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/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 299 | 301 | |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | rcases (mem_nhdsGE_iff_exists_Ico_subset' zero_lt_one).1 hle with ⟨r, hr0, hr⟩
refine fun s =>
le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s)
(ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc))
(mem_of_superset (Ioo_mem_nhdsGT hr0) fun r' hr' => ?_)
simp only [mem_setOf_eq, m... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 321 | 331 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | def OrderIso.mulRight₀' {a : α} (ha : a ≠ 0) : α ≃o α := .mulRight₀ a (zero_lt_iff.2 ha)
set_option linter.deprecated false in
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 196 | 198 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
/-!
# Schwarz lemma
In this file we prove several versions of the Schw... | /-- The **Schwarz Lemma**: if `f : ℂ → E` sends an open disk with center `c` and radius `R₁` to an
open ball with center `f c` and radius `R₂`, then for any `z` in the former disk we have
`dist (f z) (f c) ≤ (R₂ / R₁) * dist z c`. -/
| Mathlib/Analysis/Complex/Schwarz.lean | 146 | 148 |
/-
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.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
/-!
# Kolmogorov's 0-1 law
Let `s : ι → MeasurableSpace Ω` be an indep... |
@[deprecated (since := "2025-01-21")]
alias condexp_zero_or_one_of_measurableSet_limsup_atTop :=
condExp_zero_or_one_of_measurableSet_limsup_atTop
end AtTop
section AtBot
variable [SemilatticeInf ι] [NoMinOrder ι] [Nonempty ι]
theorem Kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα... | Mathlib/Probability/Independence/ZeroOne.lean | 294 | 306 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | theorem pentagon_inv_hom_hom_hom_hom :
(α_ W X Y).inv ▷ Z ≫ (α_ (W ⊗ X) Y Z).hom ≫ (α_ W X (Y ⊗ Z)).hom =
(α_ W (X ⊗ Y) Z).hom ≫ W ◁ (α_ X Y Z).hom := by
simp [← cancel_epi ((α_ W X Y).hom ▷ Z)]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 543 | 546 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... |
theorem updateRow_conjTranspose [DecidableEq n] [Star α] :
updateRow Mᴴ j (star c) = (updateCol M j c)ᴴ := by
rw [conjTranspose, conjTranspose, transpose_map, transpose_map, updateRow_transpose,
map_updateCol]
| Mathlib/Data/Matrix/RowCol.lean | 314 | 318 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,488 | 1,491 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... |
/-- See also `Order.le_succ_of_wcovBy`. -/
| Mathlib/Order/SuccPred/Basic.lean | 339 | 340 |
/-
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... | theorem top_bihimp : ⊤ ⇔ a = a := by rw [bihimp_comm, bihimp_top]
| Mathlib/Order/SymmDiff.lean | 218 | 219 |
/-
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... | where `ψ(x,y) = 2y + a₁x + a₃`. -/
lemma addY_sub_negY_addY {x₁ x₂ : F} (y₁ y₂ : F) (hx : x₁ ≠ x₂) :
let x₃ := W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)
let y₃ := W.addY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂)
y₃ - W.negY x₃ y₃ =
((y₂ - W.negY x₂ y₂) * (x₁ - x₃) - (y₁ - W.negY x₁ y₁) * (x₂ - x₃)) / (x₂ - x₁) := by
... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 598 | 605 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.AtTopBot.BigOperators
import Mathlib.Topology.Separation.Hausdorff
/-!
# Infinite sum and pro... | simp [tprod_def, h]
| Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 124 | 125 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | Mathlib/Data/Real/Sqrt.lean | 220 | 220 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.LinearAlgebra.Aff... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 275 | 278 | |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | /-- For a surjective `f : N →ₗ[R] P`,
the natural equivalence between `N ⊗[R] Q ⧸ (range (rTensor Q f))` and `P ⊗[R] Q` -/
noncomputable def rTensor.equiv :
((N ⊗[R] Q) ⧸ (LinearMap.range (rTensor Q f))) ≃ₗ[R] (P ⊗[R] Q) :=
rTensor.linearEquiv_of_rightInverse Q hfg (Function.rightInverse_surjInv hg)
include hf... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 381 | 387 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... |
open NormalExpr
| Mathlib/Tactic/Abel.lean | 159 | 160 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | lt_iff_lt_of_le_iff_le (inv_le_of_neg hb ha)
| Mathlib/Algebra/Order/Field/Basic.lean | 384 | 384 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | continuous_quotient_mk'.comp continuous_subtype_val
variable [OrderTopology 𝕜] {x : AddCircle p}
| Mathlib/Topology/Instances/AddCircle.lean | 240 | 243 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Set.Sigma
import Mathlib.Order.CompleteLattice.Finset
/-!
# Finite sets in a ... |
theorem not_mem_sigmaLift_of_ne_right (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) {a : Sigma α}
(b : Sigma β) {x : Sigma γ} (h : b.1 ≠ x.1) : x ∉ sigmaLift f a b := by
rw [mem_sigmaLift]
exact fun H => h H.snd.fst
| Mathlib/Data/Finset/Sigma.lean | 176 | 181 |
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | suffices h : σ.sign = σ.signAux by
rw [h, Finset.prod_sigma', Equiv.Perm.signAux]
| Mathlib/GroupTheory/Perm/Fin.lean | 288 | 289 |
/-
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.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A c₁ c₂ c₃`: a basis... | theorem lift_one : q.lift (1 : ℍ[R,c₁,c₂,c₃]) = 1 := by simp [lift]
| Mathlib/Algebra/QuaternionBasis.lean | 114 | 114 |
/-
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... | to the associativity, commutativity, distributivity, ... of `Finset.image₂` of those operations.
The proof pattern is `image₂_lemma operation_lemma`. For example, `image₂_comm mul_comm` proves that
| Mathlib/Data/Finset/NAry.lean | 248 | 250 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,407 | 1,410 | |
/-
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.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | right_inv := fun x => Subtype.ext <| coe_toNNReal (ne_top_of_le_ne_top coe_ne_top x.2)
map_rel_iff' := fun {_ _} => by
simp only [Equiv.coe_fn_mk, Subtype.mk_le_mk, coe_le_coe, Subtype.coe_le_coe] }
| Mathlib/Data/ENNReal/Inv.lean | 571 | 574 |
/-
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.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | theorem mul_geom_sum [Ring R] (x : R) (n : ℕ) : ((x - 1) * ∑ i ∈ range n, x ^ i) = x ^ n - 1 :=
op_injective <| by simpa using geom_sum_mul (op x) n
| Mathlib/Algebra/GeomSum.lean | 236 | 237 |
/-
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.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | (Ideal.mul_eq_inf_of_coprime coprime.sup_eq).symm
theorem isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by
simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
mem_span_singleton]
| Mathlib/RingTheory/Ideal/Operations.lean | 673 | 678 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | theorem CovBy.eq_of_between {x : α} (hab : a ⋖ b) (hbc : b ⋖ c) (hax : a < x) (hxc : x < c) :
x = b :=
| Mathlib/Order/Cover.lean | 394 | 395 |
/-
Copyright (c) 2018 Mario Carneiro. 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.CauSeq.Completion
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.Defs
/-!
# Real numbers from Cauc... | induction b using Real.ind_mk
| Mathlib/Data/Real/Basic.lean | 317 | 317 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
/-!
# Grönwall's inequality
The main technical result of this file is the Grönwall-like inequality
`norm_le_gron... | simp only [gronwallBound_ε0, zero_mul]
theorem gronwallBound_continuous_ε (δ K x : ℝ) : Continuous fun ε => gronwallBound δ K ε x := by
by_cases hK : K = 0
· simp only [gronwallBound_K0, hK]
| Mathlib/Analysis/ODE/Gronwall.lean | 79 | 83 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | namespace IsComplement
/-- The equivalence `G ≃ S × T`, such that the inverse is `(*) : S × T → G` -/
noncomputable def equiv {S T : Set G} (hST : IsComplement S T) : G ≃ S × T :=
(Equiv.ofBijective (fun x : S × T => x.1.1 * x.2.1) hST).symm
| Mathlib/GroupTheory/Complement.lean | 441 | 445 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | def spanFinset {ι : Type*} (s : Finset ι) (f : ι → K) : FractionalIdeal R₁⁰ K :=
⟨Submodule.span R₁ (f '' s), by
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 517 | 518 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... |
theorem lintegral_eq_of_lmarginal_eq [Fintype δ] (s : Finset δ) {f g : (∀ i, X i) → ℝ≥0∞}
(hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ) :
∫⁻ x, f x ∂Measure.pi μ = ∫⁻ x, g x ∂Measure.pi μ := by
rcases isEmpty_or_nonempty (∀ i, X i) with h|⟨⟨x⟩⟩
· simp_rw [lintegral_of_isEmpty]... | Mathlib/MeasureTheory/Integral/Marginal.lean | 236 | 241 |
/-
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.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 937 | 950 | |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Al... | end
/-- The composition of non-unital ⋆-algebra homomorphisms, as a non-unital ⋆-algebra
homomorphism. -/
| Mathlib/Algebra/Star/StarAlgHom.lean | 179 | 182 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... | HasProd (fun b ↦ f b / g b) (a₁ / a₂) := by
simp only [div_eq_mul_inv]
exact hf.mul hg.inv
| Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 50 | 53 |
/-
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 ... | lemma support_subgraphOfAdj {u v : V} (h : G.Adj u v) :
(G.subgraphOfAdj h).support = {u , v} := by
ext
rw [Subgraph.mem_support]
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 958 | 961 |
/-
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.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... |
instance : Mod ℤ[i] :=
⟨fun x y => x - y * (x / y)⟩
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 195 | 197 |
/-
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.... | Mathlib/Algebra/Group/Basic.lean | 1,059 | 1,060 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | end DescFactorial
lemma factorial_two_mul_le (n : ℕ) : (2 * n)! ≤ (2 * n) ^ n * n ! := by
rw [Nat.two_mul, ← factorial_mul_ascFactorial, Nat.mul_comm]
exact Nat.mul_le_mul_right _ (ascFactorial_le_pow_add _ _)
| Mathlib/Data/Nat/Factorial/Basic.lean | 445 | 449 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α :=
mem_uniformity_dist.2 ⟨ε, ε0, fun _ _ ↦ id⟩
theorem uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 697 | 700 |
/-
Copyright (c) 2022 Joanna Choules. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joanna Choules
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Homomorphisms from finite subgraphs
This file defines the typ... | @[simp, norm_cast]
lemma coe_sSup (s : Set G.Finsubgraph) : sSup s = (⨆ G ∈ s, G : G.Subgraph) := rfl
| Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 98 | 100 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,445 | 2,447 | |
/-
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... | section CommMonoid
variable [CommMonoid α] {a b u : α}
| Mathlib/Algebra/Divisibility/Units.lean | 94 | 96 |
/-
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 | 467 | 471 | |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
/-!
# Strict initial objects
This file sets up the basic... |
theorem IsInitial.subsingleton_to (hI : IsInitial I) {A : C} : Subsingleton (A ⟶ I) :=
⟨hI.strict_hom_ext⟩
| Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 74 | 77 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | · simp [hx, hxs]
theorem indicator_ae_eq_of_ae_eq_set (hst : s =ᵐ[μ] t) : s.indicator f =ᵐ[μ] t.indicator f := by
classical exact piecewise_ae_eq_of_ae_eq_set hst
| Mathlib/MeasureTheory/Measure/Restrict.lean | 955 | 958 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | Mathlib/Data/List/Range.lean | 278 | 282 | |
/-
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.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... |
end
namespace HilbertBasis
instance {ι : Type*} : Inhabited (HilbertBasis ι 𝕜 ℓ²(ι, 𝕜)) :=
⟨ofRepr (LinearIsometryEquiv.refl 𝕜 _)⟩
open Classical in
/-- `b i` is the `i`th basis vector. -/
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 374 | 383 |
/-
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.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | lemma RespectsIso.mk (P : MorphismProperty C)
(hprecomp : ∀ {X Y Z : C} (e : X ≅ Y) (f : Y ⟶ Z) (_ : P f), P (e.hom ≫ f))
(hpostcomp : ∀ {X Y Z : C} (e : Y ≅ Z) (f : X ⟶ Y) (_ : P f), P (f ≫ e.hom)) :
| Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 225 | 227 |
/-
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.... | IsRat a n d → IsRat b n d → a = b
| _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩ => by congr; apply Subsingleton.elim
| Mathlib/Tactic/NormNum/Basic.lean | 487 | 489 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... | end ContinuousMultilinearApplyConst
section SMul
| Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 224 | 227 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [← smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm 𝕜 E) x]
| Mathlib/Analysis/Seminorm.lean | 513 | 517 |
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Defs
import Mathlib.Data.Set.Function
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variab... | | succ n ih => rw [iterate, add_right_comm, iterate, ih, Nat.iterate, cons_append]
theorem take_iterate (f : α → α) (a : α) (m n : ℕ) :
| Mathlib/Data/List/Iterate.lean | 52 | 54 |
/-
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.Functor.KanExtension.Basic
import Mathlib.CategoryTheory.Localization.Predicate
/-!
# Right derived functors
In this file, given a functor `F : ... | rightDerivedNatIso RF RF' α α'₂ W (Iso.refl F)
lemma isRightDerivedFunctor_iff_isIso_rightDerivedDesc (G : D ⥤ H) (β : F ⟶ L ⋙ G) :
G.IsRightDerivedFunctor β W ↔ IsIso (RF.rightDerivedDesc α W G β) := by
rw [isRightDerivedFunctor_iff_isLeftKanExtension]
| Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean | 140 | 144 |
/-
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.HomotopyCategory.MappingCone
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift
import Mathlib.CategoryTheory.Triangulated.Functor... |
section map
variable {K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ} (φ₁ : K₁ ⟶ L₁) (φ₂ : K₂ ⟶ L₂) (φ₃ : K₃ ⟶ L₃)
(a : K₁ ⟶ K₂) (b : L₁ ⟶ L₂)
| Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean | 141 | 145 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Wieser
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
impor... | theorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s := by
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddBelow s
· exa... | Mathlib/Data/Real/Pointwise.lean | 37 | 46 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... |
theorem inf_left_eq_bot_iff (h : IsCompl y z) : x ⊓ y = ⊥ ↔ x ≤ z := by
rw [← le_bot_iff, ← h.le_sup_right_iff_inf_left_le, bot_sup_eq]
| Mathlib/Order/Disjoint.lean | 459 | 461 |
/-
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.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | rw [← Nat.cast_one, preNormEDS_ofNat, preNormEDS'_one]
@[simp]
lemma preNormEDS_two : preNormEDS b c d 2 = 1 := by
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 218 | 221 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | | [], _, _ => rfl
| List.cons a l₁, l₂, s => by
| Mathlib/Data/Stream/Init.lean | 450 | 451 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Adam Topaz, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.FreeMonoid.UniqueProds
imp... | simp [AlgHom.ext_iff] at of_id
exact of_id a
@[simp]
theorem adjoin_range_ι : Algebra.adjoin R (Set.range (ι R : X → FreeAlgebra R X)) = ⊤ := by
set S := Algebra.adjoin R (Set.range (ι R : X → FreeAlgebra R X))
refine top_unique fun x hx => ?_; clear hx
induction x with
| grade0 => exact S.algebraMap_mem _... | Mathlib/Algebra/FreeAlgebra.lean | 567 | 587 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | simp_rw [ite_smul, one_smul, zero_smul, ite_and, Finset.sum_ite_eq, Finset.mem_univ, if_true]
/-- The standard basis of the endomorphism algebra of a module
induced by a basis of the module.
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 960 | 964 |
/-
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.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | simp [Cochain.comp_v _ _ (add_zero (-1)) p q q (by omega) (by omega)]
@[simp]
lemma inr_fst :
(Cochain.ofHom (inr φ)).comp (fst φ).1 (zero_add 1) = 0 := by
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 104 | 108 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 446 | 451 | |
/-
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.Order.Interval.Set.IsoIoo
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.UrysohnsBounded
/-!
# Tietze extension theore... | exact ⟨g, hg, DFunLike.coe_injective hgf⟩
end BoundedContinuousFunction
namespace ContinuousMap
/-- **Tietze extension theorem** for real-valued continuous maps, a version for a closed
| Mathlib/Topology/TietzeExtension.lean | 445 | 451 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | rw [← whiskerLeft_iff, whiskerLeft_comp, ← cancel_epi (α_ _ _ _).hom, ←
cancel_epi ((α_ _ _ _).hom ▷ _), pentagon_assoc, triangle, ← associator_naturality_middle, ←
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 364 | 365 |
/-
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.RingTheory.LocalProperties.Basic
/-!
# The meta properties of surjective ring homomorphisms.
## Main results
Let `R` be a commutative ring, `M` be a subm... |
universe u
local notation "surjective" => fun {X Y : Type _} [CommRing X] [CommRing Y] => fun f : X →+* Y =>
| Mathlib/RingTheory/RingHom/Surjective.lean | 30 | 33 |
/-
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.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | /-- A weak version of Carathéodory's theorem. -/
theorem convexHull_eq_union_convexHull_finite_subsets (s : Set E) :
convexHull R s = ⋃ (t : Finset E) (_ : ↑t ⊆ s), convexHull R ↑t := by
classical
refine Subset.antisymm ?_ ?_
· rw [_root_.convexHull_eq]
rintro x ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩
simp on... | Mathlib/Analysis/Convex/Combination.lean | 395 | 407 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | Mathlib/Data/Nat/Totient.lean | 359 | 367 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
import Mathlib.Topology.FiberBundl... | Mathlib/Topology/VectorBundle/Basic.lean | 1,029 | 1,037 | |
/-
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.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... | @[simp]
theorem divXPowOrder_X :
divXPowOrder X = (1 : R⟦X⟧) := by
| Mathlib/RingTheory/PowerSeries/Order.lean | 339 | 341 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Mathlib/Algebra/Ring/Ext.lean | 528 | 532 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
/-!
# Left and right limits
We define the (strict) left and right limits of a f... | refine tendsto_order.2 ⟨fun m hm => ?_, fun u hu => ?_⟩
· filter_upwards [@self_mem_nhdsWithin _ _ x (Ioi x)] with y hy using hm.trans_le
(hf (le_of_lt hy))
rcases hx u hu with ⟨v, xv, fvu⟩
have : Ioo x v ∈ 𝓝[>] x := Ioo_mem_nhdsGT xv
filter_upwards [this] with y hy
apply (hf hy.2.le).trans_lt fvu
/... | Mathlib/Topology/Order/LeftRightLim.lean | 220 | 234 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | theorem EqOn.comp_right (hg : t.EqOn g₁ g₂) (hf : s.MapsTo f t) : s.EqOn (g₁ ∘ f) (g₂ ∘ f) :=
fun _ ha => hg <| hf ha
theorem EqOn.mapsTo_iff (H : EqOn f₁ f₂ s) : MapsTo f₁ s t ↔ MapsTo f₂ s t :=
⟨fun h => h.congr H, fun h => h.congr H.symm⟩
| Mathlib/Data/Set/Function.lean | 139 | 143 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... | rfl
@[simp]
| Mathlib/Data/Fin/VecNotation.lean | 302 | 304 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | @[simp]
theorem range_fst : range (fst R M M₂) = ⊤ := by rw [range_eq_map, ← prod_top, prod_map_fst]
@[simp]
theorem range_snd : range (snd R M M₂) = ⊤ := by rw [range_eq_map, ← prod_top, prod_map_snd]
variable (R M M₂)
/-- `M` as a submodule of `M × N`. -/
def fst : Submodule R (M × M₂) :=
(⊥ : Submodule R M₂).co... | Mathlib/LinearAlgebra/Prod.lean | 515 | 527 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... | rw [M.dest_map]; congr
apply eq_of_drop_last_eq (by simp)
simp only [lastFun_appendFun]
ext1; apply h
end MvPFunctor
| Mathlib/Data/PFunctor/Multivariate/M.lean | 302 | 314 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... |
@[to_additive (attr := simp) lt_add_iff_pos_right]
theorem lt_mul_iff_one_lt_right' [MulLeftStrictMono α]
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 534 | 536 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | all_goals
intro H
| Mathlib/RingTheory/WittVector/WittPolynomial.lean | 239 | 240 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | · have := R.b_pos i
positivity
· exact h_sumTransform n hn i
_ = (∑ i, C * (1 - ε n) * ((asympBound g a b n - c₁ * g n))
* (a i * (b i) ^ (p a b))) + g n := by
congr; ext; ring
_ = C * (1 - ε n) * (asympBound g a b n - c₁ * g n) + g n := ... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 1,292 | 1,437 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import... |
/-- The converse: if the additive character takes the value `1` only at `0`,
then it is primitive. -/
theorem zmod_char_primitive_of_eq_one_only_at_zero (n : ℕ) (ψ : AddChar (ZMod n) C)
(hψ : ∀ a, ψ a = 1 → a = 0) : IsPrimitive ψ := by
refine fun a ha hf => ?_
have h : mulShift ψ a 1 = (1 : AddChar (ZMod n) C)... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 166 | 173 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | simpa [coe_S, num, denom] using (inv_lt_inv₀ z.normSq_pos zero_lt_one).mpr h
/-- If `|z| < 1`, then applying `S` strictly decreases `im`. -/
theorem im_lt_im_S_smul (h : normSq z < 1) : z.im < (S • z).im := by
have : z.im < z.im / normSq (z : ℂ) := by
have imz : 0 < z.im := im_pos z
apply (lt_div_iff₀ z.no... | Mathlib/NumberTheory/Modular.lean | 349 | 361 |
/-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
/-!
# Independence in Projective Space
In this file we define independence and dependence of families of elements ... | /-- Two points in a projective space are independent if and only if the points are not equal. -/
@[simp]
theorem independent_pair_iff_ne (u v : ℙ K V) : Independent ![u, v] ↔ u ≠ v := by
rw [independent_iff_not_dependent, dependent_pair_iff_eq u v]
@[deprecated (since := "2025-04-27")]
| Mathlib/LinearAlgebra/Projectivization/Independence.lean | 109 | 114 |
/-
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 ... | @[reassoc (attr := simp)]
lemma singleObjCyclesSelfIso_hom_naturality :
cyclesMap ((single C c j).map f) j ≫ (singleObjCyclesSelfIso c j B).hom =
(singleObjCyclesSelfIso c j A).hom ≫ f := by
rw [← cancel_mono (singleObjCyclesSelfIso c j B).inv, assoc, assoc, Iso.hom_inv_id, comp_id,
← cancel_mono (iCycl... | Mathlib/Algebra/Homology/SingleHomology.lean | 129 | 134 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,539 | 1,541 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | rotation of another by `π / 2`. -/
theorem tan_oangle_add_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
Real.Angle.tan (o.oangle (x + r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x)) =
r⁻¹ := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 488 | 491 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | a `MetricSpace` instance, as other distances may be used on these spaces, but we register them as
local instances in this section.
-/
open Classical in
| Mathlib/Topology/MetricSpace/PiNat.lean | 239 | 243 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | x • ![a₀, a₁] = ![x • a₀, x • a₁] := by rw [smul_cons, smul_cons, smul_empty]
theorem smul_vec3 {R : Type*} [SMul R α] (x : R) (a₀ a₁ a₂ : α) :
| Mathlib/Data/Matrix/Notation.lean | 499 | 501 |
/-
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.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | intro h₀ hpos hmem
have zi : z i ∈ s := hmem _ (mem_insert_self _ _)
have hs₀ : ∀ j ∈ t, 0 ≤ w j := fun j hj => h₀ j <| mem_insert_of_mem hj
rw [sum_insert hi] at hpos
| Mathlib/Analysis/Convex/Combination.lean | 161 | 164 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Ring.Commute
/-!
# Cast of natural numbers: lemmas about `Commute`
-/
variable {α : Type*}
nam... | | zero => rw [Nat.cast_zero]; exact Commute.zero_left x
| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)
theorem _root_.Commute.ofNat_left (n : ℕ) [n.AtLeastTwo] (x : α) : Commute (OfNat.ofNat n) x :=
| Mathlib/Data/Nat/Cast/Commute.lean | 24 | 27 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | ext a; exact h.lift_self_apply a
@[simp]
theorem aequiv_symm (h : IsAdjoinRoot S f) (h' : IsAdjoinRoot T f) :
(h.aequiv h').symm = h'.aequiv h := by ext; rfl
@[simp]
| Mathlib/RingTheory/IsAdjoinRoot.lean | 573 | 579 |
/-
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]
theorem mapDomain.addMonoidHom_id : mapDomain.addMonoidHom id = AddMonoidHom.id (α →₀ M) :=
AddMonoidHom.ext fun _ => mapDomain_id
| Mathlib/Data/Finsupp/Basic.lean | 460 | 463 |
/-
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... |
theorem withDensity_add_left (κ η : Kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η]
(f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f := by
by_cases hf : Measurable (Function.uncurry f)
· ext a s
simp only [Kernel.withDensity_apply _ hf, coe_add, Pi.add_apply, withDensity_a... | Mathlib/Probability/Kernel/WithDensity.lean | 116 | 122 |
/-
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 | 2,237 | 2,240 | |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file deals with the following version... | simp only [cutExpand_iff, mem_singleton]
rintro ⟨t, a, hr, rfl, rfl⟩
refine acc_of_singleton fun a' ↦ ?_
rw [erase_singleton, zero_add]
exact ih a' ∘ hr a'
/-- `CutExpand r` is well-founded when `r` is. -/
theorem _root_.WellFounded.cutExpand (hr : WellFounded r) : WellFounded (CutExpand r) :=
⟨have := hr.... | Mathlib/Logic/Hydra.lean | 169 | 177 |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FDRep
/-!
# Subspace of invariants a group representation
This file introduces the s... | Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
show ⅟ (Fintype.card G : k) • ∑ x : G, f (x * g) = ⅟ (Fintype.card G : k) • ∑ x : G, f x
rw [Function.Bijective.sum_comp (Group.mulRight_bijective g) _]
end GroupAlgebra
| Mathlib/RepresentationTheory/Invariants.lean | 54 | 59 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Algebra towers for polynomial
This file proves some basic results about t... |
@[simp]
| Mathlib/RingTheory/Polynomial/Tower.lean | 37 | 38 |
/-
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.GradedObject.Bifunctor
import Mathlib.CategoryTheory.Functor.Trifunctor
/-!
# The action of trifunctors on graded objects
Given a trifunctor `F. ... | fun j => ⟨_, isColimitCofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j⟩
variable (F₁₂ G ρ₁₂ X₁ X₂ X₃)
section
variable [HasMap ((((mapTrifunctor (bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃) r]
/-- The action on graded objects of a trifunctor obtained by composition of two
bifunctors can be com... | Mathlib/CategoryTheory/GradedObject/Trifunctor.lean | 533 | 541 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,250 | 1,269 | |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x) := by
letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
rw [SeminormedAddGroup.uniformCauchySeqO... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 112 | 163 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... |
theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 340 | 341 |
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