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) 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.SpecialFunctions.Pow.Complex
/-!
... | constructor <;> · rintro ⟨k, rfl⟩; use -k; simp
_ ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x := exists_or.symm
theorem sin_eq_sin_iff {x y : ℂ} :
sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x := by
simp only [← Complex.cos_sub_pi_div_two, cos_eq_cos_iff, sub_eq_iff_eq_add]
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 90 | 101 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 837 | 840 | |
/-
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.Mul
import Mathlib.Data.PEquiv
/-!
# partial equivalences for matrices
Using partial equivalences to represent matrices.
This file introduces... | (1 : Matrix n n α) - (single i i).toMatrix - (single j j).toMatrix + (single i j).toMatrix +
(single j i).toMatrix := by
ext
dsimp [toMatrix, single, Equiv.swap_apply_def, Equiv.toPEquiv, one_apply]
| Mathlib/Data/Matrix/PEquiv.lean | 165 | 168 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | rw [← Polynomial.aeval_algHom_apply, ← Polynomial.aeval_algHom_apply, h]
open Ideal Finset Submodule in
| Mathlib/RingTheory/PowerBasis.lean | 128 | 130 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | @[norm_cast, simp]
theorem coe_zero : ((0 : L₁ →ₗ⁅R⁆ L₂) : L₁ → L₂) = 0 :=
rfl
| Mathlib/Algebra/Lie/Basic.lean | 391 | 393 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 992 | 996 | |
/-
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... | theorem div_def : p / q = C (leadingCoeff q)⁻¹ * (p /ₘ (q * C (leadingCoeff q)⁻¹)) :=
rfl
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 321 | 323 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Inv
/-! # Equivalence between fintypes
This file contains some basic results on equivalences wher... | Mathlib/Logic/Equiv/Fintype.lean | 138 | 142 | |
/-
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.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... |
theorem Monic.add_of_right (hq : Monic q) (hpq : degree p < degree q) : Monic (p + q) := by
rwa [Monic, leadingCoeff_add_of_degree_lt hpq]
theorem Monic.of_mul_monic_left (hp : p.Monic) (hpq : (p * q).Monic) : q.Monic := by
contrapose! hpq
rw [Monic.def] at hpq ⊢
rwa [leadingCoeff_monic_mul hp]
| Mathlib/Algebra/Polynomial/Monic.lean | 117 | 125 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.ColimitLimit
import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Finite
imp... | simp only [← Category.assoc, Iso.comp_inv_eq,
Limits.colimitObjIsoColimitCompEvaluation_ι_app_hom,
Limits.HasColimit.isoOfNatIso_ι_hom, NatIso.ofComponents_hom_app]
dsimp
simp
end
end CategoryTheory.Limits
| Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean | 397 | 410 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finite.Card
import Mathlib.Data.Set.Finite.Range
/-!
# ... |
@[to_additive]
| Mathlib/Algebra/Group/Subgroup/Finite.lean | 142 | 143 |
/-
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.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
/-!
# Finite intervals in a disjoint union
This file provides the... | exact inr_ne_inl h
theorem sumLift₂_eq_empty :
sumLift₂ f g a b = ∅ ↔
(∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧
∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by
| Mathlib/Data/Sum/Interval.lean | 68 | 73 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | letI := hp
rcases FiniteField.card K p with ⟨⟨n, npos⟩, ⟨hp, hn⟩⟩
haveI : Fact p.Prime := ⟨hp⟩
dsimp at hn
| Mathlib/FieldTheory/Finite/Basic.lean | 459 | 462 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.Differentiation
/-!
# Besicovitch covering theorems
The topological Besicovitch covering theorem ensures that, in a nice... | obtain ⟨r, t, rfl⟩ : ∃ (r : α → ℝ) (t : Set α), v = graphOn r t := by
have I : ∀ p ∈ v, 0 ≤ p.2 := fun p hp => (vg p hp).2.1.le
rw [exists_eq_graphOn]
refine fun x hx y hy heq ↦ v_disj.eq hx hy <| not_disjoint_iff.2 ⟨x.1, ?_⟩
simp [*]
have hinj : InjOn (fun x ↦ (x, r x)) t := LeftInvOn.injOn (f₁' :=... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 843 | 853 |
/-
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.Single
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# The homology of single complexes
The main definition in this file ... | Mathlib/Algebra/Homology/SingleHomology.lean | 260 | 262 | |
/-
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... | (exp_bound hx (by decide))
_ ≤ ‖x‖ ^ 2 * 1 := by gcongr; norm_num [Nat.factorial]
| Mathlib/Data/Complex/Exponential.lean | 445 | 446 |
/-
Copyright (c) 2023 Yaël Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
lemma UniformConvexOn.convexOn (hf : UniformConvexOn s φ f) (hφ : 0 ≤ φ) : ConvexOn ℝ s f := by
| Mathlib/Analysis/Convex/Strong.lean | 63 | 64 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma P4_neg_left : P4 x₁ x₂ y ↔ P4 (-x₂) (-x₁) y := by
simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg, ← P3_neg]
| Mathlib/SetTheory/Surreal/Multiplication.lean | 121 | 122 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | instance [IsNoetherian R M] :
IsNilpotent L (genWeightSpace M (0 : L → R)) :=
isNilpotent_iff_forall'.mpr <| isNilpotent_toEnd_genWeightSpace_zero M
| Mathlib/Algebra/Lie/Weights/Basic.lean | 358 | 360 |
/-
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.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | lemma tilted_apply (μ : Measure α) [SFinite μ] (f : α → ℝ) (s : Set α) :
μ.tilted f s = ∫⁻ a in s, ENNReal.ofReal (exp (f a) / ∫ x, exp (f x) ∂μ) ∂μ := by
rw [Measure.tilted, withDensity_apply' _ s]
| Mathlib/MeasureTheory/Measure/Tilted.lean | 101 | 103 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | exact h_mul _ ih
| Mathlib/Algebra/Group/Basic.lean | 895 | 895 |
/-
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.... | simp only [Matrix.cons_val_fin_one]
| Mathlib/ModelTheory/Semantics.lean | 281 | 282 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... | end DecidableEq
@[simp]
theorem linearEquivFunOnFintype_symm_coe [Fintype ι] (f : ⨁ i, M i) :
(linearEquivFunOnFintype R ι M).symm f = f := by
| Mathlib/Algebra/DirectSum/Module.lean | 172 | 176 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Binom... |
@[simp]
theorem single_order_mul_powerSeriesPart (x : R⸨X⸩) :
(single x.order 1 : R⸨X⸩) * x.powerSeriesPart = x := by
ext n
rw [← sub_add_cancel n x.order, coeff_single_mul_add, sub_add_cancel, one_mul]
by_cases h : x.order ≤ n
· rw [Int.eq_natAbs_of_nonneg (sub_nonneg_of_le h), coeff_coe_powerSeries,
... | Mathlib/RingTheory/LaurentSeries.lean | 231 | 240 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Continuity
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.MetricSpace... | @[to_additive (attr := simp) norm_mk]
theorem norm_mk' (p : E) : ‖mk p‖ = ‖p‖ := rfl
@[to_additive]
instance : NormedCommGroup (SeparationQuotient E) where
__ : CommGroup (SeparationQuotient E) := instCommGroup
dist_eq := Quotient.ind₂ dist_eq_norm_div
@[to_additive]
theorem mk_eq_one_iff {p : E} : mk p = 1 ↔ ‖p‖... | Mathlib/Analysis/Normed/Group/Uniform.lean | 372 | 383 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
/-... | (h : angle x (x - y) = angle y (y - x)) (hpi : angle x y ≠ π) : ‖x‖ = ‖y‖ := by
replace h := Real.arccos_injOn (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x (x - y)))
(abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one y (y - x))) h
by_cases hxy : x = y
· rw [hxy]
| Mathlib/Geometry/Euclidean/Triangle.lean | 71 | 75 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | -/
instance image.preComp_epi_of_epi [HasImage g] [HasImage (f ≫ g)] [Epi f] :
Epi (image.preComp f g) := by
apply @epi_of_epi_fac _ _ _ _ _ _ _ _ ?_ (image.factorThruImage_preComp _ _)
exact epi_comp _ _
| Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 523 | 527 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | unfold truncatedInf
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 197 | 197 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | simp only [length, sublistsLen_succ_cons, sublistsLen_length, map,
sublistsLen_of_length_lt (lt_succ_self _), nil_append]
open Function
theorem Pairwise.sublists' {R} :
∀ {l : List α}, Pairwise R l → Pairwise (Lex (swap R)) (sublists' l)
| Mathlib/Data/List/Sublists.lean | 295 | 301 |
/-
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.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# With Density
For an s-finite kernel `κ : K... | protected theorem withDensity_apply (κ : Kernel α β) [IsSFiniteKernel κ]
(hf : Measurable (Function.uncurry f)) (a : α) :
| Mathlib/Probability/Kernel/WithDensity.lean | 56 | 57 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | Nat.cast_add, coeff_X_add_one_pow]
theorem Monic.geom_sum {P : R[X]} (hP : P.Monic) (hdeg : 0 < P.natDegree) {n : ℕ} (hn : n ≠ 0) :
(∑ i ∈ range n, P ^ i).Monic := by
| Mathlib/RingTheory/Polynomial/Basic.lean | 271 | 274 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | simp only [segment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]
theorem openSegment_eq_image₂ (x y : E) :
openSegment 𝕜 x y =
| Mathlib/Analysis/Convex/Segment.lean | 62 | 65 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | end BooleanAlgebra
variable [DecidableEq α] [Fintype α]
lemma card_truncatedSup_union_add_card_truncatedSup_infs (𝒜 ℬ : Finset (Finset α)) (s : Finset α) :
#(truncatedSup (𝒜 ∪ ℬ) s) + #(truncatedSup (𝒜 ⊼ ℬ) s) =
#(truncatedSup 𝒜 s) + #(truncatedSup ℬ s) := by
by_cases h𝒜 : s ∈ lowerClosure (𝒜 : Set ... | Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 292 | 304 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | mul_one, choose_zero_right, cast_zero, if_false, zero_add, succ_succ_ne_one]
ring
| Mathlib/NumberTheory/Bernoulli.lean | 204 | 205 |
/-
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.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Formal power se... | contradiction
· rfl
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 84 | 86 |
/-
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.Homology.Embedding.IsSupported
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.Opposite
/-!
# The extension of a homological co... | obtain hi'|⟨i, hi⟩ := (e.r i').eq_none_or_eq_some
· rw [extend.d_none_eq_zero K _ _ hi']
· obtain hj'|⟨j, hj⟩ := (e.r j').eq_none_or_eq_some
· rw [extend.d_none_eq_zero' K _ _ hj']
· rw [extend.d_eq K hi hj,K.shape, zero_comp, comp_zero]
obtain rfl := e.f_eq_of_r_eq_some hi
obtai... | Mathlib/Algebra/Homology/Embedding/Extend.lean | 114 | 121 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | theorem det_zero_of_row_eq (i_ne_j : i ≠ j) (hij : M i = M j) : M.det = 0 :=
(detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_eq_zero_of_eq M hij i_ne_j
/-- If a matrix has a repeated column, the determinant will be zero. -/
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 360 | 363 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
/-!
# Integrals against peak functions
A sequence of peak functions is a sequence of functions with a... | obtain ⟨δ, hδ, δpos, δone⟩ : ∃ δ, (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ < ε ∧ 0 < δ ∧ δ < 1 := by
have A :
Tendsto (fun δ => (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ) (𝓝[>] 0)
(𝓝 ((0 * ∫ x in s, ‖g x‖ ∂μ) + 2 * 0)) := by
apply Tendsto.mono_left _ nhdsWithin_le_nhds
exact (tendsto_id.mul tendsto_con... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 99 | 182 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | apply Units.mul_right_dvd
theorem isPrimal (hu : IsUnit u) : IsPrimal u :=
| Mathlib/Algebra/Divisibility/Units.lean | 87 | 89 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | attribute [to_additive existing] npowRec'
@[to_additive]
theorem npowRec'_succ {M : Type*} [Mul M] [One M] {k : ℕ} (_ : k ≠ 0) (m : M) :
npowRec' (k + 1) m = npowRec' k m * m :=
match k with
| Mathlib/Algebra/Group/Defs.lean | 463 | 468 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
/-!
# Local extrema of differentiable functions
## Main definitions
In a real normed space `E` we define `posTangentCo... | if hf : DifferentiableWithinAt ℝ f s a then h.hasFDerivWithinAt_nonpos hf.hasFDerivWithinAt hy
else by rw [fderivWithin_zero_of_not_differentiableWithinAt hf]; rfl
/-- If `f` has a local max on `s` at `a`, `f'` is a derivative of `f` at `a` within `s`, and
| Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 137 | 140 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Huỳnh Trần Khanh, Stuart Presnell
-/
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Math... | rw [← filter_image_mk_not_isDiag, Fintype.card_of_subtype]
rintro x
rw [mem_filter, univ_product_univ, mem_image]
obtain ⟨a, ha⟩ := Quot.exists_rep x
| Mathlib/Data/Sym/Card.lean | 167 | 170 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Submonoid.Basic
/-!
# Submonoid of opposite monoids
For every monoid `M`, we construct an... | @[to_additive]
theorem op_sInf (S : Set (Submonoid M)) : (sInf S).op = sInf (.unop ⁻¹' S) :=
opEquiv.map_sInf_eq_sInf_symm_preimage _
| Mathlib/Algebra/Group/Submonoid/MulOpposite.lean | 136 | 139 |
/-
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.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... |
lemma inf'_image₂_left (g : γ → δ) (h : (image₂ f s t).Nonempty) :
inf' (image₂ f s t) h g =
inf' s h.of_image₂_left fun x ↦ inf' t h.of_image₂_right (g <| f x ·) :=
sup'_image₂_left (δ := δᵒᵈ) g h
| Mathlib/Data/Finset/NAry.lean | 552 | 556 |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | @[simp] theorem derivative_invOf {R} [CommRing R] (f : R⟦X⟧) [Invertible f] :
d⁄dX R ⅟f = - ⅟f ^ 2 * d⁄dX R f := by
rw [Derivation.leibniz_invOf, smul_eq_mul]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 158 | 160 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 822 | 832 | |
/-
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.Ab
import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib... |
lemma ShortExact.injective_f [HasZeroObject C] (hS : S.ShortExact) :
Function.Injective ((forget₂ C Ab).map S.f) := by
rw [← Preadditive.mono_iff_injective]
| Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean | 92 | 95 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... |
section H0
instance : Mono (shortComplexH0 A).f := by
rw [ModuleCat.mono_iff_injective]
apply Submodule.injective_subtype
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 775 | 780 |
/-
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.Subgroup.Finite
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.LinearAlgebra.Alternating.... | intro hσ
obtain ⟨⟨sl, sr⟩, hσ⟩ := QuotientGroup.leftRel_apply.mp (Quotient.exact' hσ)
rcases hi : σ⁻¹ i with i' | i' <;> rcases hj : σ⁻¹ j with j' | j' <;>
rw [Perm.inv_eq_iff_eq] at hi hj <;> substs hi hj
-- the term pairs with and cancels another term
case inl.inr => simpa using Equiv.congr_fun hσ (Sum.... | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | 96 | 132 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.ShortComplex.Retract
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
#... | Mathlib/Algebra/Homology/QuasiIso.lean | 500 | 504 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 853 | 857 | |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 2,029 | 2,081 | |
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | (PartialEquiv.disjointUnion_eq_piecewise _ _ _ _).symm
end Piecewise
section Continuity
/-- Continuity within a set at a point can be read under right composition with a local
homeomorphism, if the point is in its target -/
| Mathlib/Topology/PartialHomeomorph.lean | 1,011 | 1,018 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.NumberTheory.Harmo... |
lemma hasDerivAt_Gamma_one_half : HasDerivAt Gamma (-√π * (γ + 2 * log 2)) (1 / 2) := by
have h_diff {s : ℝ} (hs : 0 < s) : DifferentiableAt ℝ Gamma s :=
differentiableAt_Gamma fun m ↦ ((neg_nonpos.mpr m.cast_nonneg).trans_lt hs).ne'
have h_diff' {s : ℝ} (hs : 0 < s) : DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)... | Mathlib/NumberTheory/Harmonic/GammaDeriv.lean | 96 | 146 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Algebra.Field.Basic
/-!
# `norm_num` plugins for `Rat.cast` and `⁻¹`.
-/
va... |
theorem isRat_inv_pos {α} [DivisionRing α] [CharZero α] {a : α} {n d : ℕ} :
IsRat a (.ofNat (Nat.succ n)) d → IsRat a⁻¹ (.ofNat d) (Nat.succ n) := by
rintro ⟨_, rfl⟩
have := invertibleOfNonzero (α := α) (Nat.cast_ne_zero.2 (Nat.succ_ne_zero n))
| Mathlib/Tactic/NormNum/Inv.lean | 106 | 110 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 261 | 283 | |
/-
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.GroupWithZero.Action.Pi
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Finsupp.Order
/-!
# Flooring,... | @[simp] lemma floorDiv_of_nonpos (ha : a ≤ 0) (b : β) : b ⌊/⌋ a = 0 := FloorDiv.floorDiv_nonpos ha _
| Mathlib/Algebra/Order/Floor/Div.lean | 94 | 94 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | · rcases comp_symm_of_uniformity (H.mem_of_mem hi) with ⟨t, ht, ht', hts⟩
rcases H.mem_iff.1 ht with ⟨j, hj, hjt⟩
refine (h j hj).imp fun N hN m hm n hn => hts ⟨u N, hjt ?_, ht' <| hjt ?_⟩
exacts [hN m hm, hN n hn]
theorem cauchySeq_of_controlled [SemilatticeSup β] [Nonempty β] (U : β → Set (α × α))
... | Mathlib/Topology/UniformSpace/Cauchy.lean | 299 | 305 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.RingTheory.Finiteness.Nakayama
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheor... |
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f ∈ F.submodule ↔ ∀ i, f i ∈ F.N i :=
Iff.rfl
theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule := by
ext
exact forall_and
variable (I M)
| Mathlib/RingTheory/Filtration.lean | 242 | 251 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.ShortComplex.Retract
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
#... | · intro
infer_instance
lemma quasiIso_iff_comp_right (φ : K ⟶ L) (φ' : L ⟶ M) [∀ i, K.HasHomology i]
[∀ i, L.HasHomology i] [∀ i, M.HasHomology i]
[hφ' : QuasiIso φ'] :
QuasiIso (φ ≫ φ') ↔ QuasiIso φ := by
| Mathlib/Algebra/Homology/QuasiIso.lean | 208 | 214 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... | coeff_smul_right hwf Set.subset_union_left]
simp_all [sum_add_distrib]
· intro b
simp_all only [Set.isPWO_union, HahnSeries.isPWO_support, and_self, of_symm_add,
HahnSeries.coeff_add', Pi.add_apply, ne_eq, Set.mem_union, HahnSeries.mem_support]
contrapose!
intro h
rw [h.1, h.2, add_zer... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 238 | 264 |
/-
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.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 500 | 503 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... |
lemma InfClosed.preimage [FunLike F β α] [InfHomClass F β α] (hs : InfClosed s) (f : F) :
| Mathlib/Order/SupClosed.lean | 138 | 139 |
/-
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.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | condExpIndL1_of_measurableSet_of_measure_ne_top
theorem condExpIndL1_of_measure_eq_top (hμs : μ s = ∞) (x : G) : condExpIndL1 hm μ s x = 0 := by
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 195 | 197 |
/-
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.CPolynomial
import Mathlib.Analysis.Analytic.Inverse
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.Deriv... | HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
simpa only [hasStrictFDerivAt_iff_isLittleO, Set.insert_eq_of_mem, Set.mem_univ,
Set.univ_prod_univ, nhdsWithin_univ]
using (h.hasFPowerSeriesWithinAt (s := Set.univ)).hasStrictFDerivWithinAt
theorem HasFPowerSeriesWithinAt.hasFD... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 91 | 101 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | theorem lt_wf {α} : WellFounded (@LT.lt (Finset α) _) :=
have H : Subrelation (@LT.lt (Finset α) _) (InvImage (· < ·) card) := fun {_ _} hxy =>
| Mathlib/Data/Finset/Card.lean | 818 | 819 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | theorem prod_dvd_iff' {u : PrimeMultiset} {n : ℕ+} : u.prod ∣ n ↔ u ≤ n.factorMultiset := by
let h := @prod_dvd_iff u n.factorMultiset
rw [n.prod_factorMultiset] at h
exact h
end PrimeMultiset
| Mathlib/Data/PNat/Factors.lean | 302 | 307 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Module.Basic
/-!
# Group and ring filter bases
A `GroupFilterBasis` is a `FilterBasis` on a group with some properties relating
the ... | Example : if `R` is a topological ring then the neighbourhoods of the identity are a
`RingFilterBasis`. Conversely given a `RingFilterBasis` on a ring `R`, one can define a
topology on `R` which is compatible with the ring structure. -/
class RingFilterBasis (R : Type u) [Ring R] extends AddGroupFilterBasis R whe... | Mathlib/Topology/Algebra/FilterBasis.lean | 213 | 216 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... |
/-- The bilinear map `kahler` is invariant under pullback by a positively-oriented isometric
automorphism. -/
theorem kahler_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x y : E) :
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 501 | 505 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.Finsupp.Defs
/-!
# Locus of unequal values of finitely supported functions
Let `α N` be two Types, assume that `N` has a `0` and let `f g : α →₀ N... | Mathlib/Data/Finsupp/NeLocus.lean | 174 | 175 | |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
/-!
# Diagonal matrices
This file co... | @[simp]
theorem isDiag_neg_iff [SubtractionMonoid α] {A : Matrix n n α} : (-A).IsDiag ↔ A.IsDiag :=
⟨fun ha _ _ h => neg_eq_zero.1 (ha h), IsDiag.neg⟩
| Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 76 | 79 |
/-
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 support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support :=
Rel.dom_mono h.2
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 568 | 570 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | · exact Or.inl (infinitePos_iff_infinitesimal_inv_pos.mpr ⟨hi, inv_pos.mpr hp⟩)
theorem infinite_iff_infinitesimal_inv {x : ℝ*} (h0 : x ≠ 0) : Infinite x ↔ Infinitesimal x⁻¹ :=
| Mathlib/Data/Real/Hyperreal.lean | 619 | 621 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | (⟨x, hx⟩ : Subtype p) ≤ ⟨y, hy⟩ ↔ x ≤ y :=
| Mathlib/Order/Basic.lean | 1,005 | 1,005 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,643 | 1,644 | |
/-
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... | def cos (θ : Angle) : ℝ :=
cos_periodic.lift θ
@[simp]
theorem cos_coe (x : ℝ) : cos (x : Angle) = Real.cos x :=
rfl
@[continuity]
theorem continuous_cos : Continuous cos :=
Real.continuous_cos.quotient_liftOn' _
theorem cos_eq_real_cos_iff_eq_or_eq_neg {θ : Angle} {ψ : ℝ} :
cos θ = Real.cos ψ ↔ θ = ψ ∨ θ ... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 267 | 289 |
/-
Copyright (c) 2022 Anand Rao, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anand Rao, Rémi Bottinelli
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Finite.Set
/-!
# Ends
This ... | exact Adj.reachable cd.symm⟩
/--
| Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 125 | 127 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathl... | (hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) :
hf.implicitFunctionOfComplemented f f' hf' hker (f a) 0 = a := by
simpa only [implicitToPartialHomeomorphOfComplemented_self] using
(hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker).left_inv
(hf.mem_implicitToPartialHomeomorphO... | Mathlib/Analysis/Calculus/Implicit.lean | 313 | 318 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
/-!
#... | · simp [add_smul, ih, ← Multiset.smul_sum]
theorem Finset.sum_smul {f : ι → R} {s : Finset ι} {x : M} :
(∑ i ∈ s, f i) • x = ∑ i ∈ s, f i • x := map_sum ((smulAddHom R M).flip x) f s
| Mathlib/Algebra/Module/BigOperators.lean | 30 | 34 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.MonoidLocalization.Away
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Ma... | · intro x hx
apply Submodule.mem_of_span_eq_top_of_smul_pow_mem _ _ hS
rintro ⟨s, hs⟩
simp only [Ideal.mem_iInf, Ideal.mem_comap] at hx
obtain ⟨⟨y, ⟨_, n, rfl⟩⟩, e⟩ :=
(IsLocalization.mem_map_algebraMap_iff (.powers s) _).mp (hx s hs)
| Mathlib/RingTheory/Localization/Ideal.lean | 209 | 214 |
/-
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.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,020 | 2,021 | |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | calc
2 * a + 1 ≤ a ^ 2 * a := by nlinarith
_ = a ^ 3 := by rw [Nat.pow_succ a 2]
_ ≤ a ^ b := pow_right_mono₀ (Nat.le_of_succ_le ha) hb
| Mathlib/NumberTheory/FermatPsp.lean | 136 | 140 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | subspace with `p` added. -/
theorem existsUnique_dist_eq_of_insert {s : AffineSubspace ℝ P}
[s.direction.HasOrthogonalProjection] {ps : Set P} (hnps : ps.Nonempty) {p : P} (hps : ps ⊆ s)
(hp : p ∉ s) (hu : ∃! cs : Sphere P, cs.center ∈ s ∧ ps ⊆ (cs : Set P)) :
∃! cs₂ : Sphere P,
cs₂.center ∈ affineSpa... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 48 | 56 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | the norm of a matrix. -/
@[local instance]
protected def linftyOpNonUnitalSemiNormedRing [NonUnitalSeminormedRing α] :
NonUnitalSeminormedRing (Matrix n n α) :=
{ Matrix.linftyOpSeminormedAddCommGroup, Matrix.instNonUnitalRing with
norm_mul_le := linfty_opNorm_mul }
/-- The `L₁-L∞` norm preserves one on non-... | Mathlib/Analysis/Matrix.lean | 340 | 355 |
/-
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... |
theorem exp_sub : exp (x - y) = exp x / exp y := by
simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]
| Mathlib/Data/Complex/Exponential.lean | 230 | 232 |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | TendstoUniformlyOnFilter (fun n : ι => fun _ : α => g n) (fun _ : α => b) p p' := by
simpa only [nhds_eq_comap_uniformity, tendsto_comap_iff] using hg.comp (tendsto_fst (g := p'))
/-- If a sequence `g` converges to some `b`, then the sequence of constant functions
| Mathlib/Topology/UniformSpace/UniformConvergence.lean | 323 | 326 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | lemma hasSum_int_evenKernel₀ (a : ℝ) {t : ℝ} (ht : 0 < t) :
HasSum (fun n : ℤ ↦ if n + a = 0 then 0 else rexp (-π * (n + a) ^ 2 * t))
(evenKernel a t - if (a : UnitAddCircle) = 0 then 1 else 0) := by
haveI := Classical.propDecidable -- speed up instance search for `if / then / else`
simp_rw [AddCircle.coe_e... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 180 | 190 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... | lemma natDegree_Ψ₃_le : W.Ψ₃.natDegree ≤ 4 := by
rw [Ψ₃]
compute_degree
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 90 | 92 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 803 | 805 | |
/-
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 | 938 | 939 | |
/-
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... | Mathlib/Algebra/ModEq.lean | 298 | 300 | |
/-
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.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | swap
· rw [h]
exact natDegree_one
have : p = C (p.coeff 0) := by
| Mathlib/Algebra/Polynomial/Monic.lean | 149 | 152 |
/-
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.Data.Nat.Fib.Basic
/-!
# Zeckendorf's Theorem
This file proves Zeckendorf's theorem: Every natural number can be written uniquely as a sum of
distinct no... | @[simp] lemma greatestFib_pos : 0 < greatestFib n ↔ 0 < n := by simp [pos_iff_ne_zero]
| Mathlib/Data/Nat/Fib/Zeckendorf.lean | 101 | 101 |
/-
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.CategoryTheory.Shift.Induced
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Localization.LocalizerMorphism
/-!
# The s... | rfl
lemma shiftFunctor_comp_inverts (a : A) :
| Mathlib/CategoryTheory/Shift/Localization.lean | 48 | 50 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | theorem toGame_inj {a b : Ordinal} : a.toGame = b.toGame ↔ a = b :=
toGame.inj
@[deprecated (since := "2024-12-29")] alias toGame_eq_iff := toGame_inj
/-- The natural addition of ordinals corresponds to their sum as games. -/
theorem toPGame_nadd (a b : Ordinal) : (a ♯ b).toPGame ≈ a.toPGame + b.toPGame := by
ref... | Mathlib/SetTheory/Game/Ordinal.lean | 188 | 207 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | section TopologicalSpace
variable [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [DenselyOrdered 𝕜]
| Mathlib/Analysis/Convex/Topology.lean | 95 | 97 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... | rw [← Nat.cast_two, CharP.cast_eq_zero]
| Mathlib/Algebra/CharP/Two.lean | 29 | 29 |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | /-- An ordinal lifted to the zero universe equals itself. -/
@[simp]
theorem lift_uzero (a : Ordinal.{u}) : lift.{0} a = a :=
lift_id' a
| Mathlib/SetTheory/Ordinal/Basic.lean | 641 | 644 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | theorem baseChange_smul : (r • f).baseChange A = r • f.baseChange A := by
ext
simp [baseChange_tmul]
| Mathlib/RingTheory/TensorProduct/Basic.lean | 90 | 92 |
/-
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.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | simp
/-- To check that a function is `n` times continuously differentiable, it suffices to check that its
first `n` derivatives are differentiable. This is slightly too strong as the condition we
require on the `n`-th derivative is differentiability instead of continuity, but it has the
| Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 256 | 260 |
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