Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Action.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Order.Module.Defs
import Mathli... | section DivisionSemiring
variable (α : Type*) {β : Type*}
variable [DivisionSemiring α] [NeZero (2 : α)] [Lattice β] [AddCommGroup β] [Module α β]
| Mathlib/Algebra/Order/Module/OrderedSMul.lean | 154 | 156 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... |
lemma toMvPolynomial_mul (M : Matrix m n R) (N : Matrix n o R) (i : m) :
(M * N).toMvPolynomial i = bind₁ N.toMvPolynomial (M.toMvPolynomial i) := by
simp only [toMvPolynomial, mul_apply, map_sum, Finset.sum_comm (γ := o), bind₁, aeval,
AlgHom.coe_mk, coe_eval₂Hom, eval₂_monomial, algebraMap_apply, Algebra.i... | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 128 | 133 |
/-
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... | Mathlib/Algebra/Order/Field/Basic.lean | 366 | 366 | |
/-
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.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deri... |
protected theorem polarCoord_source : Complex.polarCoord.source = slitPlane := rfl
protected theorem polarCoord_target :
Complex.polarCoord.target = Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-π) π := rfl
| Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 185 | 190 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.LinearAlgebra.Prod
/-!
# Trivial Square-Ze... | theorem fst_map (f : M →ₗ[R'] N) (x : TrivSqZeroExt R' M) : fst (map f x) = fst x := by
simp [map, lift_def, Algebra.ofId_apply, algebraMap_eq_inl]
| Mathlib/Algebra/TrivSqZeroExt.lean | 1,062 | 1,063 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.BigOperators.Fin
/-!
# Lemmas for tuples `Fin m → α`
This file contains alternative definitions of common opera... | | n + 1, f, v =>
funext fun i => by
simp_rw [seq, seq_eq]
refine i.cases ?_ fun i => ?_
· rfl
· rw [Matrix.cons_val_succ]
rfl
example {f₁ f₂ : α → β} (a₁ a₂ : α) : seq ![f₁, f₂] ![a₁, a₂] = ![f₁ a₁, f₂ a₂] := rfl
| Mathlib/Data/Fin/Tuple/Reflection.lean | 44 | 52 |
/-
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 | 2,228 | 2,231 | |
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 557 | 559 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | @[simp]
theorem toIcoMod_add_toIcoDiv_zsmul (a b : α) : toIcoMod hp a b + toIcoDiv hp a b • p = b := by
| Mathlib/Algebra/Order/ToIntervalMod.lean | 128 | 129 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,394 | 1,400 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | simp only [FormalMultilinearSeries.comp, compAlongComposition,
ContinuousMultilinearMap.compAlongComposition_apply, ContinuousMultilinearMap.sum_apply]
| Mathlib/Analysis/Analytic/Composition.lean | 266 | 267 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Anne Baanen
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection.Basic
import Mathlib.Order.Hom.Basic
/-!
# Lattice structure on order homomorph... |
theorem iterate_sup_le_sup_iff {α : Type*} [SemilatticeSup α] (f : α →o α) :
(∀ n₁ n₂ a₁ a₂, f^[n₁ + n₂] (a₁ ⊔ a₂) ≤ f^[n₁] a₁ ⊔ f^[n₂] a₂) ↔
| Mathlib/Order/Hom/Order.lean | 117 | 119 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | Mathlib/SetTheory/Game/Nim.lean | 119 | 119 | |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | · simp only [not_exists, not_lt] at hx₁ hx₂
apply Filter.Eventually.of_forall
intro z
have : (f x, g x) ∈ u ×ˢ v := ⟨xu, xv⟩
calc
y < f x + g x := h this
_ ≤ f z + g z := add_le_add (hx₁ (f z)) (hx₂ (g z))
/-- The sum of two lower semicontinuous functions is lower semicontin... | Mathlib/Topology/Semicontinuous.lean | 469 | 536 |
/-
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... | end OneLtB
section BPosAndBLtOne
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 262 | 265 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | (p.1 ^ p.2 * log p.1 - exp (log p.1 * p.2) * sin (p.2 * π) * π) •
ContinuousLinearMap.snd ℝ ℝ ℝ) p := by
have : (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) * cos (x.2 * π) :=
(continuousAt_fst.eventually (gt_mem_nhds hp)).mono fun p hp => rpow_def_of_neg hp _
refine HasS... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 329 | 335 |
/-
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.... | theorem eq_div_iff_mul_eq' : a = b / c ↔ a * c = b := by rw [div_eq_mul_inv, eq_mul_inv_iff_mul_eq]
| Mathlib/Algebra/Group/Basic.lean | 781 | 781 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... |
/-- Structure containing the minimal axioms required to check that an order is a complete
distributive lattice. Do NOT use, except for implementing `CompleteDistribLattice` via
`CompleteDistribLattice.ofMinimalAxioms`.
This structure omits the `himp`, `compl`, `sdiff`, `hnot` fields, which can be recovered using
`Com... | Mathlib/Order/CompleteBooleanAlgebra.lean | 92 | 101 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
/-!
# Strong reducibility and degrees.
This file defines the notions of computable many-one reduction and one-one
reduc... | rfl
| Mathlib/Computability/Reduce.lean | 337 | 338 |
/-
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.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
/-! # Subtypes of cond... | end Set.Iic
| Mathlib/Order/CompleteLatticeIntervals.lean | 270 | 271 |
/-
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, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 375 | 387 | |
/-
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.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 569 | 571 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... |
theorem IsHermitian.ext_iff {A : Matrix n n α} : A.IsHermitian ↔ ∀ i j, star (A j i) = A i j :=
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 56 | 57 |
/-
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.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | _ = (‖·‖) ⁻¹' ((‖·‖ * r) '' s) := by ext; simp [eq_comm]
/-- Image of a bounded set in a normed space under scalar multiplication by a constant is
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 104 | 106 |
/-
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... | /-- `repr` preserves addition, up to multiples of `f` -/
| Mathlib/RingTheory/IsAdjoinRoot.lean | 158 | 158 |
/-
Copyright (c) 2021 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Jeremy Tan
-/
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
/-!
# Strongly regular graphs
## Main definitio... | · intro u
simp only [mem_union, mem_compl, mem_neighborFinset, mem_inter, mem_singleton]
rintro (rfl | rfl) <;> simpa [adj_comm] using ha.2
| Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 137 | 140 |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | /-- The Schnirelmann density of a set not containing `1` is `0`. -/
lemma schnirelmannDensity_eq_zero_of_one_not_mem (h : 1 ∉ A) : schnirelmannDensity A = 0 :=
((schnirelmannDensity_le_of_not_mem h).trans (by simp)).antisymm schnirelmannDensity_nonneg
| Mathlib/Combinatorics/Schnirelmann.lean | 103 | 105 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
/-!
# Recurrence Lemmas for the Continuants (`conts`) Function of Continued Fractions
## Summary
Given a... | exists_conts_a_of_num nth_num_eq
obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.conts (n + 1) = conts ∧ conts.a = predA :=
exists_conts_a_of_num succ_nth_num_eq
rw [num_eq_conts_a, conts_recurrence succ_nth_s_eq nth_conts_eq succ_nth_conts_eq]
/-- Shows that `Bₙ = bₙ * Bₙ₋₁ + aₙ * Bₙ₋₂`. -... | Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean | 50 | 59 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | theorem le_separableClosure' {L : IntermediateField F E} (hs : ∀ x : L, IsSeparable F x) :
L ≤ separableClosure F E := fun x h ↦ by simpa only [IsSeparable, minpoly_eq] using hs ⟨x, h⟩
| Mathlib/FieldTheory/SeparableClosure.lean | 150 | 151 |
/-
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.Algebra.Group.Embedding
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Order.Interval.Finset.Basic
/-!
# Finite... | · simp
ext i
| Mathlib/Data/Int/Interval.lean | 173 | 174 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... | calc
c ^ log b n ≤ b ^ log b n := Nat.pow_le_pow_left hb _
_ ≤ n := pow_log_le_self _ hn
theorem log_antitone_left {n : ℕ} : AntitoneOn (fun b => log b n) (Set.Ioi 1) := fun _ hc _ _ hb =>
log_anti_left (Set.mem_Iio.1 hc) hb
| Mathlib/Data/Nat/Log.lean | 183 | 188 |
/-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.NormedSpace.OperatorNor... | r_pos := lt_min hf.r_pos hg.r_pos
hasSum := by
intro y h'y hy
simp_rw [FormalMultilinearSeries.prod, ContinuousMultilinearMap.prod_apply]
| Mathlib/Analysis/Analytic/Constructions.lean | 289 | 292 |
/-
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... | protected theorem inv_pow : ∀ {a : ℝ≥0∞} {n : ℕ}, (a ^ n)⁻¹ = a⁻¹ ^ n
| Mathlib/Data/ENNReal/Inv.lean | 79 | 79 |
/-
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.Calculus.FDeriv.Basic
/-!
# The derivative of a composition (chain rule)
For detailed documentation of the... | rw [Function.iterate_succ, pow_succ]
rw [← hx] at ihn
exact ihn.comp x hf hL
@[fun_prop]
protected theorem HasFDerivAt.iterate {f : E → E} {f' : E →L[𝕜] E} (hf : HasFDerivAt f f' x)
| Mathlib/Analysis/Calculus/FDeriv/Comp.lean | 224 | 229 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
/-!
# The rational numbers form a linear ordered field
This f... | Mathlib/Algebra/Order/Ring/Rat.lean | 93 | 131 | |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | Function.Surjective A.mulVec ↔ IsUnit A := by
rw [mulVec_surjective_iff_exists_right_inverse, exists_right_inverse_iff_isUnit]
theorem vecMul_injective_iff_isUnit {A : Matrix m m K} :
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 341 | 344 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | induction c using limitRecOn with
| zero => simp only [succ_zero, mul_one]
| succ c IH =>
rw [mul_succ, IH, ← add_assoc, add_assoc _ b, ba, ← mul_succ]
| isLimit c l IH =>
rw [mul_succ, add_mul_limit_aux ba l IH, mul_succ, add_assoc]
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 799 | 805 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
⟨C, hC, fun n => Iff.mpr (le_div_iff₀ (pow_pos h0 _)) (hp n)⟩
/-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
theorem nnnorm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) ... | Mathlib/Analysis/Analytic/Basic.lean | 230 | 243 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Co... | refine continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => ?_
refine (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul ?_
rw [contDiffOn_nat_iff_continuousOn_differentiableOn_deriv hs] at hf
simp only [Finset.mem_range] at hi
refine hf.1 i ?_
simp only [WithTop.coe_le_coe... | Mathlib/Analysis/Calculus/Taylor.lean | 125 | 136 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory... | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | 879 | 883 | |
/-
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.Sites.Sheaf
/-!
# 1-hypercovers
Given a Grothendieck topology `J` on a category `C`, we define the type of
`1`-hypercovers of an object `S : C`.... | lemma sieve₁_eq_pullback_sieve₁' {W : C} (p₁ : W ⟶ E.X i₁) (p₂ : W ⟶ E.X i₂)
(w : p₁ ≫ E.f i₁ = p₂ ≫ E.f i₂) :
E.sieve₁ p₁ p₂ = (E.sieve₁' i₁ i₂).pullback (pullback.lift _ _ w) := by
ext Z g
constructor
· rintro ⟨j, h, fac₁, fac₂⟩
exact ⟨_, h, _, ⟨j⟩, by aesop_cat⟩
· rintro ⟨_, h, w, ⟨j⟩, fac⟩
e... | Mathlib/CategoryTheory/Sites/OneHypercover.lean | 94 | 103 |
/-
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.Algebra.ZMod
import Mathlib.Data.Nat.Multiplicity
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.... | rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
Int.cast_natCast, frobeniusPolyRat]
refine Nat.strong_induction_on n ?_; clear n
intro n IH
rw [xInTermsOfW_eq]
simp only [map_sum, map_sub, map_mul, map_pow (bind₁ _), bind₁_C_right]
have h1 : (p : ℚ) ^ n * ... | Mathlib/RingTheory/WittVector/Frobenius.lean | 131 | 140 |
/-
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.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | refine h.isBigO_sub.trans_tendsto (Tendsto.mono_left ?_ hL)
rw [← sub_self x]
exact tendsto_id.sub tendsto_const_nhds
have := this.add (tendsto_const_nhds (x := f x))
rw [zero_add (f x)] at this
exact this.congr (by simp only [sub_add_cancel, eq_self_iff_true, forall_const])
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 737 | 742 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... | right_inv' w _ := stereo_right_inv hv w
open_source := isOpen_compl_singleton
open_target := isOpen_univ
continuousOn_toFun :=
continuousOn_stereoToFun.comp continuous_subtype_val.continuousOn fun w h => by
dsimp
exact
h ∘ Subtype.ext ∘ Eq.symm ∘ (inner_eq_one_iff_of_norm_one hv (by simp... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 257 | 268 |
/-
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.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... |
theorem RightInverse.iterate {g : α → α} (hg : RightInverse g f) (n : ℕ) :
RightInverse g^[n] f^[n] :=
| Mathlib/Logic/Function/Iterate.lean | 191 | 193 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | conj_mem x hx y := by rw [mul_assoc]; exact Sn.conj' y hx
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 328 | 329 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,132 | 1,133 | |
/-
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, Yuyang Zhao
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra... | le_mul_self.trans'
@[to_additive] alias le_mul_left := le_mul_of_le_right
| Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 148 | 150 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Module.NatInt
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# M... | @[scoped simp]
theorem map_sub_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (x : G) : f (x - a) = f x - b := by
| Mathlib/Algebra/AddConstMap/Basic.lean | 166 | 168 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Setoid.Basic
import Mathlib.Order.Atoms
impor... |
section Atomise
/-- Cuts `s` along the finsets in `F`: Two elements of `s` will be in the same part if they are
in the same finsets of `F`. -/
def atomise (s : Finset α) (F : Finset (Finset α)) : Finpartition s :=
ofErase (F.powerset.image fun Q ↦ {i ∈ s | ∀ t ∈ F, t ∈ Q ↔ i ∈ t})
(Set.PairwiseDisjoint.supIndep... | Mathlib/Order/Partition/Finpartition.lean | 684 | 692 |
/-
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
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.CategoryTheory.Shift.Quotient
import Mathlib.CategoryTheory.Linear.LinearF... | lemma XIsoOfEq_shift (K : CochainComplex C ℤ) (n : ℤ) {p q : ℤ} (hpq : p = q) :
(K⟦n⟧).XIsoOfEq hpq = K.XIsoOfEq (show p + n = q + n by rw [hpq]) := rfl
variable (C)
| Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | 168 | 172 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Al... | IsLocalization.mk'_eq_of_eq (by rw [mul_comm, Subtype.coe_mk, ← h, mul_comm, Subtype.coe_mk])
theorem const_congr {f₁ f₂ g₁ g₂ : R} {U hu} (hf : f₁ = f₂) (hg : g₁ = g₂) :
const R f₁ g₁ U hu = const R f₂ g₂ U (hg ▸ hu) := by substs hf hg; rfl
| Mathlib/AlgebraicGeometry/StructureSheaf.lean | 356 | 359 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 398 | 402 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,773 | 2,818 | |
/-
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 | 102 | 102 | |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | (f : X ⟶ Y) : (tensorRightHomEquiv _ _ _ _) ((Yᘁ : C) ◁ f ≫ ε_ _ _) = fᘁ ≫ (λ_ _).inv := by
dsimp [tensorRightHomEquiv, rightAdjointMate]
simp
@[simp]
theorem tensorRightHomEquiv_whiskerRight_comp_evaluation {X Y : C} [HasRightDual X] (f : Y ⟶ Xᘁ) :
(tensorRightHomEquiv _ _ _ _) (f ▷ X ≫ ε_ X (Xᘁ)) = f ≫ (... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 438 | 446 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 970 | 972 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | Subtype.ext <| dist_conj_comm _ _
| Mathlib/Analysis/Complex/Basic.lean | 224 | 225 |
/-
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.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | HasFDerivWithinAt f₀ f' s x ↔ HasFDerivWithinAt f₁ f' s x :=
h.hasFDerivWithinAt_iff (h.eq_of_nhdsWithin hx)
theorem Filter.EventuallyEq.differentiableWithinAt_iff (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
DifferentiableWithinAt 𝕜 f₀ s x ↔ DifferentiableWithinAt 𝕜 f₁ s x :=
exists_congr fun _ => h.has... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 877 | 883 |
/-
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... |
theorem not_real_of_infinitesimal_ne_zero (x : ℝ*) : Infinitesimal x → x ≠ 0 → ∀ r : ℝ, x ≠ r :=
fun hi hx r hr =>
hx <| hr.trans <| coe_eq_zero.2 <| IsSt.unique (hr.symm ▸ isSt_refl_real r : IsSt x r) hi
theorem IsSt.infinitesimal_sub {x : ℝ*} {r : ℝ} (hxr : IsSt x r) : Infinitesimal (x - ↑r) := by
simpa only ... | Mathlib/Data/Real/Hyperreal.lean | 583 | 591 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖ₑ ∂μ) ≤ ENNReal.ofReal ε :=
let ⟨M, hM⟩ := hf.integral_indicator_norm_ge_le hmeas hε
⟨max M 0, le_max_right _ _, by simpa⟩
theorem MemLp.integral_indicator_norm_ge_nonneg_le (hf : MemLp f 1 μ) {ε : ℝ} (hε : 0 < ε) :
∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 231 | 254 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Seminorm
import Mathlib.GroupTheory.GroupAction.Pointwise
/-!
# The Minkowski functional, normed field version
In this file we define `(eg... | {c : 𝕜} {s : Set E} {x : E}
/-- If `c • x ∈ s` and `c ≠ 0`, then `egauge 𝕜 s x` is at most `(‖c‖₊⁻¹ : ℝ≥0)`.
| Mathlib/Analysis/Convex/EGauge.lean | 105 | 108 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.Galois.Basic
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and prove some of thei... |
variable (k) in
/-- A separably closed perfect field is also algebraically closed. -/
instance (priority := 100) isAlgClosed_of_perfectField [IsSepClosed k] [PerfectField k] :
IsAlgClosed k :=
IsAlgClosed.of_exists_root k fun p _ h ↦ exists_root p ((degree_pos_of_irreducible h).ne')
(PerfectField.separable_o... | Mathlib/FieldTheory/IsSepClosed.lean | 122 | 129 |
/-
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.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... | @[to_additive "\"iff\" version of `Summable.of_nat_of_neg`."]
lemma multipliable_int_iff_multipliable_nat_and_neg {f : ℤ → G} :
Multipliable f ↔ (Multipliable fun n : ℕ ↦ f n) ∧ (Multipliable fun n : ℕ ↦ f (-n)) := by
refine ⟨fun p ↦ ⟨?_, ?_⟩, fun ⟨hf₁, hf₂⟩ ↦ Multipliable.of_nat_of_neg hf₁ hf₂⟩ <;>
apply p.com... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 529 | 535 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... |
lemma norm_log_one_add_le {z : ℂ} (hz : ‖z‖ < 1) :
‖log (1 + z)‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2 + ‖z‖ := by
rw [← sub_add_cancel (log (1 + z)) z]
apply le_trans (norm_add_le _ _)
exact add_le_add_right (Complex.norm_log_one_add_sub_self_le hz) ‖z‖
/-- For `‖z‖ ≤ 1/2`, the complex logarithm is bounded by `(3/2)... | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 181 | 188 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... |
theorem roots_expand : (expand R p f).roots = p • f.roots.map (frobeniusEquiv R p).symm := by
conv_lhs => rw [← pow_one p, roots_expand_pow, iterateFrobeniusEquiv_eq_pow, pow_one]
rfl
| Mathlib/FieldTheory/Perfect.lean | 285 | 289 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Facts about algebras involving bilinear maps and tensor pro... | map_mul' := mulLeft_mul _ _
map_zero' := mulLeft_zero_eq_zero _ _
| Mathlib/Algebra/Algebra/Bilinear.lean | 146 | 148 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | xgcdAux r s t r' s' t' = xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t := by
obtain ⟨r, rfl⟩ := Nat.exists_eq_succ_of_ne_zero h.ne'
simp [xgcdAux]
| Mathlib/Data/Int/GCD.lean | 51 | 54 |
/-
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.Heyting.Basic
import Mathlib.Order.Hom.Basic
import Mathlib.Order.WithBot
/-!
# Orders on a sum type
This file defines the disjoint sum and the lin... | | inr _, inr _, LiftRel.inr h =>
let ⟨c, ha, hb⟩ := exists_between h
⟨toLex (inr c), LiftRel.inr ha, LiftRel.inr hb⟩⟩
@[simp]
theorem denselyOrdered_iff [LT α] [LT β] :
DenselyOrdered (α ⊕ β) ↔ DenselyOrdered α ∧ DenselyOrdered β :=
⟨fun _ =>
⟨⟨fun a b h => by
obtain ⟨c | c, ha, hb⟩ :... | Mathlib/Data/Sum/Order.lean | 240 | 250 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Contractions
Given modules $M, N$ over a commutative ring $R$, t... | fst_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
@[simp]
theorem zero_prodMap_dualTensorHom (g : Module.Dual R N) (q : Q) :
(0 : M →ₗ[R] P).prodMap ((dualTensorHom R N Q) (g ⊗ₜ[R] q)) =
dualTensorHom R (M × N) (P × Q) ((g ∘ₗ snd R M N) ⊗ₜ inr R P Q q) := by
ext <;>
simp only [coe_comp, c... | Mathlib/LinearAlgebra/Contraction.lean | 85 | 92 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | traverse_eq_map_id := @Vector.traverse_eq_map_id n
naturality := Vector.naturality
id_map := by intro _ x; cases x; simp! [(· <$> ·)]
| Mathlib/Data/Vector/Basic.lean | 695 | 697 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | (hf' : IntervalIntegrable f' volume a b) :
fourierCoeffOn hab f n = 1 / (-2 * π * I * n) *
(fourier (-n) (a : AddCircle (b - a)) * (f b - f a) - (b - a) * fourierCoeffOn hab f' n) :=
fourierCoeffOn_of_hasDeriv_right hab hn hf (fun x hx ↦ hff' x hx |>.hasDerivWithinAt) hf'
| Mathlib/Analysis/Fourier/AddCircle.lean | 498 | 501 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Junyan Xu, Sophie Morel
-/
import Mathlib.CategoryTheory.Limits.Creates
import Mathlib.CategoryTheory.Limits.Types.Limits
import Mathlib.CategoryTheory.Limits.Types... | Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean | 99 | 99 | |
/-
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... | empty_subset _
@[simp] theorem mapsTo_empty_iff : MapsTo f s ∅ ↔ s = ∅ := by
simp [mapsTo', subset_empty_iff]
| Mathlib/Data/Set/Function.lean | 124 | 127 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Hom.Basic
/-!
# Unbounded lattice homomorphisms
This file defines unbounded lattice homomorphisms. _Bounded_ lattice homomorphisms are defined in
`... |
@[simp, norm_cast]
lemma coe_evalLatticeHom (i : ι) : ⇑(evalLatticeHom (α := α) i) = Function.eval i := rfl
| Mathlib/Order/Hom/Lattice.lean | 831 | 833 |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | theorem IsOpen.upperSemicontinuous_indicator (hs : IsOpen s) (hy : y ≤ 0) :
UpperSemicontinuous (indicator s fun _x => y) :=
@IsOpen.lowerSemicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy
theorem IsOpen.upperSemicontinuousOn_indicator (hs : IsOpen s) (hy : y ≤ 0) :
UpperSemicontinuousOn (indicator s fun _x => y... | Mathlib/Topology/Semicontinuous.lean | 727 | 732 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 411 | 414 | |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... |
@[simp] theorem rightInvSeq_singleton (i : B) : ris [i] = [s i] := by simp [rightInvSeq]
@[simp] theorem leftInvSeq_singleton (i : B) : lis [i] = [s i] := rfl
theorem rightInvSeq_concat (ω : List B) (i : B) :
ris (ω.concat i) = (List.map (MulAut.conj (s i)) (ris ω)).concat (s i) := by
induction' ω with j ω ih
... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 206 | 214 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.LocalizerMorphism
/-!
# Resolutions for a morphism of localizers
Given a morphism of localizers `Φ : LocalizerMorphism W₁ W₂` (i.e.... | lemma essSurj_of_hasLeftResolutions [Φ.HasLeftResolutions] : (Φ.functor ⋙ L₂).EssSurj where
mem_essImage X₂ := by
have := Localization.essSurj L₂ W₂
have L : Φ.LeftResolution (L₂.objPreimage X₂) := Classical.arbitrary _
exact ⟨L.X₁, ⟨Localization.isoOfHom L₂ W₂ _ L.hw ≪≫ L₂.objObjPreimageIso X₂⟩⟩
lemma i... | Mathlib/CategoryTheory/Localization/Resolution.lean | 302 | 312 |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... | /-- This lemma is superseded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
allow `p = ∞` and only require `AEStronglyMeasurable`. -/
theorem tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasura... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 314 | 333 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | therefore noncomputable. -/
noncomputable def invertibleOfIsUnitDet (h : IsUnit A.det) : Invertible A :=
⟨A⁻¹, nonsing_inv_mul A h, mul_nonsing_inv A h⟩
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 436 | 439 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | have : encode (pair cf cg) < encode (prec cf cg) := by simp [encodeCode_eq, encodeCode]
exact (encode_lt_pair cf cg).imp (fun h => lt_trans h this) fun h => lt_trans h this
theorem encode_lt_rfind' (cf) : encode cf < encode (rfind' cf) := by
simp only [encodeCode_eq, encodeCode]
omega
| Mathlib/Computability/PartrecCode.lean | 197 | 203 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.Spe... | theorem HasStrictDerivAt.cosh (hf : HasStrictDerivAt f f' x) :
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 715 | 715 |
/-
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.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 93 | 95 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | constructor
· rintro ⟨t, ht, hn⟩
exact ⟨⟨t, eq_univ_iff_forall.1 ht⟩, hn⟩
· rintro ⟨⟨t, ht⟩, hn⟩
exact ⟨t, eq_univ_iff_forall.2 ht, hn⟩
theorem mk_union_add_mk_inter {α : Type u} {S T : Set α} :
#(S ∪ T : Set α) + #(S ∩ T : Set α) = #S + #T := by
classical
exact Quot.sound ⟨Equiv.Set.unionSumInte... | Mathlib/SetTheory/Cardinal/Basic.lean | 769 | 779 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... | simpa only [one_mul] using mul_of'_divOf (1 : k[G]) a
end divOf
/-- The remainder upon division by `of' k G g`. -/
noncomputable def modOf (x : k[G]) (g : G) : k[G] :=
| Mathlib/Algebra/MonoidAlgebra/Division.lean | 112 | 117 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | hf.2.1
theorem blsub_eq (hf : IsFundamentalSequence a o f) : blsub.{u, u} o f = a :=
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 446 | 448 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | ∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) := by
classical
| Mathlib/Data/Fintype/Basic.lean | 250 | 251 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
/-!... | theorem QInfty_f_naturality (n : ℕ) {X Y : SimplicialObject C} (f : X ⟶ Y) :
f.app (op ⦋n⦌) ≫ QInfty.f n = QInfty.f n ≫ f.app (op ⦋n⦌) :=
Q_f_naturality n n f
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 78 | 80 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | rw [closure_Ioo hab.ne, Icc_eq_Ioo_same_iff] at this
exact this hab.le
end DenselyOrdered
| Mathlib/Topology/Order/DenselyOrdered.lean | 408 | 417 |
/-
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.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.Localization.CalculusOfFractions
import Mathlib.CategoryTheory.Localization.T... | end
lemma ext₂ [S.P.IsClosedUnderIsomorphisms]
(T : Triangle C) (hT : T ∈ distTriang C) (h₁ : S.P T.obj₁)
| Mathlib/CategoryTheory/Triangulated/Subcategory.lean | 114 | 117 |
/-
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 | 1,476 | 1,516 | |
/-
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.... | induction h generalizing hn with
| refl => exact this hn
| step hnk ih => exact lt_trans (ih hn) <| this <| lt_trans hn <| lt_of_succ_le hnk
@[gcongr]
lemma factorial_lt_of_lt {m n : ℕ} (hn : 0 < n) (h : n < m) : n ! < m ! := (factorial_lt hn).mpr h
@[simp] lemma one_lt_factorial : 1 < n ! ↔ 1 < n := factorial_... | Mathlib/Data/Nat/Factorial/Basic.lean | 95 | 103 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... |
/-- If `gcd a (m * n) = 1`, then `gcd a m = 1`. -/
theorem gcd_eq_one_of_gcd_mul_right_eq_one_left {a : ℤ} {m n : ℕ} (h : a.gcd (m * n) = 1) :
| Mathlib/Data/Int/GCD.lean | 254 | 256 |
/-
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.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | refine ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condExp.integrableOn) (fun s hs hμs => ?_) hgm
(StronglyMeasurable.aestronglyMeasurable stronglyMeasurable_condExp)
rw [hg_eq s hs hμs, setIntegral_condExp hm hf hs]
| Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 280 | 284 |
/-
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.Fin.Fin2
import Mathlib.Data.TypeVec
import Mathlib.Logic.Equiv.Defs
/-!
# Functors between the category of tuples of... |
section LiftPLastPredIff
variable {F : TypeVec.{u} (n + 1) → Type*} [MvFunctor F] [LawfulMvFunctor F] {α : TypeVec.{u} n}
open MvFunctor
variable {β : Type u}
| Mathlib/Control/Functor/Multivariate.lean | 141 | 148 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Nat
import Mathli... | Subset.antisymm (range_subset_iff.2 Int.ofNat_nonneg) CanLift.prf
| Mathlib/Data/Nat/Cast/SetInterval.lean | 25 | 26 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# GCD... | simp
@[simp]
theorem lcm_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).lcm = GCDMonoid.lcm a s.lcm := by
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_cons]
simp
| Mathlib/Algebra/GCDMonoid/Multiset.lean | 95 | 100 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... |
theorem IsClosed.isCompact [CompactSpace X] (h : IsClosed s) : IsCompact s :=
| Mathlib/Topology/Compactness/Compact.lean | 695 | 696 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | Mathlib/Algebra/Module/Defs.lean | 376 | 377 |
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