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) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 399 | 410 | theorem exists_null_frontier_thickening (μ : Measure Ω) [SigmaFinite μ] (s : Set Ω) {a b : ℝ}
(hab : a < b) : ∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0 := by |
have mbles : ∀ r : ℝ, MeasurableSet (frontier (Metric.thickening r s)) :=
fun r => isClosed_frontier.measurableSet
have disjs := Metric.frontier_thickening_disjoint s
have key := Measure.countable_meas_pos_of_disjoint_iUnion (μ := μ) mbles disjs
have aux := measure_diff_null (s := Ioo a b) (Set.Countable.m... |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 238 | 239 | theorem cthickening_empty (δ : ℝ) : cthickening δ (∅ : Set α) = ∅ := by |
simp only [cthickening, ENNReal.ofReal_ne_top, setOf_false, infEdist_empty, top_le_iff]
|
/-
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.Analysis.Convex.Function
#align_import analysis.convex.quasiconvex from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Q... | Mathlib/Analysis/Convex/Quasiconvex.lean | 146 | 150 | theorem QuasiconvexOn.convex_lt (hf : QuasiconvexOn 𝕜 s f) (r : β) :
Convex 𝕜 ({ x ∈ s | f x < r }) := by |
refine fun x hx y hy a b ha hb hab => ?_
have h := hf _ ⟨hx.1, le_max_left _ _⟩ ⟨hy.1, le_max_right _ _⟩ ha hb hab
exact ⟨h.1, h.2.trans_lt <| max_lt hx.2 hy.2⟩
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 563 | 568 | theorem epi_iff_surjective {n m : SimplexCategory} {f : n ⟶ m} :
Epi f ↔ Function.Surjective f.toOrderHom := by |
rw [← Functor.epi_map_iff_epi skeletalEquivalence.functor]
dsimp only [skeletalEquivalence, Functor.asEquivalence_functor]
simp only [skeletalFunctor_obj, skeletalFunctor_map,
NonemptyFinLinOrd.epi_iff_surjective, NonemptyFinLinOrd.coe_of]
|
/-
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.Topology.Basic
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_import topology.omega_complete_partial_order from... | Mathlib/Topology/OmegaCompletePartialOrder.lean | 110 | 113 | theorem isωSup_ωSup {α} [OmegaCompletePartialOrder α] (c : Chain α) : IsωSup c (ωSup c) := by |
constructor
· apply le_ωSup
· apply ωSup_le
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.adjunction.basic from "leanprover-community/mathlib"@"d101e93197bb5f6... | Mathlib/CategoryTheory/Adjunction/Basic.lean | 221 | 224 | theorem right_triangle : whiskerLeft G adj.unit ≫ whiskerRight adj.counit G = 𝟙 _ := by |
ext; dsimp
erw [← adj.homEquiv_unit, ← Equiv.eq_symm_apply, adj.homEquiv_counit]
simp
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 244 | 245 | theorem adjMatrix_mul_self_apply_self [NonAssocSemiring α] (i : V) :
(G.adjMatrix α * G.adjMatrix α) i i = degree G i := by | simp [filter_true_of_mem]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 713 | 716 | theorem lift_comp (g : M →ₗ[R] M'') (h : ∀ x : S, IsUnit ((algebraMap R (Module.End R M'')) x)) :
(lift S g h).comp (mkLinearMap S M) = g := by |
ext x; dsimp; rw [LocalizedModule.lift_mk]
erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, one_smul]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 118 | 118 | theorem sphere_zero_right (n k : ℕ) : sphere (n + 1) 0 k = ∅ := by | simp [sphere]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 411 | 423 | theorem IsIntegralClosure.exists_smul_eq_mul {L : Type*} [Field L] [Algebra R S] [Algebra S L]
[Algebra R L] [IsScalarTower R S L] [IsIntegralClosure S R L] [Algebra.IsAlgebraic R L]
(inj : Function.Injective (algebraMap R L)) (a : S) {b : S} (hb : b ≠ 0) :
∃ᵉ (c : S) (d ≠ (0 : R)), d • a = b * c := by |
obtain ⟨c, d, d_ne, hx⟩ :=
exists_integral_multiple (Algebra.IsAlgebraic.isAlgebraic (algebraMap _ L a / algebraMap _ L b))
((injective_iff_map_eq_zero _).mp inj)
refine
⟨IsIntegralClosure.mk' S (c : L) c.2, d, d_ne, IsIntegralClosure.algebraMap_injective S R L ?_⟩
simp only [Algebra.smul_def, 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.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 565 | 568 | theorem abs_sub_le_of_uIcc_subset_uIcc (h : [[c, d]] ⊆ [[a, b]]) : |d - c| ≤ |b - a| := by |
rw [← max_sub_min_eq_abs, ← max_sub_min_eq_abs]
rw [uIcc_subset_uIcc_iff_le] at h
exact sub_le_sub h.2 h.1
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 232 | 239 | theorem IndepFun.mgf_add' {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
(hY : AEStronglyMeasurable Y μ) : mgf (X + Y) μ t = mgf X μ t * mgf Y μ t := by |
have A : Continuous fun x : ℝ => exp (t * x) := by fun_prop
have h'X : AEStronglyMeasurable (fun ω => exp (t * X ω)) μ :=
A.aestronglyMeasurable.comp_aemeasurable hX.aemeasurable
have h'Y : AEStronglyMeasurable (fun ω => exp (t * Y ω)) μ :=
A.aestronglyMeasurable.comp_aemeasurable hY.aemeasurable
exact... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 514 | 516 | theorem smul_eq_C_mul (f : MvPowerSeries σ R) (a : R) : a • f = C σ R a * f := by |
ext
simp
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 599 | 649 | theorem GoodProducts.spanFin : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (· ∈ s)))) := by |
rw [span_iff_products]
refine le_trans (spanFinBasis.span C s) ?_
rw [Submodule.span_le]
rintro _ ⟨x, rfl⟩
rw [← factors_prod_eq_basis]
let l := s.sort (·≥·)
dsimp [factors]
suffices l.Chain' (·>·) → (l.map (fun i ↦ if x.val i = true then e (π C (· ∈ s)) i
else (1 - (e (π C (· ∈ s)) i)))).prod ∈
... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 316 | 318 | theorem dual_node3L (l : Ordnode α) (x : α) (m : Ordnode α) (y : α) (r : Ordnode α) :
dual (node3L l x m y r) = node3R (dual r) y (dual m) x (dual l) := by |
simp [node3L, node3R, dual_node', add_comm]
|
/-
Copyright (c) 2023 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a... | Mathlib/Analysis/ConstantSpeed.lean | 222 | 235 | theorem unique_unit_speed_on_Icc_zero {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) {φ : ℝ → ℝ}
(φm : MonotoneOn φ <| Icc 0 s) (φst : φ '' Icc 0 s = Icc 0 t)
(hfφ : HasUnitSpeedOn (f ∘ φ) (Icc 0 s)) (hf : HasUnitSpeedOn f (Icc 0 t)) :
EqOn φ id (Icc 0 s) := by |
rw [← φst] at hf
convert unique_unit_speed φm hfφ hf ⟨le_rfl, hs⟩ using 1
have : φ 0 = 0 := by
have hm : 0 ∈ φ '' Icc 0 s := by simp only [φst, ht, mem_Icc, le_refl, and_self]
obtain ⟨x, xs, hx⟩ := hm
apply le_antisymm ((φm ⟨le_rfl, hs⟩ xs xs.1).trans_eq hx) _
have := φst ▸ mapsTo_image φ (Icc 0 ... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 972 | 974 | theorem uIcc_subset_Icc (ha : a₁ ∈ Icc a₂ b₂) (hb : b₁ ∈ Icc a₂ b₂) : [[a₁, b₁]] ⊆ Icc a₂ b₂ := by |
rw [mem_Icc] at ha hb
exact Icc_subset_Icc (_root_.le_inf ha.1 hb.1) (_root_.sup_le ha.2 hb.2)
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_ze... | Mathlib/Topology/Algebra/GroupWithZero.lean | 69 | 71 | theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : G₀) :
ContinuousOn (fun x => f x / y) s := by |
simpa only [div_eq_mul_inv] using hf.mul continuousOn_const
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 705 | 709 | theorem total_surjective (h : Function.Surjective v) :
Function.Surjective (Finsupp.total α M R v) := by |
intro x
obtain ⟨y, hy⟩ := h x
exact ⟨Finsupp.single y 1, by simp [hy]⟩
|
/-
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 | 1,437 | 1,447 | theorem _root_.MeasurableEmbedding.lintegral_map [MeasurableSpace β] {g : α → β}
(hg : MeasurableEmbedding g) (f : β → ℝ≥0∞) : ∫⁻ a, f a ∂map g μ = ∫⁻ a, f (g a) ∂μ := by |
rw [lintegral, lintegral]
refine le_antisymm (iSup₂_le fun f₀ hf₀ => ?_) (iSup₂_le fun f₀ hf₀ => ?_)
· rw [SimpleFunc.lintegral_map _ hg.measurable]
have : (f₀.comp g hg.measurable : α → ℝ≥0∞) ≤ f ∘ g := fun x => hf₀ (g x)
exact le_iSup_of_le (comp f₀ g hg.measurable) (by exact le_iSup (α := ℝ≥0∞) _ this... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 935 | 937 | theorem bddAbove_insert [IsDirected α (· ≤ ·)] {s : Set α} {a : α} :
BddAbove (insert a s) ↔ BddAbove s := by |
simp only [insert_eq, bddAbove_union, bddAbove_singleton, true_and_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,039 | 2,046 | theorem biSup_measure_Iic [Preorder α] {s : Set α} (hsc : s.Countable)
(hst : ∀ x : α, ∃ y ∈ s, x ≤ y) (hdir : DirectedOn (· ≤ ·) s) :
⨆ x ∈ s, μ (Iic x) = μ univ := by |
rw [← measure_biUnion_eq_iSup hsc]
· congr
simp only [← bex_def] at hst
exact iUnion₂_eq_univ_iff.2 hst
· exact directedOn_iff_directed.2 (hdir.directed_val.mono_comp _ fun x y => Iic_subset_Iic.2)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_im... | Mathlib/Data/Finset/NatAntidiagonal.lean | 89 | 99 | theorem antidiagonal_succ_succ' {n : ℕ} :
antidiagonal (n + 2) =
cons (0, n + 2)
(cons (n + 2, 0)
((antidiagonal n).map
(Embedding.prodMap ⟨Nat.succ, Nat.succ_injective⟩
⟨Nat.succ, Nat.succ_injective⟩)) <|
by simp)
(by simp) := by |
simp_rw [antidiagonal_succ (n + 1), antidiagonal_succ', Finset.map_cons, map_map]
rfl
|
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 70 | 112 | theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by |
-- we start by expanding d ≫ d as a double sum
dsimp
simp only [comp_sum, sum_comp, ← Finset.sum_product']
-- then, we decompose the index set P into a subset S and its complement Sᶜ
let P := Fin (n + 2) × Fin (n + 3)
let S := Finset.univ.filter fun ij : P => (ij.2 : ℕ) ≤ (ij.1 : ℕ)
erw [← Finset.sum_add... |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import m... | Mathlib/ModelTheory/Basic.lean | 999 | 1,001 | theorem self_comp_symm_toHom (f : M ≃[L] N) :
f.toHom.comp f.symm.toHom = Hom.id L N := by |
rw [← comp_toHom, self_comp_symm, refl_toHom]
|
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e962... | Mathlib/Order/Height.lean | 333 | 349 | theorem chainHeight_union_eq (s t : Set α) (H : ∀ a ∈ s, ∀ b ∈ t, a < b) :
(s ∪ t).chainHeight = s.chainHeight + t.chainHeight := by |
cases h : t.chainHeight
· rw [add_top, eq_top_iff, ← h]
exact Set.chainHeight_mono subset_union_right
apply le_antisymm
· rw [← h]
exact chainHeight_union_le
rw [← add_zero (s ∪ t).chainHeight, ← WithTop.coe_zero,
ENat.some_eq_coe, chainHeight_add_le_chainHeight_add]
intro l hl
obtain ⟨l', hl... |
/-
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 | 819 | 819 | theorem disjoint_left : Disjoint l₁ l₂ ↔ ∀ ⦃a⦄, a ∈ l₁ → a ∉ l₂ := by | simp [Disjoint]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "lea... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 145 | 147 | theorem snorm_sub_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(hp : 1 ≤ p) : snorm (f - g) p μ ≤ snorm f p μ + snorm g p μ := by |
simpa [LpAddConst_of_one_le hp] using snorm_sub_le' hf hg p
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 421 | 424 | theorem snd_of_isColimit {X Y : C} {t : BinaryBicone X Y} (ht : IsColimit t.toCocone) :
t.snd = ht.desc (BinaryCofan.mk 0 (𝟙 Y)) := by |
apply ht.uniq (BinaryCofan.mk 0 (𝟙 Y))
rintro ⟨⟨⟩⟩ <;> dsimp <;> simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 319 | 322 | theorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x := by |
refine sdiff_le.lt_of_ne fun h => hy ?_
rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h
rw [← h, inf_eq_right.mpr hx]
|
/-
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 | 402 | 402 | theorem arccos_eq_pi_div_two {x} : arccos x = π / 2 ↔ x = 0 := by | simp [arccos]
|
/-
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 | 1,222 | 1,226 | theorem prod_ite_of_false {p : α → Prop} {hp : DecidablePred p} (f g : α → β) (h : ∀ x ∈ s, ¬p x) :
∏ x ∈ s, (if p x then f x else g x) = ∏ x ∈ s, g x := by |
rw [prod_ite, filter_false_of_mem, filter_true_of_mem]
· simp only [prod_empty, one_mul]
all_goals intros; apply h; assumption
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 639 | 645 | theorem dvd_ofDigits_sub_ofDigits {α : Type*} [CommRing α] {a b k : α} (h : k ∣ a - b)
(L : List ℕ) : k ∣ ofDigits a L - ofDigits b L := by |
induction' L with d L ih
· change k ∣ 0 - 0
simp
· simp only [ofDigits, add_sub_add_left_eq_sub]
exact dvd_mul_sub_mul h ih
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 1,119 | 1,120 | theorem model_infiniteTheory_iff : M ⊨ L.infiniteTheory ↔ Infinite M := by |
simp [infiniteTheory, infinite_iff, aleph0_le]
|
/-
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.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 246 | 258 | theorem StrictConvex.add (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) :
StrictConvex 𝕜 (s + t) := by |
rintro _ ⟨v, hv, w, hw, rfl⟩ _ ⟨x, hx, y, hy, rfl⟩ h a b ha hb hab
rw [smul_add, smul_add, add_add_add_comm]
obtain rfl | hvx := eq_or_ne v x
· refine interior_mono (add_subset_add (singleton_subset_iff.2 hv) Subset.rfl) ?_
rw [Convex.combo_self hab, singleton_add]
exact
(isOpenMap_add_left _).im... |
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 317 | 320 | theorem Basis.opNorm_le {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} {M : ℝ}
(hM : 0 ≤ M) (hu : ∀ i, ‖u (v i)‖ ≤ M) :
‖u‖ ≤ Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖ * M := by |
simpa using NNReal.coe_le_coe.mpr (v.opNNNorm_le ⟨M, hM⟩ hu)
|
/-
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 | 199 | 220 | theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ =
⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ x, ↑(φ x) ≤ f x), (φ.map ((↑) : ℝ≥0 → ℝ≥0∞)).lintegral μ := by |
rw [lintegral]
refine
le_antisymm (iSup₂_le fun φ hφ => ?_) (iSup_mono' fun φ => ⟨φ.map ((↑) : ℝ≥0 → ℝ≥0∞), le_rfl⟩)
by_cases h : ∀ᵐ a ∂μ, φ a ≠ ∞
· let ψ := φ.map ENNReal.toNNReal
replace h : ψ.map ((↑) : ℝ≥0 → ℝ≥0∞) =ᵐ[μ] φ := h.mono fun a => ENNReal.coe_toNNReal
have : ∀ x, ↑(ψ x) ≤ f x := fun x... |
/-
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.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 222 | 222 | theorem arg_neg_I : arg (-I) = -(π / 2) := by | simp [arg, le_refl]
|
/-
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.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 145 | 146 | theorem image₂_eq_empty_iff : image₂ f s t = ∅ ↔ s = ∅ ∨ t = ∅ := by |
simp_rw [← not_nonempty_iff_eq_empty, image₂_nonempty_iff, not_and_or]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 267 | 287 | theorem ofDigits_digits (b n : ℕ) : ofDigits b (digits b n) = n := by |
cases' b with b
· cases' n with n
· rfl
· change ofDigits 0 [n + 1] = n + 1
dsimp [ofDigits]
· cases' b with b
· induction' n with n ih
· rfl
· rw [Nat.zero_add] at ih ⊢
simp only [ih, add_comm 1, ofDigits_one_cons, Nat.cast_id, digits_one_succ]
· apply Nat.strongInducti... |
/-
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 | 928 | 930 | theorem succ_le_or_le_castSucc (p : Fin (n + 1)) (i : Fin n) : succ i ≤ p ∨ p ≤ i.castSucc := by |
rw [le_castSucc_iff, ← castSucc_lt_iff_succ_le]
exact p.castSucc_lt_or_lt_succ i
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.doubling from "leanprover-commu... | Mathlib/MeasureTheory/Measure/Doubling.lean | 113 | 129 | theorem eventually_measure_mul_le_scalingConstantOf_mul (K : ℝ) :
∃ R : ℝ,
0 < R ∧
∀ x t r, t ∈ Ioc 0 K → r ≤ R →
μ (closedBall x (t * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) := by |
have h := Classical.choose_spec (exists_eventually_forall_measure_closedBall_le_mul μ K)
rcases mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 h with ⟨R, Rpos, hR⟩
refine ⟨R, Rpos, fun x t r ht hr => ?_⟩
rcases lt_trichotomy r 0 with (rneg | rfl | rpos)
· have : t * r < 0 := mul_neg_of_pos_of_neg ht.1 rneg
s... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,695 | 1,697 | theorem comp_swap_eq_update (i j : α) (f : α → β) :
f ∘ Equiv.swap i j = update (update f j (f i)) i (f j) := by |
rw [swap_eq_update, comp_update, comp_update, comp_id]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 906 | 914 | theorem translationNumber_lt_of_forall_lt_add (hf : Continuous f) {z : ℝ} (hz : ∀ x, f x < x + z) :
τ f < z := by |
obtain ⟨x, -, hx⟩ : ∃ x ∈ Icc (0 : ℝ) 1, ∀ y ∈ Icc (0 : ℝ) 1, f y - y ≤ f x - x :=
isCompact_Icc.exists_isMaxOn (nonempty_Icc.2 zero_le_one)
(hf.sub continuous_id).continuousOn
refine lt_of_le_of_lt ?_ (sub_lt_iff_lt_add'.2 <| hz x)
apply translationNumber_le_of_le_add
simp only [← sub_le_iff_le_add'... |
/-
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.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.CompleteMetrizable
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathli... | Mathlib/Analysis/NormedSpace/Banach.lean | 164 | 226 | theorem exists_preimage_norm_le (surj : Surjective f) :
∃ C > 0, ∀ y, ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖ := by |
obtain ⟨C, C0, hC⟩ := exists_approx_preimage_norm_le f surj
/- Second step of the proof: starting from `y`, we want an exact preimage of `y`. Let `g y` be
the approximate preimage of `y` given by the first step, and `h y = y - f(g y)` the part that
has no preimage yet. We will iterate this process, taking ... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yaël Dillies
-/
import Mathlib.Topology.Sets.Opens
#align_import topology.sets.closeds from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298... | Mathlib/Topology/Sets/Closeds.lean | 110 | 111 | theorem coe_sup (s t : Closeds α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t := by |
rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 833 | 844 | theorem isTopologicalBasis_basic_opens :
TopologicalSpace.IsTopologicalBasis
(Set.range fun r : R => (basicOpen r : Set (PrimeSpectrum R))) := by |
apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
· rintro _ ⟨r, rfl⟩
exact isOpen_basicOpen
· rintro p U hp ⟨s, hs⟩
rw [← compl_compl U, Set.mem_compl_iff, ← hs, mem_zeroLocus, Set.not_subset] at hp
obtain ⟨f, hfs, hfp⟩ := hp
refine ⟨basicOpen f, ⟨f, rfl⟩, hfp, ?_⟩
rw [← Set.compl_... |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 125 | 127 | theorem card_edgeFinset_top_eq_card_choose_two [DecidableEq V] :
(⊤ : SimpleGraph V).edgeFinset.card = (Fintype.card V).choose 2 := by |
simp_rw [Set.toFinset_card, edgeSet_top, Set.coe_setOf, ← Sym2.card_subtype_not_diag]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 893 | 896 | theorem get?_terminates_le {s : WSeq α} {m n} (h : m ≤ n) :
Terminates (get? s n) → Terminates (get? s m) := by |
induction' h with m' _ IH
exacts [id, fun T => IH (@head_terminates_of_head_tail_terminates _ _ T)]
|
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib... | Mathlib/NumberTheory/Multiplicity.lean | 56 | 71 | theorem sq_dvd_add_pow_sub_sub (p x : R) (n : ℕ) :
p ^ 2 ∣ (x + p) ^ n - x ^ (n - 1) * p * n - x ^ n := by |
cases' n with n n
· simp only [pow_zero, Nat.cast_zero, sub_zero, sub_self, dvd_zero, Nat.zero_eq, mul_zero]
· simp only [Nat.succ_sub_succ_eq_sub, tsub_zero, Nat.cast_succ, add_pow, Finset.sum_range_succ,
Nat.choose_self, Nat.succ_sub _, tsub_self, pow_one, Nat.choose_succ_self_right, pow_zero,
mul_... |
/-
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 | 246 | 252 | theorem eval_basis_self (hvs : Set.InjOn v s) (hi : i ∈ s) :
(Lagrange.basis s v i).eval (v i) = 1 := by |
rw [Lagrange.basis, eval_prod]
refine prod_eq_one fun j H => ?_
rw [eval_basisDivisor_left_of_ne]
rcases mem_erase.mp H with ⟨hij, hj⟩
exact mt (hvs hi hj) hij.symm
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,723 | 2,724 | theorem count_eq_card_filter_eq [DecidableEq α] (s : Multiset α) (a : α) :
s.count a = card (s.filter (a = ·)) := by | rw [count, countP_eq_card_filter]
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Nilpotent
/-!
# Polynomials over an irreducible ring
This file contains results about the polyn... | Mathlib/RingTheory/Polynomial/IrreducibleRing.lean | 37 | 61 | theorem Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical
{R S : Type*} [CommRing R] [(nilradical R).IsPrime] [CommRing S] [IsDomain S]
(φ : R →+* S) (f : R[X]) (hm : f.Monic) (hi : Irreducible (f.map φ)) : Irreducible f := by |
let R' := R ⧸ nilradical R
let ψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ
(haveI := RingHom.ker_isPrime φ; nilradical_le_prime (RingHom.ker φ))
let ι := algebraMap R R'
rw [show φ = ψ.comp ι from rfl, ← map_map] at hi
replace hi := hm.map ι |>.irreducible_of_irreducible_map _ _ hi
refine ⟨fun... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,986 | 1,987 | theorem ae_eq_bot : ae μ = ⊥ ↔ μ = 0 := by |
rw [← empty_mem_iff_bot, mem_ae_iff, compl_empty, measure_univ_eq_zero]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 666 | 667 | theorem ofDigits_mod_eq_head! (b : ℕ) (l : List ℕ) : ofDigits b l % b = l.head! % b := by |
induction l <;> simp [Nat.ofDigits, Int.ModEq]
|
/-
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 | 1,963 | 1,964 | theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by |
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
|
/-
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 | 1,041 | 1,044 | theorem extend_image_nhd_mem_nhds_of_boundaryless [I.Boundaryless] {x} (hx : x ∈ f.source)
{s : Set M} (h : s ∈ 𝓝 x) : (f.extend I) '' s ∈ 𝓝 ((f.extend I) x) := by |
rw [← f.map_extend_nhds_of_boundaryless _ hx, Filter.mem_map]
filter_upwards [h] using subset_preimage_image (f.extend I) s
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 670 | 674 | theorem eq_spanSingleton_of_principal (I : FractionalIdeal S P) [IsPrincipal (I : Submodule R P)] :
I = spanSingleton S (generator (I : Submodule R P)) := by |
-- Porting note: this used to be `coeToSubmodule_injective (span_singleton_generator ↑I).symm`
-- but Lean 4 struggled to unify everything. Turned it into an explicit `rw`.
rw [spanSingleton, ← coeToSubmodule_inj, coe_mk, span_singleton_generator]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Co... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 426 | 430 | theorem norm_iteratedFDeriv_le (n : ℕ) (x : (i : ι) → E i) :
‖iteratedFDeriv 𝕜 n f x‖
≤ Nat.descFactorial (Fintype.card ι) n * ‖f‖ * ‖x‖ ^ (Fintype.card ι - n) := by |
rw [f.iteratedFDeriv_eq]
exact f.norm_iteratedFDeriv_le' n x
|
/-
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,266 | 1,270 | theorem tsum_eq_add_tsum_ite {f : α → ℝ≥0} (hf : Summable f) (i : α) :
∑' x, f x = f i + ∑' x, ite (x = i) 0 (f x) := by |
refine tsum_eq_add_tsum_ite' i (NNReal.summable_of_le (fun i' => ?_) hf)
rw [Function.update_apply]
split_ifs <;> simp only [zero_le', le_rfl]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.sh... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 225 | 228 | theorem PreservesPushout.inl_iso_hom :
pushout.inl ≫ (PreservesPushout.iso G f g).hom = G.map pushout.inl := by |
delta PreservesPushout.iso
simp
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 160 | 162 | theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
sSup (s / t) = sSup s / sInf t := by |
rw [div_eq_mul_inv, csSup_mul hs₀ hs₁ ht₀.inv ht₁.inv, csSup_inv ht₀ ht₁, div_eq_mul_inv]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
#align_import number_theory.p... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 303 | 303 | theorem padic_norm_e_of_padicInt (z : ℤ_[p]) : ‖(z : ℚ_[p])‖ = ‖z‖ := by | simp [norm_def]
|
/-
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 | 221 | 223 | theorem floor_pos : 0 < ⌊a⌋₊ ↔ 1 ≤ a := by |
-- Porting note: broken `convert le_floor_iff' Nat.one_ne_zero`
rw [Nat.lt_iff_add_one_le, zero_add, le_floor_iff' Nat.one_ne_zero, cast_one]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,077 | 1,081 | theorem lift_unique {j : N →* P} (hj : ∀ x, j (f.toMap x) = g x) : f.lift hg = j := by |
ext
rw [lift_spec, ← hj, ← hj, ← j.map_mul]
apply congr_arg
rw [← sec_spec']
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 493 | 494 | theorem mem_toList {a : α} {s : Multiset α} : a ∈ s.toList ↔ a ∈ s := by |
rw [← mem_coe, coe_toList]
|
/-
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 | 41 | 44 | theorem not_isFiniteMeasure_iff : ¬IsFiniteMeasure μ ↔ μ Set.univ = ∞ := by |
refine ⟨fun h => ?_, fun h => fun h' => h'.measure_univ_lt_top.ne h⟩
by_contra h'
exact h ⟨lt_top_iff_ne_top.mpr h'⟩
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import M... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 243 | 244 | theorem le_div_iff₀ (hc : c ≠ 0) : a ≤ b / c ↔ a * c ≤ b := by |
rw [div_eq_mul_inv, le_mul_inv_iff₀ hc]
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 415 | 415 | theorem whiskerRight_iff {f g : a ⟶ b} (η θ : f ⟶ g) : η ▷ 𝟙 b = θ ▷ 𝟙 b ↔ η = θ := by | simp
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.fold from "leanprover-community/mathlib"@"446eb51ce0a90f8385f260d2b5... | Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean | 115 | 117 | theorem foldl_algebraMap (f : M →ₗ[R] N →ₗ[R] N) (hf) (n : N) (r : R) :
foldl Q f hf n (algebraMap R _ r) = r • n := by |
rw [← foldr_reverse, reverse.commutes, foldr_algebraMap]
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 205 | 212 | theorem polyOfInterest_vars_eq (n : ℕ) : (polyOfInterest p n).vars =
((p : 𝕄) ^ (n + 1) * (wittMul p (n + 1) + (p : 𝕄) ^ (n + 1) * X (0, n + 1) * X (1, n + 1) -
X (0, n + 1) * rename (Prod.mk (1 : Fin 2)) (wittPolynomial p ℤ (n + 1)) -
X (1, n + 1) * rename (Prod.mk (0 : Fin 2)) (wittPolynomial p ℤ (n... |
have : (p : 𝕄) ^ (n + 1) = C ((p : ℤ) ^ (n + 1)) := by norm_cast
rw [polyOfInterest, this, vars_C_mul]
apply pow_ne_zero
exact mod_cast hp.out.ne_zero
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 508 | 511 | theorem coe_toIcoMod (θ ψ : ℝ) : ↑(toIcoMod two_pi_pos ψ θ) = (θ : Angle) := by |
rw [angle_eq_iff_two_pi_dvd_sub]
refine ⟨-toIcoDiv two_pi_pos ψ θ, ?_⟩
rw [toIcoMod_sub_self, zsmul_eq_mul, mul_comm]
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.cha... | Mathlib/RepresentationTheory/Character.lean | 106 | 109 | theorem average_char_eq_finrank_invariants (V : FdRep k G) :
⅟ (Fintype.card G : k) • ∑ g : G, V.character g = finrank k (invariants V.ρ) := by |
erw [← (isProj_averageMap V.ρ).trace] -- Porting note: Changed `rw` to `erw`
simp [character, GroupAlgebra.average, _root_.map_sum]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 250 | 259 | theorem one_act_left' : (R.one ▷ _) ≫ actLeft P Q = (λ_ _).hom := by |
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_
dsimp [X]
-- Porting note: had to replace `rw` by `erw`
slice_lhs 1 2 => erw [whisker_exchange]
slice_lhs 2 3 => rw [whiskerLeft_π_actLeft]
slice_lhs 1 2 => rw [associator_inv_naturality_left]
slice_lhs 2 3 => rw [← comp_whiskerRight, one_... |
/-
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.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set α` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 347 | 351 | theorem trim_anti_measurableSpace (m : OuterMeasure α) {m0 m1 : MeasurableSpace α}
(h : m0 ≤ m1) : @trim _ m1 m ≤ @trim _ m0 m := by |
simp only [le_trim_iff]
intro s hs
rw [trim_eq _ (h s hs)]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 1,101 | 1,108 | theorem LinearIndependent.inl_union_inr {s : Set M} {t : Set M'}
(hs : LinearIndependent R (fun x => x : s → M))
(ht : LinearIndependent R (fun x => x : t → M')) :
LinearIndependent R (fun x => x : ↥(inl R M M' '' s ∪ inr R M M' '' t) → M × M') := by |
refine (hs.image_subtype ?_).union (ht.image_subtype ?_) ?_ <;> [simp; simp; skip]
-- Note: #8386 had to change `span_image` into `span_image _`
simp only [span_image _]
simp [disjoint_iff, prod_inf_prod]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 940 | 946 | theorem mem_insert [@TransCmp α cmp] {t : RBSet α cmp} :
v' ∈ t.insert v ↔ v' ∈ t ∨ cmp v v' = .eq := by |
refine ⟨fun h => ?_, fun | .inl h => mem_insert_of_mem _ h | .inr h => mem_insert_of_eq _ h⟩
let ⟨_, h₁, h₂⟩ := mem_iff_mem_toList.1 h
match mem_toList_insert.1 h₁ with
| .inl ⟨h₃, _⟩ => exact .inl <| mem_iff_mem_toList.2 ⟨_, h₃, h₂⟩
| .inr rfl => exact .inr <| OrientedCmp.cmp_eq_eq_symm.1 h₂
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 968 | 970 | theorem rank_eq_one_iff : Module.rank F K = 1 ↔ K = ⊥ := by |
rw [← toSubalgebra_eq_iff, ← rank_eq_rank_subalgebra, Subalgebra.rank_eq_one_iff,
bot_toSubalgebra]
|
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib... | Mathlib/NumberTheory/Multiplicity.lean | 39 | 43 | theorem dvd_geom_sum₂_iff_of_dvd_sub {x y p : R} (h : p ∣ x - y) :
(p ∣ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) ↔ p ∣ n * y ^ (n - 1) := by |
rw [← mem_span_singleton, ← Ideal.Quotient.eq] at h
simp only [← mem_span_singleton, ← eq_zero_iff_mem, RingHom.map_geom_sum₂, h, geom_sum₂_self,
_root_.map_mul, map_pow, map_natCast]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 634 | 635 | theorem preimage_mul_const_Icc (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c) := by | simp [← Ici_inter_Iic, h]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,644 | 2,652 | theorem inr_inr_pushoutAssoc_inv [HasPushout (g₃ ≫ (pushout.inr : X₂ ⟶ Y₁)) g₄]
[HasPushout g₁ (g₂ ≫ (pushout.inl : X₂ ⟶ Y₂))] :
pushout.inr ≫ pushout.inr ≫ (pushoutAssoc g₁ g₂ g₃ g₄).inv = pushout.inr := by |
trans f₄ ≫ l₂'
· congr 1
exact
(pushoutPushoutLeftIsPushout g₁ g₂ g₃ g₄).comp_coconePointUniqueUpToIso_inv
(pushoutPushoutRightIsPushout g₁ g₂ g₃ g₄) WalkingCospan.right
· exact pushout.inr_desc _ _ _
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 172 | 175 | theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (h₁ : p ∣ q) (h₂ : natDegree q < natDegree p) :
q = 0 := by |
by_contra hc
exact (lt_iff_not_ge _ _).mp h₂ (natDegree_le_of_dvd h₁ hc)
|
/-
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 | 1,113 | 1,116 | theorem prod.triangle [HasBinaryProducts C] (X Y : C) :
(prod.associator X (⊤_ C) Y).hom ≫ prod.map (𝟙 X) (prod.leftUnitor Y).hom =
prod.map (prod.rightUnitor X).hom (𝟙 Y) := by |
ext <;> simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 452 | 457 | theorem coe_rpow_of_nonneg (x : ℝ≥0) {y : ℝ} (h : 0 ≤ y) : (x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) := by |
by_cases hx : x = 0
· rcases le_iff_eq_or_lt.1 h with (H | H)
· simp [hx, H.symm]
· simp [hx, zero_rpow_of_pos H, NNReal.zero_rpow (ne_of_gt H)]
· exact coe_rpow_of_ne_zero hx _
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cf... | Mathlib/Data/Nat/Pairing.lean | 64 | 73 | theorem unpair_pair (a b : ℕ) : unpair (pair a b) = (a, b) := by |
dsimp only [pair]; split_ifs with h
· show unpair (b * b + a) = (a, b)
have be : sqrt (b * b + a) = b := sqrt_add_eq _ (le_trans (le_of_lt h) (Nat.le_add_left _ _))
simp [unpair, be, Nat.add_sub_cancel_left, h]
· show unpair (a * a + a + b) = (a, b)
have ae : sqrt (a * a + (a + b)) = a := by
rw... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 574 | 575 | theorem pos_size_of_mem [LE α] [@DecidableRel α (· ≤ ·)] {x : α} {t : Ordnode α} (h : Sized t)
(h_mem : x ∈ t) : 0 < size t := by | cases t; · { contradiction }; · { simp [h.1] }
|
/-
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.Complex
#align_import analysis.special_function... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 142 | 143 | theorem sin_arctan (x : ℝ) : sin (arctan x) = x / √(1 + x ^ 2) := by |
rw_mod_cast [← tan_div_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | Mathlib/RingTheory/Kaehler.lean | 565 | 567 | theorem KaehlerDifferential.total_surjective :
Function.Surjective (Finsupp.total S (Ω[S⁄R]) S (KaehlerDifferential.D R S)) := by |
rw [← LinearMap.range_eq_top, Finsupp.range_total, KaehlerDifferential.span_range_derivation]
|
/-
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.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 976 | 980 | theorem HasColimit.isoOfEquivalence_inv_π {F : J ⥤ C} [HasColimit F] {G : K ⥤ C} [HasColimit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (k : K) :
colimit.ι G k ≫ (HasColimit.isoOfEquivalence e w).inv =
G.map (e.counitInv.app k) ≫ w.hom.app (e.inverse.obj k) ≫ colimit.ι F (e.inverse.obj k) := by |
simp [HasColimit.isoOfEquivalence, IsColimit.coconePointsIsoOfEquivalence_inv]
|
/-
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,330 | 1,335 | theorem Summable.of_nonneg_of_le {f g : β → ℝ} (hg : ∀ b, 0 ≤ g b) (hgf : ∀ b, g b ≤ f b)
(hf : Summable f) : Summable g := by |
lift f to β → ℝ≥0 using fun b => (hg b).trans (hgf b)
lift g to β → ℝ≥0 using hg
rw [NNReal.summable_coe] at hf ⊢
exact NNReal.summable_of_le (fun b => NNReal.coe_le_coe.1 (hgf b)) hf
|
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousFunction.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#alig... | Mathlib/Topology/Homotopy/Basic.lean | 172 | 175 | theorem extend_apply_of_one_le (F : Homotopy f₀ f₁) {t : ℝ} (ht : 1 ≤ t) (x : X) :
F.extend t x = f₁ x := by |
rw [← F.apply_one]
exact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' ℝ) F.curry ht) x
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_impor... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 38 | 39 | theorem HasSum.mul_right (a₂) (hf : HasSum f a₁) : HasSum (fun i ↦ f i * a₂) (a₁ * a₂) := by |
simpa only using hf.map (AddMonoidHom.mulRight a₂) (continuous_id.mul continuous_const)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.pi... | Mathlib/CategoryTheory/Pi/Basic.lean | 226 | 229 | theorem eqToHom_proj {x x' : ∀ i, C i} (h : x = x') (i : I) :
(eqToHom h : x ⟶ x') i = eqToHom (Function.funext_iff.mp h i) := by |
subst h
rfl
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#... | Mathlib/Data/Complex/Basic.lean | 262 | 262 | theorem im_ofReal_mul (r : ℝ) (z : ℂ) : (r * z).im = r * z.im := by | simp [ofReal']
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 114 | 116 | theorem cos_eq_neg_one_iff {x : ℂ} : cos x = -1 ↔ ∃ k : ℤ, π + k * (2 * π) = x := by |
rw [← neg_eq_iff_eq_neg, ← cos_sub_pi, cos_eq_one_iff]
simp only [eq_sub_iff_add_eq']
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearF... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 97 | 110 | theorem IsSymmetric.continuous [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) :
Continuous T := by |
-- We prove it by using the closed graph theorem
refine T.continuous_of_seq_closed_graph fun u x y hu hTu => ?_
rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜]
have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by
intro k
rw [← T.map_sub, hT]
refine tendsto_nhds_unique ((hTu.sub_c... |
/-
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.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa... | Mathlib/Data/Matrix/PEquiv.lean | 123 | 139 | theorem toMatrix_injective [DecidableEq n] [MonoidWithZero α] [Nontrivial α] :
Function.Injective (@toMatrix m n α _ _ _) := by |
classical
intro f g
refine not_imp_not.1 ?_
simp only [Matrix.ext_iff.symm, toMatrix_apply, PEquiv.ext_iff, not_forall, exists_imp]
intro i hi
use i
cases' hf : f i with fi
· cases' hg : g i with gi
-- Porting note: was `cc`
· rw [hf, hg] at hi
exact (hi rfl).elim
... |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import prob... | Mathlib/Probability/Martingale/BorelCantelli.lean | 287 | 287 | theorem process_zero : process s 0 = 0 := by | rw [process, Finset.range_zero, Finset.sum_empty]
|
/-
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.Computability.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 729 | 795 | theorem evaln_complete {c n x} : x ∈ eval c n ↔ ∃ k, x ∈ evaln k c n := by |
refine ⟨fun h => ?_, fun ⟨k, h⟩ => evaln_sound h⟩
rsuffices ⟨k, h⟩ : ∃ k, x ∈ evaln (k + 1) c n
· exact ⟨k + 1, h⟩
induction c generalizing n x with
simp [eval, evaln, pure, PFun.pure, Seq.seq, Option.bind_eq_some] at h ⊢
| pair cf cg hf hg =>
rcases h with ⟨x, hx, y, hy, rfl⟩
rcases hf hx with... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
#align_import numb... | Mathlib/NumberTheory/GaussSum.lean | 151 | 155 | theorem gaussSum_frob (χ : MulChar R R') (ψ : AddChar R R') :
gaussSum χ ψ ^ p = gaussSum (χ ^ p) (ψ ^ p) := by |
rw [← frobenius_def, gaussSum, gaussSum, map_sum]
simp_rw [pow_apply' χ fp.1.ne_zero, map_mul, frobenius_def]
rfl
|
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