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/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... | exists_pair_ne := ⟨0, 1, zero_ne_one⟩
inv_zero := inv_zero
mul_inv_cancel _ := CauSeq.Completion.mul_inv_cancel
nnqsmul := (· • ·)
| Mathlib/Algebra/Order/CauSeq/Completion.lean | 239 | 242 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,581 | 1,599 | |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | Mathlib/Data/List/Sublists.lean | 435 | 453 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 328 | 331 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Set.Accumulate
import Mathlib.Data.Set.Pairwise.Lattice
import Mathlib.MeasureTheory.PiSystem
/-!... | simp only [disjointOfDiffUnion, mem_sdiff, Finset.mem_singleton, eq_self_iff_true,
not_true, and_false, not_false_iff]
lemma disjointOfDiffUnion_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
↑(hC.disjointOfDiffUnion hs hI) ⊆ C := by
classical
| Mathlib/MeasureTheory/SetSemiring.lean | 227 | 232 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | intro x
conv => congr
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 473 | 474 |
/-
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... | f ∈ discrete.arrows c d ↔ ∃ h : c = d, f = eqToHom h :=
⟨by rintro ⟨⟩; exact ⟨rfl, rfl⟩, by rintro ⟨rfl, rfl⟩; constructor⟩
/-- A subgroupoid is wide if its carrier set is all of `C`. -/
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 272 | 275 |
/-
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.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | This is similar to `Polynomial.natDegree` but not to `Polynomial.degree`, namely, the separable
degree of `0` is `0`, not negative infinity. -/
| Mathlib/FieldTheory/SeparableDegree.lean | 295 | 296 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Hom.ContinuousEval
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Separation.Regular
/-!
# The compact-open topology
In this ... | -- So glue the `f s hs` together and prove that this glued function `f₀` is a limit on each
-- compact set `s`
refine ⟨liftCover' _ _ h exists_compact_mem_nhds, ?_⟩
rw [tendsto_compactOpen_iff_forall]
intro s hs
rw [liftCover_restrict']
exact hf s hs
end InfInduced
section Coev
variable (... | Mathlib/Topology/CompactOpen.lean | 312 | 338 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... |
theorem C_mul_dvd (ha : a ≠ 0) : C a * p ∣ q ↔ p ∣ q :=
⟨fun h => dvd_trans (dvd_mul_left _ _) h, fun ⟨r, hr⟩ =>
⟨C a⁻¹ * r, by
rw [mul_assoc, mul_left_comm p, ← mul_assoc, ← C.map_mul, mul_inv_cancel₀ ha, C.map_one,
one_mul, hr]⟩⟩
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 538 | 543 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | rw [← mul_smul, mul_inv_cancel_left₀ ha.ne']
theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s) :
1 ≤ gauge s x :=
le_gauge_of_not_mem hs₁ hs₂ <| by rwa [one_smul]
section LinearOrderedField
| Mathlib/Analysis/Convex/Gauge.lean | 218 | 225 |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,
Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser
-/
import Mathlib.NumberTheory.FLT.Basic
import... | lemma a_cube_b_cube_congr_one_or_neg_one :
λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1 := by
obtain ⟨z, hz⟩ := S'.hcdvd
rcases lambda_pow_four_dvd_cube_sub_one_or_add_one_of_lambda_not_dvd hζ S'.ha with
⟨x, hx⟩ | ⟨x, hx⟩ <;>
rcases lambda_pow_four_dvd_cube_sub... | Mathlib/NumberTheory/FLT/Three.lean | 240 | 255 |
/-
Copyright (c) 2024 Frédéric Marbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Marbach
-/
import Mathlib.Algebra.Lie.Derivation.AdjointAction
import Mathlib.Algebra.Lie.Killing
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Derivations of ... | variable (R L)
variable [LieAlgebra.IsKilling R L]
@[simp] lemma ad_apply_eq_zero_iff (x : L) : ad R L x = 0 ↔ x = 0 := by
| Mathlib/Algebra/Lie/Derivation/Killing.lean | 76 | 79 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Algebra.EuclideanDomain.Field
... | dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n := by rw [dickson]
| Mathlib/RingTheory/Polynomial/Dickson.lean | 77 | 78 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem Set.Countable.isLindelof_sUnion {S : Set (Set X)} (hf : S.Countable)
(hc : ∀ s ∈ S, IsLindelof s) : IsLindelof (⋃₀ S) := by
rw [sUnion_eq_biUnion]; exact hf.isLindelof_biUnion hc
| Mathlib/Topology/Compactness/Lindelof.lean | 347 | 349 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Types.Shapes
import Mathlib.Tactic.ApplyFun
/-!
# The equalizer diagram sheaf conditi... | @[stacks 00VM "The rightmost object of the fork diagram there."]
def SecondObj : Type w :=
∏ᶜ (fun (ij : I × I) ↦ P.obj (op (pullback (π ij.1) (π ij.2))))
@[ext]
lemma SecondObj.ext (z₁ z₂ : SecondObj P X π) (h : ∀ ij, (Pi.π _ ij : SecondObj P X π ⟶ _) z₁ =
| Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 281 | 286 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | rw [condIndepFun_iff_condIndep, condIndep_iff _ _ _ _ hf.comap_le hg.comap_le]
end DefinitionLemmas
section CondIndepSets
variable {m' : MeasurableSpace Ω} {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω]
| Mathlib/Probability/Independence/Conditional.lean | 343 | 349 |
/-
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.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix... | end BasisToMatrix
| Mathlib/LinearAlgebra/Matrix/Basis.lean | 278 | 282 |
/-
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.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | /-- Negating the first vector produces the same angle as negating the second vector. -/
theorem oangle_neg_left_eq_neg_right (x y : V) : o.oangle (-x) y = o.oangle x (-y) := by
rw [← neg_neg y, oangle_neg_neg, neg_neg]
/-- The angle between the negation of a nonzero vector and that vector is `π`. -/
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 212 | 216 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | have : n - #t < n - #s := by omega
strongDownwardInduction H t ht
termination_by s => n - #s
@[nolint unusedHavesSuffices] -- Porting note: false positive
theorem strongDownwardInduction_eq {p : Finset α → Sort*}
(H : ∀ t₁, (∀ {t₂ : Finset α}, #t₂ ≤ n → t₁ ⊂ t₂ → p t₂) → #t₁ ≤ n → p t₁)
(s : Fins... | Mathlib/Data/Finset/Card.lean | 793 | 802 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | theorem symmDiff_eq_sup_sdiff_inf : a ∆ b = (a ⊔ b) \ (a ⊓ b) := by simp [sup_sdiff, symmDiff]
| Mathlib/Order/SymmDiff.lean | 137 | 138 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... |
lemma RightHomologyData.exact_iff [S.HasHomology]
(h : S.RightHomologyData) :
S.Exact ↔ IsZero h.H := by
rw [S.exact_iff_isZero_homology]
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 70 | 74 |
/-
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... | rpow_natCast _ _ ▸ le_rpow_iff_log_le hx hy
lemma le_zpow_iff_log_le {n : ℤ} (hx : 0 < x) (hy : 0 < y) : x ≤ y ^ n ↔ log x ≤ n * log y :=
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 773 | 775 |
/-
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... | rw [← map_map, ← map_map]
refine hg.map_prod_eq_map_prod ?_ ?_ (by rwa [card_map]) (by rwa [card_map])
(hf.map_prod_eq_map_prod hsA htA hs ht h)
· simpa using fun a h ↦ hf.mapsTo (hsA h)
| Mathlib/Combinatorics/Additive/FreimanHom.lean | 175 | 178 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
forms a measura... | apply hgf
/-- Monadic bind on `Measure`, only works in the category of measurable spaces and measurable
functions. When the function `f` is not measurable the result is not well defined. -/
| Mathlib/MeasureTheory/Measure/GiryMonad.lean | 205 | 208 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... | theorem truncate_comp {n₁ n₂ n₃ : ℕ} (h1 : n₁ ≤ n₂) (h2 : n₂ ≤ n₃) :
(truncate (p := p) (R := R) h1).comp (truncate h2) = truncate (h1.trans h2) := by
ext1 x; simp only [truncate_truncate, Function.comp_apply, RingHom.coe_comp]
| Mathlib/RingTheory/WittVector/Truncated.lean | 356 | 359 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | (ht : ∀ᵐ x ∂μ.restrict t, p x) (htc : ∀ᵐ x ∂μ.restrict tᶜ, p x) : ∀ᵐ x ∂μ, p x :=
nonpos_iff_eq_zero.1 <|
calc
| Mathlib/MeasureTheory/Measure/Restrict.lean | 602 | 604 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | /-- `OuterMeasure.pi m` is the finite product of the outer measures `{m i | i : ι}`.
It is defined to be the maximal outer measure `n` with the property that
`n (pi univ s) ≤ ∏ i, m i (s i)`, where `pi univ s` is the product of the sets
`{s i | i : ι}`. -/
protected def pi (m : ∀ i, OuterMeasure (α i)) : OuterMea... | Mathlib/MeasureTheory/Constructions/Pi.lean | 100 | 127 |
/-
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
/-... | variable [CommSemiring R] [StarRing R]
variable [NonUnitalSemiring A] [StarRing A] [Module R A]
variable [StarModule R A]
/-- The pointwise `star` of a non-unital subalgebra is a non-unital subalgebra. -/
instance instInvolutiveStar : InvolutiveStar (NonUnitalSubalgebra R A) where
star S :=
| Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 557 | 563 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | rw [twoTorsionPolynomial]
ext <;> dsimp
· linear_combination CharP.cast_eq_zero R 3
· linear_combination W.b₄ * CharP.cast_eq_zero R 3
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 325 | 328 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,061 | 1,062 | |
/-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Option.Basic
import Mathlib.SetTheory.Cardinal.Basic
impo... | decode_encode : ∀ x, decode (encode x) = some x
theorem Encoding.encode_injective {α : Type u} (e : Encoding α) : Function.Injective e.encode := by
| Mathlib/Computability/Encoding.lean | 43 | 45 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... |
/-- Composition by continuous linear equivs on the left respects higher differentiability on
domains. -/
theorem ContinuousLinearEquiv.comp_contDiffOn_iff (e : F ≃L[𝕜] G) :
ContDiffOn 𝕜 n (e ∘ f) s ↔ ContDiffOn 𝕜 n f s := by
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 306 | 310 |
/-
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... | @[fun_prop]
alias ⟨_, HasFDerivWithinAt.sub_const⟩ := hasFDerivWithinAt_sub_const_iff
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 605 | 607 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | theorem symmDiff_symmDiff_cancel_left : a ∆ (a ∆ b) = b := by simp [← symmDiff_assoc]
| Mathlib/Order/SymmDiff.lean | 413 | 414 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... | end Prod
section AlgebraInverse
variable {R : Type*} [NormedRing R] [HasSummableGeomSeries R] [NormedAlgebra 𝕜 R]
open NormedRing ContinuousLinearMap Ring
/-- At an invertible element `x` of a normed algebra `R`, the Fréchet derivative of the inversion
| Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 798 | 806 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | theorem sign_neg_coe_nonpos_of_nonneg_of_le_pi {θ : ℝ} (h0 : 0 ≤ θ) (hpi : θ ≤ π) :
(-θ : Angle).sign ≤ 0 := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 814 | 815 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.QuotientGroup.Defs
/-!
# Finitely gene... | Mathlib/GroupTheory/Finiteness.lean | 458 | 462 | |
/-
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.BoxIntegral.Box.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Tagged
/-!
# Induction on subboxes
In this file we prove (se... |
/-- Given a prepartition `π` of a box `I` and a function `r : ℝⁿ → (0, ∞)`, `π.toSubordinate r`
is a tagged partition `π'` such that
* each box of `π'` is included in some box of `π`;
* `π'` is a Henstock partition;
* `π'` is subordinate to `r`;
* `π'` covers exactly the same part of `I` as `π`;
* the distortion of `... | Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean | 154 | 163 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,463 | 1,478 | |
/-
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... | then the filter is less than or equal to `𝓝 y`. -/
lemma le_nhds_of_unique_clusterPt [CompactSpace X] {l : Filter X} {y : X}
| Mathlib/Topology/Compactness/Compact.lean | 700 | 701 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
open Finset Relation
set_option linter.unusedVariables false in -- `have` for wf induction triggers linter
lemma transGen_wcovBy_of_le [Preorder α] [LocallyFiniteOrder α] {x y : α} (hxy : x ≤ y) :
TransGen (· ⩿ ·) x y := by
-- We proceed by well-founded induction on the cardinality of `Icc x y`.
-- It's impos... | Mathlib/Order/Interval/Finset/Basic.lean | 1,078 | 1,101 |
/-
Copyright (c) 2023 Mantas Bakšys, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys, Yaël Dillies
-/
import Mathlib.Algebra.Order.Monovary
import Mathlib.Algebra.Order.Rearrangement
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.... | product. -/
theorem AntivaryOn.card_mul_sum_le_sum_mul_sum (hfg : AntivaryOn f g s) :
(#s : α) * ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i) * ∑ i ∈ s, g i := by
rw [← nsmul_eq_mul]
| Mathlib/Algebra/Order/Chebyshev.lean | 109 | 112 |
/-
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.Field.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
/-!
# Casts of rational numbers in... | cast_strictMono.monotone
| Mathlib/Data/Rat/Cast/Order.lean | 37 | 38 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... |
/-- If `s` is a convex set, then `a • closure s + b • interior s ⊆ interior s` for all `0 ≤ a`,
`0 < b`, `a + b = 1`. See also `Convex.combo_self_interior_subset_interior` for a weaker version. -/
theorem Convex.combo_closure_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
| Mathlib/Analysis/Convex/Topology.lean | 149 | 152 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | refine
p.prime_of_coprime hp fun n hn hnz => Finset.filter_card_eq h n <| Finset.mem_Ico.mpr ⟨?_, hn⟩
rwa [succ_le_iff, pos_iff_ne_zero]
| Mathlib/Data/Nat/Totient.lean | 217 | 220 |
/-
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... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 865 | 868 | |
/-
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.BigOperators.Finsupp.Fin
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.MvPolynomial.Rename... | (p : MvPolynomial (Option S₁) R) (i : ℕ) :
((optionEquivLeft R S₁ p).coeff i).totalDegree ≤ p.totalDegree := by
classical
by_cases hpi : (optionEquivLeft R S₁ p).coeff i = 0
· rw [hpi]; simp
rw [totalDegree, Finset.sup_le_iff]
intro σ hσ
refine le_trans ?_ (Finset.le_sup (b := σ.embDomain .some + .s... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 446 | 464 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,125 | 1,129 | |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | open Submodule
lemma span_selfAdjoint : span ℂ (selfAdjoint A : Set A) = ⊤ := by
refine eq_top_iff'.mpr fun x ↦ ?_
rw [← realPart_add_I_smul_imaginaryPart x]
exact add_mem (subset_span (ℜ x).property) <|
| Mathlib/Data/Complex/Module.lean | 455 | 460 |
/-
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... | lift δ to ℝ≥0 using hδpos.le
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ eLpNorm (f (ℐ C)) p μ := by
intro C
calc
C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ C • μ { x | C ≤ ‖f (ℐ C) x‖₊ } ^ (1 / p.toReal) := by
rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 835 | 851 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | (orthogonalProjection W s.circumcenter : P) =
↑((s.face hc).orthogonalProjectionSpan s.circumcenter) := by
apply eq_orthogonalProjection_of_eq_subspace
simp [W]
rw [EuclideanGeometry.reflection_apply, h_faces, s.orthogonalProjection_circumcenter hc,
| Mathlib/Geometry/Euclidean/Circumcenter.lean | 580 | 584 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | by simp_all [funext_iff, of_smul_eq_smul])
namespace ReducedWord
theorem exists_normalWord_prod_eq
(d : TransversalPair G A B) (w : ReducedWord G A B) :
∃ w' : NormalWord d, w'.prod φ = w.prod φ ∧
| Mathlib/GroupTheory/HNNExtension.lean | 593 | 599 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | @[simp]
theorem hnot_symmDiff_self : (￢a) ∆ a = ⊤ := by
| Mathlib/Order/SymmDiff.lean | 292 | 293 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,918 | 1,921 | |
/-
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.Data.Finset.Lattice.Fold
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
/-!
# Symmetric powers of a finset
This file defines the sym... | rwa [val_le_iff]
theorem monotone_sym2 : Monotone (Finset.sym2 : Finset α → _) := fun _ _ => sym2_mono
theorem injective_sym2 : Function.Injective (Finset.sym2 : Finset α → _) := by
| Mathlib/Data/Finset/Sym.lean | 85 | 89 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,776 | 1,777 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Free
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Data.Matrix.Notation
/-!
# Lie algebras from Cartan matrices
Split semi-simple Lie alge... | (Set.range <| HH R) ∪ (Set.range <| EF R) ∪ (Set.range <| HE R A) ∪ (Set.range <| HF R A) ∪
(Set.range <| adE R A) ∪
(Set.range <| adF R A)
| Mathlib/Algebra/Lie/CartanMatrix.lean | 154 | 156 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... |
theorem le_toMeasure_apply (m : OuterMeasure α) (h : ms ≤ m.caratheodory) (s : Set α) :
m s ≤ m.toMeasure h s :=
m.le_trim s
theorem toMeasure_apply₀ (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α}
(hs : NullMeasurableSet s (m.toMeasure h)) : m.toMeasure h s = m s := by
refine le_antisymm ?_ (l... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 685 | 694 |
/-
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.Order.Hom.CompleteLattice
import Mathlib.Topology.Compactness.Bases
import Mathlib.Topology.ContinuousMap.Basic
impor... | · exact hb'
lemma IsBasis.le_iff {α} {t₁ t₂ : TopologicalSpace α}
{Us : Set (Opens α)} (hUs : @IsBasis α t₂ Us) :
t₁ ≤ t₂ ↔ ∀ U ∈ Us, IsOpen[t₁] U := by
conv_lhs => rw [hUs.eq_generateFrom]
simp [Set.subset_def, le_generateFrom_iff_subset_isOpen]
@[simp]
theorem isCompactElement_iff (s : Opens α) :
... | Mathlib/Topology/Sets/Opens.lean | 314 | 327 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... |
variable [NoZeroDivisors R]
| Mathlib/Algebra/Ring/Commute.lean | 155 | 156 |
/-
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... | - (∑ i ∈ range k, p i (fun _ ↦ y) + ∑ i ∈ Ico k n, p i (fun _ ↦ y))‖ := by
simp [sum_range_add_sum_Ico _ hkn]
_ = ‖(p.partialSum k z - p.partialSum k y) + (∑ i ∈ Ico k n, p i (fun _ ↦ z))
+ (- ∑ i ∈ Ico k n, p i (fun _ ↦ y))‖ := by
congr 1
| Mathlib/Analysis/Analytic/Basic.lean | 930 | 934 |
/-
Copyright (c) 2019 Michael Howes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Howes, Newell Jensen
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup.Defs
/-!
# Defining a gr... | rename_i h1 h2
change QuotientGroup.mk _ ∈ H.carrier
rw [QuotientGroup.mk_mul]
exact Subgroup.mul_mem _ h1 h2
| Mathlib/GroupTheory/PresentedGroup.lean | 93 | 97 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | Cochain K L n →ₗ[R] Cochain (K⟦a⟧) (L⟦a⟧) n where
toAddHom := shiftAddHom K L n a
map_smul' _ _ := by dsimp; simp only [shift_smul]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 316 | 319 |
/-
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... | #{f ∈ piFinset s | f i = a} = ∏ j ∈ univ.erase i, #(s j) := by
calc
_ = ∏ j, #(Function.update s i {a} j) := by
rw [← piFinset_update_singleton_eq_filter_piFinset_eq _ _ ha, Fintype.card_piFinset]
_ = ∏ j, Function.update (fun j ↦ #(s j)) i 1 j :=
| Mathlib/Data/Fintype/BigOperators.lean | 168 | 172 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by
rintro ⟨⟩ ⟨⟩ _; congr
@[ext] theorem AddCommMonoidWithOne.ext ⦃inst₁ inst₂ : AddCommMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst... | Mathlib/Algebra/Ring/Ext.lean | 116 | 131 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Module.Projective
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# ... | exact LinearMap.map_le_range
theorem lift_rank_comp_le_right (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') :
Cardinal.lift.{v'} (rank (f.comp g)) ≤ Cardinal.lift.{v''} (rank g) := by
| Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 52 | 55 |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | Eq.symm <|
degree_map_eq_of_leadingCoeff_ne_zero _
(by rw [Monic.def.1 hq, f.map_one]; exact one_ne_zero)⟩
exact ⟨this.1.symm, this.2.symm⟩
theorem map_divByMonic [Ring S] (f : R →+* S) (hq : Monic q) :
(p /ₘ q).map f = p.map f /ₘ q.map f :=
(map_mod_divByMonic f hq).1... | Mathlib/Algebra/Polynomial/Div.lean | 371 | 385 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
/-!
# Wide equalizers and wide coequalizers
This file defines wide... | hom := k
w := by
rintro ⟨_ | _⟩
| Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean | 477 | 479 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.Invarian... | rw [Fintype.card_congr (Equiv.ofInjective b b.injective)]
exact linearIndependent_le_span v i (range b) b.span_eq
· -- and otherwise we have `linearIndependent_le_infinite_basis`.
exact linearIndependent_le_infinite_basis b v i
/-- `StrongRankCondition` implies that if there is an injective linear map `(... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 266 | 276 |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union, ← Set.toFinset_singleton]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_adj ha)
lemma disjoint_sdiff_neighborFinset_image :
Disjoint (G.edgeFinset \ G.incidenceFinset t) ((G.neighborFinset s)... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 104 | 113 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | Associates.count_some hv, Multiset.replicate_one, Multiset.count_eq_zero,
Multiset.mem_singleton]
simp only [Subtype.mk.injEq]
rw [Associates.mk_eq_mk_iff_associated, associated_iff_eq, ← HeightOneSpectrum.ext_iff]
exact Ne.symm hw
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 459 | 463 |
/-
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... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,723 | 1,726 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | Mathlib/Order/Basic.lean | 1,341 | 1,341 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | rcases h with ⟨h, b2, m0⟩
have b0 : 0 < b := by omega
have n0 : 0 < n := by linarith [mul_pos b0 m0]
refine ⟨?_, b2, n0⟩
obtain ⟨rfl, rfl⟩ := (Nat.div_mod_unique b0).2 ⟨e, hr⟩
subst h; exact Nat.digits_def' b2 n0
theorem digits_one (b n) (n0 : 0 < n) (nb : n < b) : Nat.digits b n = [n] ∧ 1 < b ∧ 0 < n := b... | Mathlib/Data/Nat/Digits.lean | 821 | 845 |
/-
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.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 785 | 787 | |
/-
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.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 564 | 564 | |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
/-!
# Type Spaces
This file defines the space of complete types over a first-order theory.
(Note that types in model theor... | theorem toList_foldr_inf_mem {p : T.CompleteType α} {t : Finset (L[[α]]).Sentence} :
t.toList.foldr (· ⊓ ·) ⊤ ∈ p ↔ (t : L[[α]].Theory) ⊆ ↑p := by
simp_rw [subset_def, ← SetLike.mem_coe, p.isMaximal.mem_iff_models, models_sentence_iff,
Sentence.Realize, Formula.Realize, BoundedFormula.realize_foldr_inf, Finse... | Mathlib/ModelTheory/Types.lean | 147 | 151 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | open scoped Classical in
@[simp]
theorem toMeasure_ofMultiset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
| Mathlib/Probability/Distributions/Uniform.lean | 375 | 377 |
/-
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 | 998 | 1,002 | |
/-
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.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by
rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)]
simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg]
theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} :
arg (-x) = arg x - π ↔ 0 < x.im... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 404 | 409 |
/-
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... | apply padicNorm.nonneg
exact lt_irrefl _ this
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 315 | 317 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | lemma Exact.isZero_X₂_iff (hS : S.Exact) : IsZero S.X₂ ↔ S.f = 0 ∧ S.g = 0 := by
constructor
· intro h
exact ⟨h.eq_of_tgt _ _, h.eq_of_src _ _⟩
· rintro ⟨hf, hg⟩
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 459 | 463 |
/-
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... | | _m + 2, _, _n + 2, _, hmn => fib_add_two_strictMono <| lt_of_add_lt_add_right hmn
lemma fib_lt_fib {m : ℕ} (hm : 2 ≤ m) : ∀ {n}, fib m < fib n ↔ m < n
| 0 => by simp [hm]
| Mathlib/Data/Nat/Fib/Basic.lean | 113 | 116 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 655 | 655 | |
/-
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 | 3,151 | 3,159 | |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 235 | 238 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | variable [Module R M]
section ULift
@[simp]
theorem rank_ulift : Module.rank R (ULift.{w} M) = Cardinal.lift.{w} (Module.rank R M) :=
Cardinal.lift_injective.{v} <| Eq.symm <| (lift_lift _).trans ULift.moduleEquiv.symm.lift_rank_eq
@[simp]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 114 | 122 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x := by
rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]
end RCLike
open Filter
section TopologicalSpace
variable [TopologicalSpace E]
theorem comap_gauge_nhds_zero_le (ha : Absorbent ℝ s) (hb... | Mathlib/Analysis/Convex/Gauge.lean | 303 | 316 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L`... | Mathlib/Algebra/Lie/IdealOperations.lean | 325 | 333 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 800 | 803 | |
/-
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.Asymptotics.Lemmas
import Mathlib.Analysis.Normed.Module.Basic
/-!
# Asymptotic equivalence up to a constant
In this file we define `Asymp... |
variable (l' : Filter β)
| Mathlib/Analysis/Asymptotics/Theta.lean | 316 | 318 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,499 | 1,500 | |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | simp only [preΨ₄, map_b₂, map_b₄, map_b₆, map_b₈]
map_simp
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 560 | 561 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Category.Cat.AsSmall
import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.IsConnected
i... | has_colimit := fun _ => hasColimit_of_comp F
include F in
theorem preservesColimitsOfShape_of_final {B : Type u₄} [Category.{v₄} B] (H : E ⥤ B)
[PreservesColimitsOfShape C H] : PreservesColimitsOfShape D H where
preservesColimit := preservesColimit_of_comp F
include F in
theorem reflectsColimitsOfShape_of_fin... | Mathlib/CategoryTheory/Limits/Final.lean | 411 | 427 |
/-
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.LinearAlgebra.Multilinear.Curry
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-! # Multilinear maps over finite dimensional spaces
The main result... | induction n with
| zero =>
haveI : IsEmpty (Fin Nat.zero) := inferInstanceAs (IsEmpty (Fin 0))
exact
⟨Module.Free.of_equiv (constLinearEquivOfIsEmpty R R N M₂),
Module.Finite.equiv (constLinearEquivOfIsEmpty R R N M₂)⟩
| succ n ih =>
suffices
Module.Free R (N 0 →ₗ[R] MultilinearMap... | Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean | 32 | 48 |
/-
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` ... | rw [← nhdsWithin_eq_iff_eventuallyEq] at hst
simpa [mem_interior_iff_mem_nhds, ← nhdsWithin_eq_nhds, hst] using h
| Mathlib/Topology/ContinuousOn.lean | 295 | 296 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-!
# Tri... | suffices ∃ μ, f.HasUnifEigenvalue μ k by
peel this with μ hμ
exact HasUnifEigenvalue.lt zero_lt_one hμ
simp [HasUnifEigenvalue, ← not_forall, ← iSup_eq_bot, hf]
| Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 51 | 54 |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.ContinuousOn
/-!
### Locally finite families of sets
We say that a family of sets in a topological s... | We formulate the conclusion in terms of the product of filter `Filter.atTop` and `𝓝 x`. -/
theorem exists_forall_eventually_eq_prod {π : X → Sort*} {f : ℕ → ∀ x : X, π x}
(hf : LocallyFinite fun n => { x | f (n + 1) x ≠ f n x }) :
| Mathlib/Topology/LocallyFinite.lean | 134 | 136 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 499 | 500 |
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