
exact le_sdiff_sup
· rw [sdiff_le_iff, ← hnot_hnot_sup_distrib]
exact hnot_anti (hnot_anti le_sup_sdiff)
|
/-
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.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
/-!
## Pushou... | Mathlib/GroupTheory/PushoutI.lean | 643 | 656 | theorem Reduced.eq_empty_of_mem_range (hφ : ∀ i, Injective (φ i))
{w : Word G} (hw : Reduced φ w)
(h : ofCoprodI w.prod ∈ (base φ).range) : w = .empty := by |
rcases transversal_nonempty φ hφ with ⟨d⟩
rcases hw.exists_normalWord_prod_eq d with ⟨w', hw'prod, hw'map⟩
rcases h with ⟨h, heq⟩
have : (NormalWord.prod (d := d) ⟨.empty, h, by simp⟩) = base φ h := by
simp [NormalWord.prod]
rw [← hw'prod, ← this] at heq
suffices w'.toWord = .empty by
simp [this, @... |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 152 | 202 | theorem norm_eval_le_injectiveSeminorm (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
‖lift f.toMultilinearMap x‖ ≤ ‖f‖ * injectiveSeminorm x := by |
/- If `F` were in `Type (max uι u𝕜 uE)` (which is the type of `⨂[𝕜] i, E i`), then the
property that we want to prove would hold by definition of `injectiveSeminorm`. This is
not necessarily true, but we will show that there exists a normed vector space `G` in
`Type (max uι u𝕜 uE)` and an injective ... |
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