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/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | (s \ t).ncard + t.ncard = (s ∪ t).ncard := by
rw [← ncard_union_eq disjoint_sdiff_left hs.diff ht, diff_union_self]
theorem diff_nonempty_of_ncard_lt_ncard (h : s.ncard < t.ncard) (hs : s.Finite := by toFinite_tac) :
| Mathlib/Data/Set/Card.lean | 899 | 902 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | simp only [o.rotation_apply, map_add, map_mul, LinearMap.map_smulₛₗ, RingHom.id_apply,
LinearMap.add_apply, LinearMap.smul_apply, real_smul, kahler_rightAngleRotation_left,
Real.Angle.coe_toCircle, Complex.conj_ofReal, conj_I]
ring
/-- Negating a rotation is equivalent to rotation by π plus the angle. -/
t... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 162 | 168 |
/-
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.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | (cancel_of_ne hb).tsub_lt_iff_right hab
| Mathlib/Data/ENNReal/Operations.lean | 379 | 380 |
/-
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.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... | ext n
dsimp [N₂Γ₂ToKaroubiIso]
simp only [comp_id, PInfty_f_idem_assoc, AlternatingFaceMapComplex.obj_X, Γ₀_obj_obj]
convert comp_id _
apply (Γ₀.splitting X).hom_ext'
intro A
rw [Splitting.ι_desc]
erw [comp_id, id_comp]
/-- The counit isomorphism of the Dold-Kan equivalence for additive categories. -/
... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 137 | 149 |
/-
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.Algebra.Rat
import Mathlib.Data.Nat.Cast.Field
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In t... | ext
simp [sin, apply_ite f]
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 212 | 214 |
/-
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... | HasFPowerSeriesWithinAt f p s x := by
rw [← hasFPowerSeriesWithinAt_univ] at hf
apply hf.mono (subset_univ _)
theorem HasFPowerSeriesWithinAt.mono_of_mem_nhdsWithin
| Mathlib/Analysis/Analytic/Basic.lean | 658 | 662 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | theorem coeff_X_pow [DecidableEq σ] (i : σ) (m) (k : ℕ) :
coeff m (X i ^ k : MvPolynomial σ R) = if Finsupp.single i k = m then 1 else 0 := by
| Mathlib/Algebra/MvPolynomial/Basic.lean | 614 | 615 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... |
section Transformations
variable {μ : ℝ} {v : ℝ≥0}
lemma _root_.MeasurableEmbedding.gaussianReal_comap_apply (hv : v ≠ 0)
{f : ℝ → ℝ} (hf : MeasurableEmbedding f)
| Mathlib/Probability/Distributions/Gaussian.lean | 246 | 252 |
/-
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... | theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by
induction' n with n ih; · simp [Set.singleton_one]
| Mathlib/RingTheory/Ideal/Operations.lean | 431 | 432 |
/-
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.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | ⟨z, hz, hzt⟩
exact ⟨z - y, by simpa using hzt, by simpa using hz⟩
choose d' hd' using this
refine ⟨c, fun n => (d n, d' n), ?_, hc, ?_⟩
· show ∀ᶠ n in atTop, (x, y) + (d n, d' n) ∈ s ×ˢ t
filter_upwards [hd] with n hn
simp [hn, (hd' n).1]
· apply Tendsto.prodMk_nhds hy _
refine squeeze_zer... | Mathlib/Analysis/Calculus/TangentCone.lean | 182 | 201 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.Logic.Equiv.Defs
/-!
# Full and faithful functors
We define typeclasses `Full` and `Faithful`, decorating functors. ... |
variable {F F'}
/-- If `F` is full, and naturally isomorphic to some `F'`, then `F'` is also full. -/
| Mathlib/CategoryTheory/Functor/FullyFaithful.lean | 267 | 270 |
/-
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.... | rw [mul_comm]; apply div_mul_div_cancel
@[to_additive (attr := simp)]
theorem mul_div_div_cancel (a b c : G) : a * b / (a / c) = b * c := by
| Mathlib/Algebra/Group/Basic.lean | 990 | 993 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | Mathlib/Logic/Function/Basic.lean | 1,062 | 1,064 | |
/-
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... | simp
/-- Transport an exact pairing across an isomorphism in the first argument. -/
def exactPairingCongrLeft {X X' Y : C} [ExactPairing X' Y] (i : X ≅ X') : ExactPairing X Y where
evaluation' := Y ◁ i.hom ≫ ε_ _ _
coevaluation' := η_ _ _ ≫ i.inv ▷ Y
evaluation_coevaluation' :=
calc
_ = η_ X' Y ▷ X ⊗... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 476 | 484 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Currying and uncurrying continuous multilinear maps
... | calc
‖f (m 0) (tail m)‖ ≤ ‖f (m 0)‖ * ∏ i, ‖(tail m) i‖ := (f (m 0)).le_opNorm _
_ ≤ ‖f‖ * ‖m 0‖ * ∏ i, ‖tail m i‖ := mul_le_mul_of_nonneg_right (f.le_opNorm _) <| by positivity
_ = ‖f‖ * (‖m 0‖ * ∏ i, ‖(tail m) i‖) := by ring
_ = ‖f‖ * ∏ i, ‖m i‖ := by
rw [prod_univ_succ]
rfl
theorem Con... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 61 | 70 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.G... | lemma IsMulFreimanIso.mul_eq_mul (hf : IsMulFreimanIso 2 A B f) {a b c d : α}
(ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) (hd : d ∈ A) :
f a * f b = f c * f d ↔ a * b = c * d := by
simp_rw [← prod_pair]
refine hf.map_prod_eq_map_prod ?_ ?_ (card_pair _ _) (card_pair _ _) <;> simp [ha, hb, hc, hd]
/-- Characteri... | Mathlib/Combinatorics/Additive/FreimanHom.lean | 141 | 149 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | · simp
· simp
| Mathlib/Data/Sign.lean | 444 | 445 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
theorem Finset.prod_nonneg_of_card_nonpos_even {α β : Type*}
[CommRing β] [LinearOrder β] [IsStrictOrderedRing β] {f : α → β}
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 48 | 54 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and lower... | @[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ := rfl
@[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) := rfl
@[simp] lemma toLowerSet_specializes_toLowerSet {a b : α} :
| Mathlib/Topology/Order/UpperLowerSetTopology.lean | 402 | 405 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.Restrict
/-! # Mutually singular measures
Two measures `μ`, `ν` are said to be mutually singular (`MeasureTheory.Mea... | (h : μ ≪ ν₁ + ν₂) (h_ms : μ ⟂ₘ ν₂) : μ ≪ ν₁ := by
refine AbsolutelyContinuous.mk fun s hs hs_zero ↦ ?_
let t := h_ms.nullSet
have ht : MeasurableSet t := h_ms.measurableSet_nullSet
| Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 154 | 157 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.Category... | theorem Presieve.isSheaf_iff_preservesFiniteProducts (F : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (extensiveTopology C) F ↔ PreservesFiniteProducts F := by
refine ⟨fun hF ↦ ⟨fun n ↦ ⟨fun {K} ↦ ?_⟩⟩, fun hF ↦ ?_⟩
· rw [extensiveTopology, isSheaf_coverage] at hF
let Z : Fin n → C := fun i ↦ unop (K.obj ⟨i⟩)
have... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 81 | 111 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,559 | 1,560 | |
/-
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... | Mathlib/Computability/PartrecCode.lean | 1,031 | 1,034 | |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Small.Module
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.TensorProduc... | (LinearMap.quotKerEquivOfSurjective _ <| LinearMap.range_eq_top.mp ?_).symm, ?_⟩
· simpa [range_linearCombination] using hι₁
· simpa [LinearMap.ker_comp, Submodule.comap_equiv_eq_map_symm] using hι₂.map _
| Mathlib/Algebra/Module/FinitePresentation.lean | 87 | 90 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,426 | 1,426 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Brendan Murphy
-/
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Function.OfArity
/-!
# Currying and uncurrying of n-ary funct... | theorem curry_uncurry (f : Function.FromTypes p τ) : curry (uncurry f) = f := by
induction n with
| zero => rfl
| succ n ih => exact funext (ih <| f ·)
| Mathlib/Data/Fin/Tuple/Curry.lean | 73 | 76 |
/-
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 [show 2 * (m + 3) = 2 * m + 1 + 5 by rfl, preNormEDS', dif_neg m.not_even_two_mul_add_one]
simp only [Nat.mul_add_div two_pos]
rfl
/-- The auxiliary sequence for a normalised EDS `W : ℤ → R`, with initial values
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 195 | 199 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | lemma abs_le_one_iff_mul_self_le_one : |a| ≤ 1 ↔ a * a ≤ 1 := by
simpa only [abs_one, one_mul] using abs_le_iff_mul_self_le (a := a) (b := 1)
| Mathlib/Algebra/Order/Ring/Abs.lean | 90 | 92 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | exact Finset.mem_map.not.mp his ⟨i.1, Finset.mem_univ _, Prod.ext rfl h⟩
@[simp]
| Mathlib/Analysis/Matrix.lean | 584 | 586 |
/-
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 | 2,240 | 2,243 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,265 | 2,266 | |
/-
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.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | have : map (a + ·) (𝓝[s] x) = 𝓝[a +ᵥ s] (a + x) := by
simp only [add_comm x a, nhdsWithin, Filter.map_inf (add_right_injective a)]
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 712 | 713 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | rw [inv_def, log_mul_ofReal, Real.log_inv, ofReal_neg, ← sub_eq_neg_add, log_conj_eq_ite]
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 116 | 116 |
/-
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.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | · simp [hz]
lemma traceForm_apply_eq_zero_of_mem_lcs_of_mem_center {x y : L}
(hx : x ∈ lowerCentralSeries R L L 1) (hy : y ∈ LieAlgebra.center R L) :
traceForm R L M x y = 0 := by
apply traceForm_eq_zero_if_mem_lcs_of_mem_ucs R L M 1
| Mathlib/Algebra/Lie/TraceForm.lean | 143 | 148 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.UniformSpace.Completion
import Mathlib.Topology.Algebra.Group.Pointwise
/-!
... | Mathlib/Topology/Algebra/GroupCompletion.lean | 288 | 298 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Nat
import Ma... | Mathlib/RingTheory/Int/Basic.lean | 147 | 152 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathlib.Algebra.Polyno... | @[simp]
theorem taylor_mul {R} [CommSemiring R] (r : R) (p q : R[X]) :
| Mathlib/Algebra/Polynomial/Taylor.lean | 88 | 89 |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | elim _b hb _a₁ _a₂ ha i := smul_le_smul_of_nonneg_right ha (hb i)
| Mathlib/Algebra/Order/Module/Defs.lean | 1,018 | 1,019 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | ‖(n : ℂ) ^ s‖ = (n : ℝ) ^ (s.re) := by
rw [← ofReal_natCast, norm_cpow_eq_rpow_re_of_nonneg n.cast_nonneg hs]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 333 | 334 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | Mathlib/Analysis/Convex/Side.lean | 936 | 945 | |
/-
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.Group.Commute.Hom
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Subgro... | (Noncommutative case; in the commutative case, use `AddHom.coprod`.)"]
def noncommCoprod : M × N →* P where
| Mathlib/GroupTheory/NoncommCoprod.lean | 81 | 82 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib... | (hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
(bound_integrable : Integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (F' a)... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 75 | 155 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | refine ⟨s ∪ r', subset_union_left, union_subset hst (hr'.trans diff_subset), ?_⟩
rw [encard_union_eq (disjoint_of_subset_right hr' disjoint_sdiff_right)]
section Function
variable {s : Set α} {t : Set β} {f : α → β}
| Mathlib/Data/Set/Card.lean | 395 | 400 |
/-
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.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.SelfAdjoint
/-... | (zero : p 0 (zero_mem _)) (mul : ∀ x y hx hy, p x hx → p y hy → p (x * y) (mul_mem hx hy))
(smul : ∀ (r : R) x hx, p x hx → p (r • x) (SMulMemClass.smul_mem r hx))
(star : ∀ x hx, p x hx → p (star x) (star_mem hx))
{a : A} (ha : a ∈ adjoin R s) : p a ha := by
| Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 688 | 691 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | protected theorem mul_left (c : ℕ) (h : a ≡ b [MOD n]) : c * a ≡ c * b [MOD n] :=
(h.mul_left' _).of_dvd (dvd_mul_left _ _)
| Mathlib/Data/Nat/ModEq.lean | 99 | 100 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 77 | 77 | |
/-
Copyright (c) 2022 Anand Rao, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anand Rao, Rémi Bottinelli
-/
import Mathlib.Combinatorics.SimpleGraph.Ends.Defs
import Mathlib.CategoryTheory.CofilteredSystem
/-!
# Properties of the ends of graphs
Thi... | ((e : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j) K).supp.Infinite := by
refine (e.val K).infinite_iff_in_all_ranges.mpr (fun L h => ?_)
change Opposite.unop K ⊆ Opposite.unop (Opposite.op L) at h
exact ⟨e.val (Opposite.op L), (e.prop (CategoryTheory.opHomOfLE h))⟩
instance compononentComplFunctor_no... | Mathlib/Combinatorics/SimpleGraph/Ends/Properties.lean | 31 | 36 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,301 | 1,334 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... |
@[deprecated (since := "2024-11-09")]
alias valuation := toValuation
| Mathlib/RingTheory/Valuation/Basic.lean | 678 | 681 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | (rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
· intro u r
| Mathlib/Algebra/MvPolynomial/Rename.lean | 284 | 288 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 1,588 | 1,590 | |
/-
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... | theorem empty_vecMul (v : Fin 0 → α) (B : Matrix (Fin 0) o' α) : v ᵥ* B = 0 :=
rfl
@[simp]
| Mathlib/Data/Matrix/Notation.lean | 291 | 294 |
/-
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.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | ⟨_, Lists'.atom a⟩
/-- Converts a proper ZFA prelist to a ZFA list. -/
@[match_pattern]
def of' (l : Lists' α true) : Lists α :=
⟨_, l⟩
/-- Converts a ZFA list to a `List` of ZFA lists. Atoms are sent to `[]`. -/
| Mathlib/SetTheory/Lists.lean | 191 | 198 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.Group.TypeTags.Basic
/-!
# Squares and even elements
This ... | @[to_additive (attr := simp) Even.nsmul_left]
lemma Even.isSquare_pow : Even n → ∀ a : α, IsSquare (a ^ n) := by
| Mathlib/Algebra/Group/Even.lean | 130 | 131 |
/-
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... | Basis.repr_self, Finsupp.smul_single_one, Finsupp.single_eq_pi_single, Matrix.diagonal_apply,
Pi.single_apply]
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 772 | 774 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 695 | 695 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | ring
theorem tan_pos_of_pos_of_lt_pi_div_two {x : ℝ} (h0x : 0 < x) (hxp : x < π / 2) : 0 < tan x := by
rw [tan_eq_sin_div_cos]
exact div_pos (sin_pos_of_pos_of_lt_pi h0x (by linarith)) (cos_pos_of_mem_Ioo ⟨by linarith, hxp⟩)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 867 | 871 |
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | dimH_image_le_of_locally_lipschitzOn fun x hx =>
let ⟨C, u, hu, hf⟩ := (hf x (ht hx)).exists_lipschitzOnWith hc
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 500 | 501 |
/-
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.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | Mathlib/Algebra/Quaternion.lean | 1,340 | 1,340 | |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | | 3 => le_rfl
| 4 => le_rfl
| _n + 5 => (le_fib_self le_add_self).trans <| le_succ _
/-- Subsequent Fibonacci numbers are coprime,
see https://proofwiki.org/wiki/Consecutive_Fibonacci_Numbers_are_Coprime -/
theorem fib_coprime_fib_succ (n : ℕ) : Nat.Coprime (fib n) (fib (n + 1)) := by
induction' n with n ih
... | Mathlib/Data/Nat/Fib/Basic.lean | 134 | 142 |
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.CardCommute
import Mathlib.Grou... | right_inv := by rintro (x | x) <;> rfl
/-- If `0 < n`, then `DihedralGroup n` is a finite group.
-/
| Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 129 | 132 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.P... | slice_lhs 1 3 => simp only []
simp [Equivalence.counitInv]
end
section
| Mathlib/CategoryTheory/Limits/Fubini.lean | 575 | 581 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... |
theorem mem_domain_of_mem_graph {f : E →ₗ.[R] F} {x : E} {y : F} (h : (x, y) ∈ f.graph) :
x ∈ f.domain := by
rw [mem_domain_iff]
exact ⟨y, h⟩
theorem image_iff {f : E →ₗ.[R] F} {x : E} {y : F} (hx : x ∈ f.domain) :
y = f ⟨x, hx⟩ ↔ (x, y) ∈ f.graph := by
rw [mem_graph_iff]
constructor <;> intro h
| Mathlib/LinearAlgebra/LinearPMap.lean | 782 | 791 |
/-
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.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Option
import Ma... | @[simp] lemma card_pi [∀ i, Fintype (α i)] : card (∀ i, α i) = ∏ i, card (α i) :=
card_piFinset _
| Mathlib/Data/Fintype/BigOperators.lean | 130 | 131 |
/-
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.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | instance instRing : Ring ℍ[R] := QuaternionAlgebra.instRing
| Mathlib/Algebra/Quaternion.lean | 767 | 767 |
/-
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.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 689 | 701 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | range of the isometry. -/
@[simps! +simpRhs toEquiv apply]
def Isometry.isometryEquivOnRange [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
| Mathlib/Topology/MetricSpace/Isometry.lean | 608 | 610 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,096 | 3,101 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | rw [γ.cyclesMap'_eq, (γ.map F).cyclesMap'_eq, ShortComplex.LeftHomologyMapData.map_φK]
lemma map_leftHomologyMap' : F.map (ShortComplex.leftHomologyMap' φ hl₁ hl₂) =
ShortComplex.leftHomologyMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F) := by
have γ : ShortComplex.LeftHomologyMapData φ hl₁ hl₂ := def... | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 365 | 369 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... |
theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α}
(hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) := by
| Mathlib/Topology/ContinuousOn.lean | 1,336 | 1,338 |
/-
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.Group.Submonoid.BigOperators
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod... | /-- Subsemiring closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. -/
@[gcongr]
theorem closure_mono ⦃s t : Set R⦄ (h : s ⊆ t) : closure s ≤ closure t :=
closure_le.2 <| Set.Subset.trans h subset_closure
| Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 368 | 372 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by
cases o
· simp only [true_and, false_or, eq_self_iff_true, none_orElse, reduceCtorEq]
· simp only [some_orElse, or_false, false_and, reduceCtorEq]
theorem orElse_eq_some' (o o' : Option α) (x : α) :
o.orElse (fun _ ↦ o') = some x ↔ o = some ... | Mathlib/Data/Option/Basic.lean | 260 | 270 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 2,184 | 2,189 | |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Constructi... | -- Let `f` be the minimal polynomial of `α ∈ E` over `F`
let f : F[X] := minpoly F α
let G := { g : E[X] // g.Monic ∧ g ∣ f.map (algebraMap F E) }
-- Then `f` has only finitely many monic factors
have hfin : Finite G := @Finite.of_fintype _ <| fintypeSubtypeMonicDvd
(f.map (algebraMap F E)) <| map_ne_zero... | Mathlib/FieldTheory/PrimitiveElement.lean | 307 | 312 |
/-
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.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | theorem lt_log_of_lt_opow {b x c : Ordinal} (hc : c ≠ 0) : x < b ^ c → log b x < c :=
lt_imp_lt_of_le_imp_le <| opow_le_of_le_log hc
theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o := by
rwa [← succ_le_iff, succ_zero, ← opow_le_iff_le_log hb ho, opow_one]
theorem log_eq_zero ... | Mathlib/SetTheory/Ordinal/Exponential.lean | 385 | 394 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | @[to_additive (attr := simp)]
theorem prod_univ_two (f : Fin 2 → M) : ∏ i, f i = f 0 * f 1 := by
| Mathlib/Algebra/BigOperators/Fin.lean | 118 | 119 |
/-
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.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | I.map_id
theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) :
(I.comap g).comap f = I.comap (g.comp f) :=
rfl
| Mathlib/RingTheory/Ideal/Maps.lean | 149 | 153 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Operator.LinearIsometry
import Mathlib.LinearAlgebra.Basis.VectorSpace
/-!
# Conformal Linear ... | rcases hf with ⟨cf, hcf, lif, rfl⟩
rcases hg with ⟨cg, hcg, lig, rfl⟩
| Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 78 | 79 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# A collection of specific asymptotic results
This fil... | end NormedLinearOrderedField
section Real
open Finset
theorem Asymptotics.IsLittleO.sum_range {α : Type*} [NormedAddCommGroup α] {f : ℕ → α} {g : ℕ → ℝ}
| Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 82 | 88 |
/-
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.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
/-!
# Remov... | trans ∮ z in C(c, R), ((z - w₀) ^ 2)⁻¹ • (f z - f w₀)
· have h1 : ContinuousOn (fun z : ℂ => ((z - w₀) ^ 2)⁻¹) (sphere c R) := by
refine ((continuous_id'.sub continuous_const).pow 2).continuousOn.inv₀ fun w hw h => ?_
exact sphere_disjoint_ball.ne_of_mem hw hw₀ (sub_eq_zero.mp (sq_eq_zero_iff.mp h))
... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 136 | 161 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | /-- A vector in `(𝕜 ∙ u)ᗮ` is orthogonal to `u`. -/
theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪v, u⟫ = 0 := by
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
theorem sub_mem_orthogonal_of_inner_left {x y : E} (h : ∀ v : K, ⟪x, v⟫ = ⟪y, v⟫) : x - y ∈ Kᗮ := by
rw... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 73 | 78 |
/-
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... | simp only [List.length_append, List.length_reverse, List.length_map, ← h,
Nat.sub_self, List.length_singleton, List.getElem_singleton,
le_refl, Nat.lt_succ_self]
| Mathlib/Computability/TuringMachine.lean | 553 | 555 |
/-
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.Interval.Set.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.SetLike.Basic
/-!
# Order intervals
This file defines (nonempty) close... | Mathlib/Order/Interval/Basic.lean | 759 | 760 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,653 | 1,657 | |
/-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
/-!
# Chinese Re... |
theorem chineseRemainderOfList_modEq_unique (l : List ι)
(co : l.Pairwise (Coprime on s)) {z} (hz : ∀ i ∈ l, z ≡ a i [MOD s i]) :
z ≡ chineseRemainderOfList a s l co [MOD (l.map s).prod] := by
induction' l with i l ih
· simp [modEq_one]
· simp only [List.map_cons, List.prod_cons, chineseRemainderOfList]
... | Mathlib/Data/Nat/ChineseRemainder.lean | 93 | 105 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 406 | 413 | |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Uniformly locally doubling measures
A uniformly locally doubl... |
theorem scalingScaleOf_pos (K : ℝ) : 0 < scalingScaleOf μ K :=
(eventually_measure_mul_le_scalingConstantOf_mul μ K).choose_spec.1
theorem measure_mul_le_scalingConstantOf_mul {K : ℝ} {x : α} {t r : ℝ} (ht : t ∈ Ioc 0 K)
| Mathlib/MeasureTheory/Measure/Doubling.lean | 139 | 143 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib... | lemma equalizerCondition_w (P : Cᵒᵖ ⥤ D) {X B : C} {π : X ⟶ B} (c : PullbackCone π π) :
P.map π.op ≫ P.map c.fst.op = P.map π.op ≫ P.map c.snd.op := by
simp only [← Functor.map_comp, ← op_comp, c.condition]
| Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 44 | 46 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | @[simp]
theorem valuation_zero : valuation (0 : ℚ_[p]) = 0 :=
dif_pos ((const_equiv p).2 rfl)
theorem norm_eq_zpow_neg_valuation {x : ℚ_[p]} : x ≠ 0 → ‖x‖ = (p : ℝ) ^ (-x.valuation) := by
refine Quotient.inductionOn' x fun f hf => ?_
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 939 | 944 |
/-
Copyright (c) 2022 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# The filter of small sets
This file defines the ... | rfl
protected theorem HasBasis.smallSets {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) :
HasBasis l.smallSets p fun i => 𝒫 s i :=
h.lift' monotone_powerset
| Mathlib/Order/Filter/SmallSets.lean | 47 | 51 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | @[simp]
theorem IntegrableAtFilter.inf_ae_iff {l : Filter α} :
IntegrableAtFilter f (l ⊓ ae μ) μ ↔ IntegrableAtFilter f l μ := by
refine ⟨?_, fun h ↦ h.filter_mono inf_le_left⟩
rintro ⟨s, ⟨t, ht, u, hu, rfl⟩, hf⟩
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 445 | 449 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Data.Set.CoeSort
import Mathlib.T... | section
| Mathlib/CategoryTheory/Types.lean | 251 | 252 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | Mathlib/CategoryTheory/Subobject/Limits.lean | 432 | 435 | |
/-
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... | intro i
simpa using IsLittleO.eventuallyLE (R.dist_r_b i)
lemma eventually_b_le_r : ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i : ℝ) * n - (n / log n ^ 2) ≤ r i n := by
filter_upwards [R.dist_r_b'] with n hn
intro i
have h₁ : 0 ≤ b i := le_of_lt <| R.b_pos _
rw [sub_le_iff_le_add, add_comm, ← sub_le_iff_le_add]
calc ... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 152 | 162 |
/-
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.Group.Commute.Hom
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Subgro... | __ := f.toMulHom.noncommCoprod g.toMulHom comm
/-- Variant of `MonoidHom.noncomCoprod_apply` with the product written in the other direction` -/
@[to_additive
| Mathlib/GroupTheory/NoncommCoprod.lean | 85 | 88 |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | Mathlib/Data/PFunctor/Univariate/M.lean | 720 | 734 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | end Functor
lemma NatTrans.app_homology {F G : C ⥤ D} (τ : F ⟶ G)
(S : ShortComplex C) [S.HasHomology] [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms]
[F.PreservesLeftHomologyOf S] [G.PreservesLeftHomologyOf S] [F.PreservesRightHomologyOf S]
[G.PreservesRightHomologyOf S] :
τ.app S.homology = (S.... | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 854 | 861 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.Normal.Defs
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
/-!
# Algebraica... | theorem exists_pow_nat_eq [IsAlgClosed k] (x : k) {n : ℕ} (hn : 0 < n) : ∃ z, z ^ n = x := by
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn x]
exact ne_of_gt (WithBot.coe_lt_coe.2 hn)
obtain ⟨z, hz⟩ := exists_root (X ^ n - C x) this
use z
simp only [eval_C, eval_X, eval_pow, eval_sub, ... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 89 | 96 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
/-!
# Racks and Quandles
This file defines racks and quandles, algebraic structu... | right_inv x := by simp
/-- An involutory rack is one for which `Rack.oppositeRack R x` is an involution for every x.
-/
def IsInvolutory (R : Type*) [Rack R] : Prop :=
| Mathlib/Algebra/Quandle.lean | 293 | 297 |
/-
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.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | theorem terminatedAt_iff_s_none : g.TerminatedAt n ↔ g.s.get? n = none := by rfl
| Mathlib/Algebra/ContinuedFractions/Translations.lean | 35 | 35 |
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