Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | (x : ℕ) < c.sizeUpTo (i₁ : ℕ).succ := (c.mem_range_embedding_iff.1 hx₁).2
_ ≤ c.sizeUpTo (i₂ : ℕ) := monotone_sum_take _ A
_ ≤ x := (c.mem_range_embedding_iff.1 hx₂).1
theorem mem_range_embedding (j : Fin n) : j ∈ Set.range (c.embedding (c.index j)) := by
have : c.embedding (c.index j) (c.invEmbedd... | Mathlib/Combinatorics/Enumerative/Composition.lean | 382 | 396 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.Algebra.Group.ULift
import Mathlib.Data.Int.ModEq
import M... |
instance Prod.charZero_of_right [CharZero S] : CharZero (R × S) where
cast_injective _ _ h := CharZero.cast_injective congr(Prod.snd $h)
end Prod
| Mathlib/Algebra/CharP/Basic.lean | 147 | 152 |
/-
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.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for
`a < o`. -/
def IsNormal (f : Ordinal → Ordinal) : Prop :=
(∀ o, f o < f (succ o)) ∧ ∀ o, IsLimit o → ∀ a, f o ≤ a ↔ ∀ b < o, f b ≤ a
theorem IsNormal.limit_le {f} (H : IsNormal f) :
∀ {o}, IsLimit o → ∀ {a}, f o ≤ a ↔ ∀ b... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 363 | 370 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | lemma add_eq (f₁ f₂ : X' ⟶ Y') :
letI := addCommGroup L W X' Y'
f₁ + f₂ = add W eX eY f₁ f₂ := by
apply add_eq_add
| Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 288 | 291 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | end LinearOrderedSemiring
theorem Convex.midpoint_mem [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
[AddCommGroup E] [Module 𝕜 E] [Invertible (2 : 𝕜)] {s : Set E} {x y : E}
(h : Convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) : midpoint 𝕜 x y ∈ s :=
h.segment_subset hx hy <| midpoint_mem_segment x y
| Mathlib/Analysis/Convex/Basic.lean | 500 | 505 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | /-- `Finset.cons` version of `Finset.Ioo_insert_right`. -/
theorem Ioc_eq_cons_Ioo (h : a < b) : Ioc a b = (Ioo a b).cons b right_not_mem_Ioo := by
| Mathlib/Order/Interval/Finset/Basic.lean | 625 | 626 |
/-
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.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,811 | 1,812 | |
/-
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.DirectSum.Internal
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.LinearAlgebra.Finsupp.SumPr... | rw [← Finset.coe_subset, Finset.coe_singleton]
rfl
| Mathlib/Algebra/MonoidAlgebra/Grading.lean | 63 | 64 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | rw [h diff_subset B' ⟨hB', disjoint_sdiff_left⟩]
· simpa [hB'.subset_ground]
| Mathlib/Data/Matroid/Basic.lean | 515 | 516 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Probability.Kernel.CondDistrib
import Mathlib.Probability.ConditionalProbability
/-!
# Kernel as... | simp_rw [condExpKernel_apply_eq_condDistrib]
refine Measurable.mono ?_ (inf_le_left : m ⊓ mΩ ≤ m) le_rfl
convert measurable_condDistrib (μ := μ) hs
rw [MeasurableSpace.comap_id]
@[deprecated (since := "2025-01-21")] alias measurable_condexpKernel := measurable_condExpKernel
| Mathlib/Probability/Kernel/Condexp.lean | 125 | 130 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Tactic.Lift... | pow_lt_one' (M := Mᵒᵈ) ha hk
@[to_additive]
lemma le_self_pow (ha : 1 ≤ a) (hn : n ≠ 0) : a ≤ a ^ n := by
simpa using pow_le_pow_right' ha (Nat.one_le_iff_ne_zero.2 hn)
end Left
| Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 71 | 77 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | simp only [sum, comapDomain_apply, (· ∘ ·), comapDomain]
exact Finset.sum_preimage_of_bij f _ hf fun x => g x (l x)
theorem eq_zero_of_comapDomain_eq_zero [Zero M] (f : α → β) (l : β →₀ M)
(hf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support) : comapDomain f l hf.injOn = 0 → l = 0 := by
rw [← support_eq_empty, ← ... | Mathlib/Data/Finsupp/Basic.lean | 626 | 632 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
/-!
# Preservati... | end CategoryTheory.Adjunction
namespace CategoryTheory.Functor
variable {C D : Type*} [Category C] [Category D] {F : C ⥤ D} {A B : C} (f : A ⟶ B)
| Mathlib/CategoryTheory/Functor/EpiMono.lean | 273 | 278 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | rw [← thickenedIndicatorAux_closure_eq, thickenedIndicatorAux_one δ (closure E) x_mem]
theorem thickenedIndicatorAux_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
| Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 79 | 81 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... |
lemma derivWithin_mul_const_field (u : 𝕜') :
derivWithin (fun y => v y * u) s x = derivWithin v s x * u := by
by_cases hv : DifferentiableWithinAt 𝕜 v s x
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 248 | 251 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,055 | 1,055 | |
/-
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.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... |
theorem transReflReparamAux_zero : transReflReparamAux 0 = 0 := by
norm_num [transReflReparamAux]
theorem transReflReparamAux_one : transReflReparamAux 1 = 1 := by
| Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 130 | 134 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | | _, _, ⟨b₁, l₁, l₂, a₁_nin_l₁, rfl, rfl⟩, _ =>
if h' : a₂ ∈ l₁.keys then by
simp [kerase_append_left h',
kerase_append_right (mt (mem_keys_kerase_of_ne h).mp a₁_nin_l₁)]
else by
simp [kerase_append_right h', kerase_append_right a₁_nin_l₁,
... | Mathlib/Data/List/Sigma.lean | 510 | 523 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Polynom... |
theorem eq_one_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (hB : B < 0) (h1 : p.Monic)
(h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) : p = 1 :=
h1.natDegree_eq_zero_iff_eq_one.mp (by
contrapose! hB
| Mathlib/Topology/Algebra/Polynomial.lean | 144 | 148 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.PFun
/-!
# `Tendsto` for relations and partial functions
This file generalizes `Filter` definitions from funct... | `Filter.Tendsto` to relations. -/
def RTendsto' (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :=
l₁ ≤ l₂.rcomap' r
theorem rtendsto'_def (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
RTendsto' r l₁ l₂ ↔ ∀ s ∈ l₂, r.preimage s ∈ l₁ := by
unfold RTendsto' rcomap'; constructor
· simpa [le_def, Rel.mem_image] ... | Mathlib/Order/Filter/Partial.lean | 165 | 173 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Insert
/-!
# Subsingleton
Defines the predicate `Subsingleton s : Prop`, saying that `s` has at most one element.
Also def... | protected theorem Nontrivial.nonempty (hs : s.Nontrivial) : s.Nonempty :=
let ⟨x, hx, _⟩ := hs
⟨x, hx⟩
protected theorem Nontrivial.ne_empty (hs : s.Nontrivial) : s ≠ ∅ :=
| Mathlib/Data/Set/Subsingleton.lean | 194 | 198 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | refine le_trans ?_ (add_le_add_left (le_of_lt hε) _)
rw [← ENNReal.tsum_add]
choose g hg using
show ∀ i, ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal (ε' i) by
intro i
have hl :=
ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 328 | 345 |
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
/-!
# Gram-Schmidt Orthogonalization and Orthonor... |
/-- Given an indexed family `f : ι → E` of vectors in an inner product space `E`, for which the
size of the index set is the dimension of `E`, the matrix of coefficients of `f` with respect to the
orthonormal basis `gramSchmidtOrthonormalBasis` constructed from `f` is upper-triangular. -/
theorem gramSchmidtOrthonorma... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 345 | 351 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex cones
In a `𝕜`-module `E`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` w... | Mathlib/Analysis/Convex/Cone/Basic.lean | 658 | 661 | |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | rw [← conj_mul, hd.cycleType, (hd.conj _).cycleType, hσ, hπ]
theorem sum_cycleType (σ : Perm α) : σ.cycleType.sum = #σ.support := by
induction σ using cycle_induction_on with
| base_one => simp
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 147 | 151 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,243 | 1,247 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | { toFun := fun z ↦ f.smulRight z
map_add' := fun x y ↦ by ext; simp
map_smul' := fun c x ↦ by ext; simp [smul_smul, mul_comm] }
map_add' := fun f g ↦ by ext; simp [add_smul]
map_smul' := fun c f ↦ by ext; simp [smul_smul] }
1 (fun f z ↦ by simp [norm_smulRight])
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 803 | 809 |
/-
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.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s :=
rfl
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,215 | 1,216 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... |
theorem measure_eq_measure_of_null_diff {s t : Set α} (hst : s ⊆ t) (h_nulldiff : μ (t \ s) = 0) :
μ s = μ t := measure_congr <|
EventuallyLE.antisymm (HasSubset.Subset.eventuallyLE hst) (ae_le_set.mpr h_nulldiff)
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 267 | 270 |
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
/-! # The Wallis formula for Pi
This file establishes the Wallis product for `π` (`Real.tendsto_prod_pi_div_two`). ... | rw [← div_le_one pi_div_two_pos, div_eq_inv_mul]
rw [W_eq_integral_sin_pow_div_integral_sin_pow, div_le_one (integral_sin_pow_pos _)]
apply integral_sin_pow_succ_le
theorem le_W (k : ℕ) : ((2 : ℝ) * k + 1) / (2 * k + 2) * (π / 2) ≤ W k := by
| Mathlib/Data/Real/Pi/Wallis.lean | 78 | 82 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Images of pairs of submodules under bilinear maps
This file provides `Subm... |
theorem map₂_iSup_right (f : M →ₗ[R] N →ₗ[R] P) (s : Submodule R M) (t : ι → Submodule R N) :
map₂ f s (⨆ i, t i) = ⨆ i, map₂ f s (t i) := by
| Mathlib/Algebra/Module/Submodule/Bilinear.lean | 135 | 137 |
/-
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.Order.Monotone.Odd
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.Spe... |
/-- The complex cosine function is everywhere differentiable, with the derivative `-sin x`. -/
theorem hasDerivAt_cos (x : ℂ) : HasDerivAt cos (-sin x) x :=
(hasStrictDerivAt_cos x).hasDerivAt
theorem contDiff_cos {n} : ContDiff ℂ n cos :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 68 | 73 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Batteries.Data.List.Basic
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two... | @[simp]
theorem levenshtein_nil_nil : levenshtein C [] [] = 0 := by
| Mathlib/Data/List/EditDistance/Defs.lean | 274 | 275 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... | · exact Finset.mem_insert_of_mem <| hn.2 _ H
/-- If `p ∉ s` is a prime and `n` is `s`-factored, then `p` and `n` are coprime. -/
lemma Prime.factoredNumbers_coprime {s : Finset ℕ} {p n : ℕ} (hp : p.Prime) (hs : p ∉ s)
(hn : n ∈ factoredNumbers s) :
Nat.Coprime p n := by
| Mathlib/NumberTheory/SmoothNumbers.lean | 196 | 201 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | theorem gold_pos : 0 < φ :=
mul_pos (by apply add_pos <;> norm_num) <| inv_pos.2 zero_lt_two
theorem gold_ne_zero : φ ≠ 0 :=
| Mathlib/Data/Real/GoldenRatio.lean | 91 | 94 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 580 | 590 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,408 | 2,432 | |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
/-!
# ω-limits
For a function `ϕ : τ → α → β` where `β` is a topological space, we
define the ω-limit under `ϕ` of... | simp only [subset_def, mem_omegaLimit_iff_frequently₂, frequently_iff]
intro _ h
rintro n hn u hu
rcases mem_nhds_iff.mp hn with ⟨o, ho₁, ho₂, ho₃⟩
| Mathlib/Dynamics/OmegaLimit.lean | 340 | 343 |
/-
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
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | /-- If `K` is compact and `V` has nonempty interior, then the index `(K : V)` is well-defined,
there is a finite set `t` satisfying the desired properties. -/
@[to_additive addIndex_defined
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 142 | 144 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.G... | mapsTo := h₁.mapsTo.prodMap h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h := by
simp only [mem_prod, forall_and, Prod.forall] at hsA htA
simp only [Prod.ext_iff, fst_prod, snd_prod, map_map, Function.comp_apply, Prod.map_fst,
Prod.map_snd] at h ⊢
rw [← Function.comp_def, ← map_map, ← map_map,... | Mathlib/Combinatorics/Additive/FreimanHom.lean | 352 | 366 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.Presheaf.Colimits
import Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits
/-!
# Free sheaves of modules
In this file, we construct ... | lemma freeHomEquiv_symm_comp {M N : SheafOfModules.{u} R} {I : Type u} (s : I → M.sections)
(p : M ⟶ N) :
M.freeHomEquiv.symm s ≫ p = N.freeHomEquiv.symm (fun i ↦ sectionsMap p (s i)) :=
N.freeHomEquiv.injective (by ext; simp [freeHomEquiv_comp_apply])
| Mathlib/Algebra/Category/ModuleCat/Sheaf/Free.lean | 50 | 53 |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,651 | 1,656 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | variable (K)
/-- Conjugation as a ring equivalence. This is used to convert the inner product into a
sesquilinear product. -/
| Mathlib/Analysis/RCLike/Basic.lean | 363 | 366 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | Mathlib/MeasureTheory/Integral/Average.lean | 315 | 315 | |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... |
theorem domCongr_toAlgHom (e : G ≃+ H) : (domCongr k A e).toAlgHom = mapDomainAlgHom k A e :=
AlgHom.ext fun _ => equivMapDomain_eq_mapDomain _ _
@[simp] theorem domCongr_apply (e : G ≃+ H) (f : MonoidAlgebra A G) (h : H) :
domCongr k A e f h = f (e.symm h) :=
rfl
@[simp] theorem domCongr_support (e : G ≃+ H... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 558 | 570 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.Subring
import Mathlib.Algebra.Polynomial.Monic
/-!
# Polynomials that lift
Gi... | Mathlib/Algebra/Polynomial/Lifts.lean | 257 | 258 | |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 1,047 | 1,048 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | ∀ (n : ℕ)
(p :
Σ' (f : P.X n ⟶ Q.X (n + 1)) (f' : P.X (n + 1) ⟶ Q.X (n + 2)),
e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)),
Σ' f'' : P.X (n + 2) ⟶ Q.X (n + 3),
| Mathlib/Algebra/Homology/Homotopy.lean | 450 | 454 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | theorem prod_inter_prod : s₁ ×ˢ t₁ ∩ s₂ ×ˢ t₂ = (s₁ ∩ s₂) ×ˢ (t₁ ∩ t₂) := by
ext ⟨x, y⟩
simp [and_assoc, and_left_comm]
| Mathlib/Data/Set/Prod.lean | 126 | 128 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# p-adic norm
This file defines the `p`-adic norm on `ℚ`.
The `p`-adic valuation on `ℚ` is the difference o... | apply absurd h
apply zpow_ne_zero
| Mathlib/NumberTheory/Padics/PadicNorm.lean | 123 | 124 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | ((ContinuousLinearEquiv.arrowCongr isoE' isoF).symm : (E' →L[𝕜] F) →L[𝕜] eE' →L[𝕜] eF) L
let R := fun q : eP × eG => (ef ⋆[eL, eμ] eg q.1) q.2
have R_contdiff : ContDiffOn 𝕜 n R ((isoP ⁻¹' s) ×ˢ univ) := by
have hek : IsCompact (isoG ⁻¹' k) := isoG.toHomeomorph.isClosedEmbedding.isCompact_preimage hk
... | Mathlib/Analysis/Convolution.lean | 1,249 | 1,292 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
/-!
# Complex roots of unity
In this file we show that th... | rw [IsPrimitiveRoot.iff_def]
simp only [← exp_nat_mul, exp_eq_one_iff]
have hn0 : (n : ℂ) ≠ 0 := mod_cast h0
constructor
· use i
field_simp [hn0, mul_comm (i : ℂ), mul_comm (n : ℂ)]
· simp only [hn0, mul_right_comm _ _ ↑n, mul_left_inj' two_pi_I_ne_zero, Ne, not_false_iff,
mul_comm _ (i : ℂ), ← mu... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 33 | 50 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... | simp at hnd
variable {x y : α}
theorem isRelPrime_zero_left : IsRelPrime 0 x ↔ IsUnit x :=
⟨(· (dvd_zero _) dvd_rfl), IsUnit.isRelPrime_right⟩
theorem isRelPrime_zero_right : IsRelPrime x 0 ↔ IsUnit x :=
| Mathlib/Algebra/GroupWithZero/Divisibility.lean | 71 | 78 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | ENNReal.toReal_nonneg
theorem infsep_pos : 0 < s.infsep ↔ 0 < s.einfsep ∧ s.einfsep < ∞ := by
simp_rw [infsep, ENNReal.toReal_pos_iff]
theorem Subsingleton.infsep_zero (hs : s.Subsingleton) : s.infsep = 0 :=
Set.infsep_zero.mpr <| Or.inr hs.einfsep
| Mathlib/Topology/MetricSpace/Infsep.lean | 287 | 293 |
/-
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.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Sequence of measurable functions associated to a sequence o... | simp only [aeSeq_eq_mk_of_mem_aeSeqSet hf hx i, mk_eq_fun_of_mem_aeSeqSet hf hx i]
theorem prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
| Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 59 | 61 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... | isUpperSet_iUnion H
lemma stableUnderGeneralization_iInter {ι : Sort*} (S : ι → Set X)
| Mathlib/Topology/Inseparable.lean | 276 | 278 |
/-
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
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | theorem e0_mul_v_mul_e0 (m : M) : e0 Q * v Q m * e0 Q = v Q m := by
rw [← neg_v_mul_e0, ← neg_mul, mul_assoc, e0_mul_e0, mul_neg_one, neg_neg]
@[simp]
theorem reverse_v (m : M) : reverse (Q := Q' Q) (v Q m) = v Q m :=
| Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 82 | 86 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... |
@[deprecated pow_right_strictAnti₀ (since := "2024-11-13")]
theorem pow_right_strictAnti (h₀ : 0 < a) (h₁ : a < 1) : StrictAnti (a ^ ·) :=
pow_right_strictAnti₀ h₀ h₁
| Mathlib/Algebra/Order/Ring/Basic.lean | 106 | 109 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | simpa only [LeftHomologyData.homologyIso_leftHomologyData, Iso.symm_inv,
RightHomologyData.homologyIso_rightHomologyData, Iso.symm_hom] using
leftRightHomologyComparison'_fac S.leftHomologyData S.rightHomologyData
variable {S}
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 679 | 684 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | section IsDomain
variable (R : Type u) [CommRing R] [IsDomain R]
| Mathlib/Algebra/Polynomial/Expand.lean | 297 | 299 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 789 | 790 | |
/-
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.Defs
import Mathlib.Analysis.NormedSpac... |
@[deprecated (since := "2025-03-18")]
alias tendsto_log_nhdsWithin_zero_right := tendsto_log_nhdsGT_zero
theorem tendsto_log_nhdsNE_zero : Tendsto log (𝓝[≠] 0) atBot := by
simpa [comp_def] using tendsto_log_nhdsGT_zero.comp tendsto_abs_nhdsNE_zero
@[deprecated (since := "2025-03-18")]
alias tendsto_log_nhdsWithin... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 327 | 339 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | h.sub modEq_rfl
protected theorem add_left_cancel' (c : α) : c + a ≡ c + b [PMOD p] → a ≡ b [PMOD p] :=
| Mathlib/Algebra/ModEq.lean | 194 | 196 |
/-
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.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | theorem castSucc_lt_or_lt_succ (p : Fin (n + 1)) (i : Fin n) : castSucc i < p ∨ p < i.succ := by
simp [Fin.lt_def, -val_fin_lt]; omega
| Mathlib/Data/Fin/Basic.lean | 611 | 612 |
/-
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.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... |
lemma Ioc_filter_modEq_eq (v : ℤ) :
{x ∈ Ioc a b | x ≡ v [ZMOD r]} =
{x ∈ Ioc (a - v) (b - v) | r ∣ x}.map ⟨(· + v), add_left_injective v⟩ := by
ext x
simp_rw [mem_map, mem_filter, mem_Ioc, Function.Embedding.coeFn_mk, ← eq_sub_iff_add_eq,
| Mathlib/Data/Int/CardIntervalMod.lean | 34 | 39 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | (ShortComplex.mk _ _ hφ).Exact ∧ Epi φ := by
suffices ∀ (i : ℕ) (hx : (ComplexShape.down ℕ).prev 0 = i)
(hφ : K.d i 0 ≫ φ = 0), IsIso (K.descOpcycles φ i hx hφ) ↔
(ShortComplex.mk _ _ hφ).Exact ∧ Epi φ from this 1 (by simp) hφ
rintro _ rfl hφ
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 748 | 752 |
/-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# Archimedean groups
This file prov... | additive commutative group is disjoint with the interval `Set.Ioo 0 a` for some positive `a`, then
this is a cyclic subgroup."]
theorem Subgroup.cyclic_of_isolated_one {H : Subgroup G} {a : G} (h₀ : 1 < a)
(hd : Disjoint (H : Set G) (Ioo 1 a)) : ∃ b, H = closure {b} := by
| Mathlib/GroupTheory/Archimedean.lean | 98 | 101 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | simp [toIcoMod_eq_add_fract_mul]
theorem toIocMod_eq_sub_fract_mul (a b : α) :
toIocMod hp a b = a + p - Int.fract ((a + p - b) / p) * p := by
rw [toIocMod, toIocDiv_eq_neg_floor, Int.fract]
field_simp
ring
| Mathlib/Algebra/Order/ToIntervalMod.lean | 885 | 892 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... |
theorem FamilyOfElements.Compatible.functorPullback (h : x.Compatible) :
(x.functorPullback F).Compatible := by
intro Z₁ Z₂ W g₁ g₂ f₁ f₂ h₁ h₂ eq
| Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 281 | 284 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Graded tensor products over graded algebras
The graded tens... | (a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) =
(-1 : ℤˣ)^(j₁ * i₂) • ((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) := by
dsimp only [mul_def, mulHom_apply, of_symm_of]
dsimp [auxEquiv, tmul]
rw [decompose_coe, decompose_coe]
simp_rw [← lof_eq_of R]
rw [tmul_of_gradedMul_of_tmul]
simp_rw [lof_eq_of R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 183 | 198 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... |
protected theorem IsCompact.insert (hs : IsCompact s) (a) : IsCompact (insert a s) :=
isCompact_singleton.union hs
-- TODO: reformulate using `𝓝ˢ`
/-- If `V : ι → Set X` is a decreasing family of closed compact sets then any neighborhood of
`⋂ i, V i` contains some `V i`. We assume each `V i` is compact *and* clos... | Mathlib/Topology/Compactness/Compact.lean | 478 | 484 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.GroupTheory.Subsemigroup.Center
/-!
# Centers of monoids
## Main definitions
* `Submonoid.center`: the c... | instance decidableMemCenter (a) [Decidable <| ∀ b : M, b * a = a * b] : Decidable (a ∈ center M) :=
decidable_of_iff' _ mem_center_iff
| Mathlib/GroupTheory/Submonoid/Center.lean | 79 | 81 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 3,208 | 3,210 | |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Prime.Defs
import Mathlib.Algebra.Ring.Units
import... |
end CommRing
| Mathlib/RingTheory/Prime.lean | 70 | 73 |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.List.Sym
/-! # Unordered tuples of elements of a multiset
Defines `Multiset.sym` and the specialized `Multiset.sym2` for computing multisets of all
un... | @[simp]
theorem sym2_zero : (0 : Multiset α).sym2 = 0 := rfl
| Mathlib/Data/Multiset/Sym.lean | 50 | 52 |
/-
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.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Logic.Equiv.Fin.... | rw [← f.quotKerEquivRange.lift_rank_eq, ← lift_add, rank_quotient_add_rank]
/-- The **rank-nullity theorem** -/
theorem LinearMap.rank_range_add_rank_ker (f : M →ₗ[R] M₁) :
| Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 75 | 78 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | contractNth_apply_of_lt]
· assumption
· rw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]
exact le_of_lt h
· rwa [succAbove_of_castSucc_lt, succAbove_of_le_castSucc, partialProd_succ,
| Mathlib/Algebra/BigOperators/Fin.lean | 259 | 263 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Mario Carneiro, Reid Barton, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Func... | Mathlib/Topology/Sheaves/Presheaf.lean | 445 | 461 | |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | let .app i (a : Q(ℤ)) ← whnfR e | failure
guard <|← withNewMCtxDepth <| isDefEq i q(Int.cast (R := $α))
| Mathlib/Tactic/NormNum/Basic.lean | 125 | 126 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... |
theorem IsRelPrime.of_mul_add_right_left (h : IsRelPrime (z * y + x) y) : IsRelPrime x y :=
(add_comm _ x ▸ h).of_add_mul_right_left
| Mathlib/RingTheory/Coprime/Basic.lean | 217 | 219 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... |
The following statements no longer any explicit interval, as they only require
conditions holding eventually.
-/
namespace HasDerivAt
/-- L'Hôpital's rule for approaching a real from the right, `HasDerivAt` version -/
theorem lhopital_zero_nhdsGT (hff' : ∀ᶠ x in 𝓝[>] a, HasDerivAt f (f' x) x)
| Mathlib/Analysis/Calculus/LHopital.lean | 256 | 265 |
/-
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.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | pure (.positive q(Real.exp_pos $a))
| Mathlib/Data/Complex/Exponential.lean | 680 | 680 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | Mathlib/Data/Nat/GCD/Basic.lean | 300 | 306 | |
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepa... | rcases h with ⟨b, hbc, hb⟩
refine ⟨⟨b, hbc, fun t ht ↦ hb.symm ▸ .basic t ht, ?_⟩⟩
rw [hb] at ‹SeparatesPoints _›
convert separating_of_generateFrom b
simp
variable (α)
| Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 232 | 239 |
/-
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.Group.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Order properties of cast of integers
This... | Mathlib/Algebra/Order/Ring/Cast.lean | 63 | 63 | |
/-
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
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | rw [ne_eq, encard_eq_zero, nonempty_iff_ne_empty]
@[simp] theorem encard_pos : 0 < s.encard ↔ s.Nonempty := by
| Mathlib/Data/Set/Card.lean | 107 | 109 |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | simp [CommGroupWithZero.coe_normUnit _ this]
theorem map_dvd_map' [Field k] (f : R →+* k) {x y : R[X]} : x.map f ∣ y.map f ↔ x ∣ y := by
by_cases H : x = 0
· rw [H, Polynomial.map_zero, zero_dvd_iff, zero_dvd_iff, Polynomial.map_eq_zero]
· classical
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 555 | 560 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Types.Shapes
import Mathlib.Tactic.ApplyFun
/-!
# The equalizer diagram sheaf conditi... | (hf : R f), (Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₁ =
(Pi.π _ ⟨Y, f, hf⟩ : FirstObj P R ⟶ _) z₂) : z₁ = z₂ := by
apply Limits.Types.limit_ext
rintro ⟨⟨Y, f, hf⟩⟩
exact h Y f hf
variable (P R)
| Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 59 | 65 |
/-
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
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | induction l with
| nil => simp
| cons hd tl ih =>
simp only [List.mem_cons, forall_eq_or_imp] at hl
rcases hl with ⟨hhd, htl⟩
rw [List.sum_cons]
exact le_trans (degree_add_le hd tl.sum) (max_le hhd (ih htl))
theorem degree_list_sum_le (l : List S[X]) : degree l.sum ≤ (l.map natDegree).maximum := ... | Mathlib/Algebra/Polynomial/BigOperators.lean | 66 | 77 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... |
theorem intDegree_add {x y : RatFunc K} (hxy : x + y ≠ 0) :
(x + y).intDegree =
(x.num * y.denom + x.denom * y.num).natDegree - (x.denom * y.denom).natDegree :=
natDegree_sub_eq_of_prod_eq (num_ne_zero hxy) (x + y).denom_ne_zero
(num_mul_denom_add_denom_mul_num_ne_zero hxy) (mul_ne_zero x.denom_ne_zero... | Mathlib/FieldTheory/RatFunc/Degree.lean | 85 | 91 |
/-
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.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
forms a measura... | exact lintegral_congr_ae <| (ae_ae_of_ae_join hfg).mono fun μ hμ ↦
.symm <| lintegral_congr_ae hμ
simp_rw [lintegral_eq_iSup_eapprox_lintegral hfm, SimpleFunc.lintegral,
| Mathlib/MeasureTheory/Measure/GiryMonad.lean | 176 | 178 |
/-
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
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | convert prod_replicate (Multiset.card t) (1 : R)
· simp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and]
rintro i hi
| Mathlib/Algebra/Polynomial/BigOperators.lean | 193 | 195 |
/-
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.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | theorem x_mk (x y : ℤ) (prop : x ^ 2 - d * y ^ 2 = 1) : (mk x y prop).x = x :=
| Mathlib/NumberTheory/Pell.lean | 137 | 137 |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,543 | 1,544 | |
/-
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.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... |
@[ext]
theorem ext {w w' : NormalWord d}
| Mathlib/GroupTheory/HNNExtension.lean | 225 | 227 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | `liminf_reparam j` is equal to `j` if `f` is bounded below on `s j`, and otherwise to some
index `k` such that `f` is bounded below on `s k` (if there exists one).
To ensure good measurability behavior, this index `k` is chosen as the minimal suitable index.
This function is used to write down a liminf in a measurable ... | Mathlib/Order/LiminfLimsup.lean | 950 | 959 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | theorem y_mul_dvd (n) : ∀ k, yn a1 n ∣ yn a1 (n * k)
| 0 => dvd_zero _
| k + 1 => by
rw [Nat.mul_succ, yn_add]; exact dvd_add (dvd_mul_left _ _) ((y_mul_dvd _ k).mul_right _)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 364 | 368 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | protected def id : StieltjesFunction where
toFun := id
mono' _ _ := id
right_continuous' _ := continuousWithinAt_id
@[simp]
theorem id_leftLim (x : ℝ) : leftLim StieltjesFunction.id x = x :=
| Mathlib/MeasureTheory/Measure/Stieltjes.lean | 83 | 89 |
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
/-!
# Gram-Schmidt Orthogonalization and Orthonor... | (gramSchmidtOrthonormalBasis h f).toBasis.det f =
∏ i, ⟪gramSchmidtOrthonormalBasis h f i, f i⟫ := by
convert Matrix.det_of_upperTriangular (gramSchmidtOrthonormalBasis_inv_blockTriangular h f)
exact ((gramSchmidtOrthonormalBasis h f).repr_apply_apply (f _) _).symm
end OrthonormalBasis
| Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 354 | 369 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# GCD... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 240 | 254 |
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