Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | Set.range (fun n : ℕ ↦ m * n + k) = {n : ℕ | (n : ZMod m) = k ∧ k ≤ n} := by
ext n
simp only [Set.mem_range, Set.mem_setOf_eq]
conv => enter [1, 1, y]; rw [add_comm, eq_comm]
| Mathlib/Data/ZMod/Basic.lean | 1,263 | 1,266 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | /-- Induction principle to build a sequence, by adding one point at a time satisfying a given
relation with respect to all the previously chosen points.
| Mathlib/Data/Fintype/Basic.lean | 241 | 243 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 604 | 607 | |
/-
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... | mul_assoc a b c :=
Quotient.inductionOn₃ a b c fun ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ =>
Eq.symm <|
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 593 | 595 |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... |
theorem right_mem_of_mk_mem_sym2 {xs : List α} {a b : α}
(h : s(a, b) ∈ xs.sym2) : b ∈ xs := by
rw [Sym2.eq_swap] at h
exact left_mem_of_mk_mem_sym2 h
theorem mk_mem_sym2 {xs : List α} {a b : α} (ha : a ∈ xs) (hb : b ∈ xs) :
s(a, b) ∈ xs.sym2 := by
induction xs with
| nil => simp at ha
| cons x xs i... | Mathlib/Data/List/Sym.lean | 68 | 79 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... |
@[simp]
theorem eqToIso.inv {X Y : C} (p : X = Y) : (eqToIso p).inv = eqToHom p.symm :=
rfl
@[simp]
theorem eqToIso_refl {X : C} (p : X = X) : eqToIso p = Iso.refl X :=
rfl
@[simp]
theorem eqToIso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToIso p ≪≫ eqToIso q = eqToIso (p.trans q) := by ext; simp
@[simp... | Mathlib/CategoryTheory/EqToHom.lean | 200 | 213 |
/-
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.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | | ε, ε0 =>
let ⟨i, hi⟩ := exists_forall_ge_and (h _ (half_pos <| half_pos ε0)) (g.cauchy₃ <| half_pos ε0)
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 450 | 451 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | theorem eq_zero_of_degree_lt_of_eval_finset_eq_zero (degree_f_lt : f.degree < #s)
(eval_f : ∀ x ∈ s, f.eval x = 0) : f = 0 := by
rw [← mem_degreeLT] at degree_f_lt
simp_rw [eval_eq_sum_degreeLTEquiv degree_f_lt] at eval_f
rw [← degreeLTEquiv_eq_zero_iff_eq_zero degree_f_lt]
exact
Matrix.eq_zero_of_foral... | Mathlib/LinearAlgebra/Lagrange.lean | 44 | 52 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | /-! ### `kerase` -/
/-- Remove the first pair with the key `a`. -/
def kerase (a : α) : List (Sigma β) → List (Sigma β) :=
| Mathlib/Data/List/Sigma.lean | 368 | 372 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | corootSpace (0 : H → K) = ⊥ := by
refine eq_bot_iff.mpr fun x hx ↦ ?_
suffices {x | ∃ y ∈ H, ∃ z ∈ H, ⁅y, z⁆ = x} = {0} by simpa [mem_corootSpace, this] using hx
refine eq_singleton_iff_unique_mem.mpr ⟨⟨0, H.zero_mem, 0, H.zero_mem, zero_lie 0⟩, ?_⟩
rintro - ⟨y, hy, z, hz, rfl⟩
suffices ⁅(⟨y, hy⟩ : H), (⟨... | Mathlib/Algebra/Lie/Weights/Killing.lean | 161 | 176 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Images of pairs of submodules under bilinear maps
This file provides `Subm... |
theorem map₂_span_singleton_eq_map_flip (f : M →ₗ[R] N →ₗ[R] P) (s : Submodule R M) (n : N) :
map₂ f s (span R {n}) = map (f.flip n) s := by rw [← map₂_span_singleton_eq_map, map₂_flip]
end Submodule
| Mathlib/Algebra/Module/Submodule/Bilinear.lean | 146 | 150 |
/-
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.Group.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... |
theorem signAux_eq_signAux2 {n : ℕ} :
∀ (l : List α) (f : Perm α) (e : α ≃ Fin n) (_h : ∀ x, f x ≠ x → x ∈ l),
signAux ((e.symm.trans f).trans e) = signAux2 l f
| [], f, e, h => by
have : f = 1 := Equiv.ext fun y => Classical.not_not.1 (mt (h y) List.not_mem_nil)
rw [this, one_def, Equiv.trans_refl... | Mathlib/GroupTheory/Perm/Sign.lean | 299 | 309 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathli... | rw [← sub_add_cancel M.charpoly (∏ i : n, (X - C (M i i)))]
-- Porting note: added `↑` in front of `Fintype.card n`
have h1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n) := by
rw [degree_eq_iff_natDegree_eq_of_pos (Nat.pos_of_ne_zero h), natDegree_prod']
· simp_rw [natDegree_X_sub_C]
rw [←... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 96 | 119 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... |
theorem mk_preimage_prod (f : γ → α) (g : γ → β) :
(fun x => (f x, g x)) ⁻¹' s ×ˢ t = f ⁻¹' s ∩ g ⁻¹' t :=
| Mathlib/Data/Set/Prod.lean | 180 | 182 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.NAry
import Mathlib.Order.SupClosed
import Mathlib.Order.UpperLower.Closure
/-!
# Set family operations
This file defines a few binary operation... | Mathlib/Data/Set/Sups.lean | 429 | 438 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... |
@[continuity]
| Mathlib/Analysis/Complex/Basic.lean | 228 | 229 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... |
theorem totallyBounded_iff_subset {s : Set α} :
TotallyBounded s ↔
| Mathlib/Topology/UniformSpace/Cauchy.lean | 465 | 467 |
/-
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... | _ ≤ gauge s y + gauge s (x - y) := gauge_add_le hc (this.mono hs) _ _
_ ≤ gauge s y + ‖x - y‖ / r :=
add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr.le _)) _
_ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul, NNReal.coe_inv]
theorem Convex.lipschitz_g... | Mathlib/Analysis/Convex/Gauge.lean | 576 | 582 |
/-
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
-/
import Mathlib.CategoryTheory.Equivalence
/-!
# Opposite categories
We provide a category instance on `Cᵒᵖ`.
The morphisms `X ⟶ Y` are defined to be the... |
/-- An equivalence between opposite categories gives an equivalence between the original categories.
-/
| Mathlib/CategoryTheory/Opposites.lean | 488 | 490 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,149 | 2,151 | |
/-
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.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | Mathlib/Topology/MetricSpace/Gluing.lean | 636 | 641 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
| Mathlib/Order/Lattice.lean | 669 | 669 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | refine isComplement_iff_existsUnique.mpr fun g => ?_
obtain ⟨h, hh⟩ := h2 g
have hh' : (↑h * g) • a = a := by rwa [mul_smul]
refine ⟨⟨h⁻¹, h * g, hh'⟩, inv_mul_cancel_left ↑h g, ?_⟩
rintro ⟨h', g, hg : g • a = a⟩ rfl
specialize h1 (h * h') (by rwa [mul_smul, smul_def h', ← hg, ← mul_smul, hg])
refine Prod... | Mathlib/GroupTheory/Complement.lean | 828 | 839 |
/-
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.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.Algebra.Module.LinearMap
/-!
# Continuous affine maps.
This file defines a type of bundled cont... |
variable {R P} {W₂ Q₂ : Type*}
variable [AddCommGroup W₂] [Module R W₂] [TopologicalSpace Q₂] [AddTorsor W₂ Q₂]
| Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 127 | 129 |
/-
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.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | AlternatingMap.ext fun _ => Basis.det_reindex _ _ _
theorem Basis.det_reindex_symm {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (b : Basis ι R M)
(v : ι → M) (e : ι' ≃ ι) : (b.reindex e.symm).det (v ∘ e) = b.det v := by
rw [Basis.det_reindex, Function.comp_assoc, e.self_comp_symm, Function.comp_id]
| Mathlib/LinearAlgebra/Determinant.lean | 601 | 606 |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | refine Eq.trans (congr rfl (Set.ext fun x => ?_)) (sSup_compact_le_eq ⊤)
exact (and_iff_left le_top).symm
theorem le_iff_compact_le_imp {a b : α} :
a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b :=
⟨fun ab _ _ ca => le_trans ca ab, fun h => by
rw [← sSup_compact_le_eq a, ← sSup_compa... | Mathlib/Order/CompactlyGenerated/Basic.lean | 367 | 380 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Nonunits
import Mathlib.RingTheory.Ideal.Quotient.De... | @Subsingleton.elim _ (@CharOne.subsingleton _ _ (ringChar.of_eq h1)) _ _
/-- If an ideal does not contain any coercions of natural numbers other than zero, then its quotient
inherits the characteristic of the underlying ring. -/
theorem quotient' {R : Type*} [CommRing R] (p : ℕ) [CharP R p] (I : Ideal R)... | Mathlib/Algebra/CharP/Quotient.lean | 37 | 43 |
/-
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.Subsemigroup.Operations
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Order.Filter.Extr
/-!
# Lemmas about the `sup` and `in... | exact (Finset.sup_mono Finsupp.support_finset_sum).trans_eq (Finset.sup_biUnion _ _)
theorem supDegree_single_ne_zero (a : A) {r : R} (hr : r ≠ 0) :
| Mathlib/Algebra/MonoidAlgebra/Degree.lean | 244 | 246 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Lie algebras with non-degenerate invariant bilinear forms are s... | variable [LieAlgebra R L]
lemma orthogonal_disjoint
(Φ : LinearMap.BilinForm R L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : Φ.lieInvariant L)
-- TODO: replace the following assumption by a typeclass assumption `[HasNonAbelianAtoms]`
(hL : ∀ I : LieIdeal R L, IsAtom I → ¬IsLieAbelian I)
(I : LieIdeal R L)... | Mathlib/Algebra/Lie/InvariantForm.lean | 90 | 108 |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Functor.EpiMono
import Mathlib.CategoryTheory.HomCongr
/-!
# Reflective functors
Ba... | as long as `B` is in the essential image of `i`.
This definition gives an equivalence: the key property that the inverse can be described
nicely is shown in `unitCompPartialBijective_symm_apply`.
| Mathlib/CategoryTheory/Adjunction/Reflective.lean | 121 | 124 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | apply measure_ball_pos
exact div_pos (mul_pos Dpos (by linarith only [hx])) B
variable (E)
theorem y_smooth : ContDiffOn ℝ ∞ (uncurry y) (Ioo (0 : ℝ) 1 ×ˢ (univ : Set E)) := by
have hs : IsOpen (Ioo (0 : ℝ) (1 : ℝ)) := isOpen_Ioo
have hk : IsCompact (closedBall (0 : E) 1) := ProperSpace.isCompact_closedBa... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 421 | 461 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Contractions
Given modules $M, N$ over a commutative ring $R$, t... |
@[simp]
theorem homTensorHomEquiv_toLinearMap :
(homTensorHomEquiv R M N P Q).toLinearMap = homTensorHomMap R M N P Q := by
ext m n
simp only [homTensorHomEquiv, compr₂_apply, mk_apply, LinearEquiv.coe_toLinearMap,
LinearEquiv.trans_apply, lift.equiv_apply, LinearEquiv.arrowCongr_apply, LinearEquiv.refl_sy... | Mathlib/LinearAlgebra/Contraction.lean | 260 | 271 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
-- Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
import Mathlib.Algebra.Group.Defs
import Mathlib.Order.Defs.Unbundled
/-... | Mathlib/Algebra/Group/Semiconj/Defs.lean | 159 | 160 | |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Data.Set.MulAntidiagonal
import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-! # Multiplicat... | rintro _ ⟨x, hx, y, hy, rfl⟩
exact mul_le_mul' (hs.min_le _ hx) (ht.min_le _ hy)
end Set
namespace Finset
| Mathlib/Data/Finset/MulAntidiagonal.lean | 40 | 45 |
/-
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.Algebra.Homology.HomologicalComplex
/-!
# Homological complexes supported in a single degree
We define `single V j c : V ⥤ HomologicalComplex V c`,
which... | to an `ℕ`-indexed cochain complex `C` are the same as morphisms `f : C.X 0 ⟶ X`.
-/
@[simps apply]
noncomputable def toSingle₀Equiv (C : CochainComplex V ℕ) (X : V) :
(C ⟶ (single₀ V).obj X) ≃ (C.X 0 ⟶ X) where
| Mathlib/Algebra/Homology/Single.lean | 298 | 302 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 537 | 552 | |
/-
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.Algebra.Order.Field.Power
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousMap... |
theorem bernstein_nonneg {n ν : ℕ} {x : I} : 0 ≤ bernstein n ν x := by
simp only [bernstein_apply]
have h₁ : (0 : ℝ) ≤ x := by unit_interval
| Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 61 | 64 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n := by
rw [firstDiff_def] at hn
split_ifs at hn with h
· convert Nat.find_min (ne_iff.1 h) hn
| Mathlib/Topology/MetricSpace/PiNat.lean | 74 | 77 |
/-
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, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | @[deprecated (since := "2024-11-30")]
alias multiplicity_eq_one_of_not_finite :=
multiplicity_eq_one_of_not_finiteMultiplicity
@[simp]
theorem multiplicity_le_emultiplicity :
| Mathlib/RingTheory/Multiplicity.lean | 129 | 134 |
/-
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.Real
/-!
# Power function... | simp [H, top_rpow_of_neg, top_rpow_of_pos, le_of_lt]
| coe x =>
by_cases h : x = 0
· rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt]
· simp [← coe_rpow_of_ne_zero h, h]
theorem rpow_eq_top_iff_of_pos {x : ℝ≥0∞} {y : ℝ} (hy : 0 < y) : x... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 543 | 551 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... |
theorem rdropWhile_singleton (x : α) : rdropWhile p [x] = if p x then [] else [x] := by
rw [← nil_append [x], rdropWhile_concat, rdropWhile_nil]
| Mathlib/Data/List/DropRight.lean | 105 | 108 |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
/-!
# List of booleans
In this file we prove lemmas about the number of `false`s and `true`s ... | hl.count_not_le_count_add_one false
theorem two_mul_count_bool_of_even (hl : Chain' (· ≠ ·) l) (h2 : Even (length l)) (b : Bool) :
2 * count b l = length l := by
rw [← count_not_add_count l b, hl.count_not_eq_count h2, two_mul]
theorem two_mul_count_bool_eq_ite (hl : Chain' (· ≠ ·) l) (b : Bool) :
2 * cou... | Mathlib/Data/Bool/Count.lean | 79 | 87 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.EulerProduct.ExpLog
import Mathlib.NumberTheory.LSeries.Dirichlet
/-!
# The Euler Product for the Riemann Zeta Function and Dirichlet L-Ser... | This version is stated in terms of `tprod`. -/
theorem riemannZeta_eulerProduct_tprod (hs : 1 < s.re) :
∏' p : Primes, (1 - (p : ℂ) ^ (-s))⁻¹ = riemannZeta s :=
(riemannZeta_eulerProduct_hasProd hs).tprod_eq
| Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 91 | 94 |
/-
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, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
/-!
# Density of si... | rcases h with ⟨l, hlN, hxl⟩
rcases hk.eq_or_lt with (rfl | hk)
exacts [(ihN hlN).trans hxl, ihN (Nat.lt_succ_iff.1 hk)]
theorem tendsto_nearestPt {e : ℕ → α} {x : α} (hx : x ∈ closure (range e)) :
Tendsto (fun N => nearestPt e N x) atTop (𝓝 x) := by
refine (atTop_basis.tendsto_iff nhds_basis_e... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 102 | 113 |
/-
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.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | norm_eq_mul_conj, eq_comm] at h⟩)
fun hx =>
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 481 | 482 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... |
lemma set_smul_mono_left {s t : Set S} (le : s ≤ t) :
s • N ≤ t • N :=
set_smul_le _ _ _ fun _ _ hr hm => mem_set_smul_of_mem_mem (mem1 := le hr)
(mem2 := hm)
| Mathlib/Algebra/Module/Submodule/Pointwise.lean | 355 | 359 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... | C_simp
ring1
lemma XYIdeal_neg_mul {x y : F} (h : W.Nonsingular x y) :
XYIdeal W x (C <| W.negY x y) * XYIdeal W x (C y) = XIdeal W x := by
have Y_rw : (Y - C (C y)) * (Y - C (C <| W.negY x y)) -
C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) =
W.pol... | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 297 | 305 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
import Mathlib.Algebra.Order.Floor.Ring
/-!
# ... | | succ n ih => exact (of (a : K)).s.prop ih
Stream'.Seq.ext fun n => (h n).trans (Stream'.Seq.get?_nil n).symm
variable {K} (v)
| Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 275 | 278 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... | @[scoped simp]
theorem sub_eq_add (x y : R) : x - y = x + y := by rw [sub_eq_add_neg, neg_eq]
| Mathlib/Algebra/CharP/Two.lean | 65 | 66 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 1,050 | 1,051 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | {c : ε} {hf : AEStronglyMeasurable f μ} {s : Set α}
lemma eLpNorm_indicator_eq_eLpNorm_restrict {f : α → ε} {s : Set α} (hs : MeasurableSet s) :
eLpNorm (s.indicator f) p μ = eLpNorm f p (μ.restrict s) := by
by_cases hp_zero : p = 0
· simp only [hp_zero, eLpNorm_exponent_zero]
by_cases hp_top : p = ∞
· s... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 634 | 648 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | contrapose! h
rw [← Sentence.realize_not] at *
exact (L.completeTheory M).realize_sentence_of_mem (mem_completeTheory.2 h)
variable {M N}
namespace BoundedFormula
@[simp]
theorem realize_alls {φ : L.BoundedFormula α n} {v : α → M} :
| Mathlib/ModelTheory/Semantics.lean | 732 | 741 |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | rw [mul_comm b, Int.mul_ediv_assoc c (dvd_refl b), Int.cast_mul,
intCast_div_self, Int.cast_mul, mul_div_assoc]
theorem natCast_div (a b : ℕ) (h : b ∣ a) : ((a / b : ℕ) : ℚ) = a / b :=
intCast_div a b (Int.ofNat_dvd.mpr h)
| Mathlib/Data/Rat/Lemmas.lean | 216 | 220 |
/-
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... | Mathlib/Topology/Compactness/Compact.lean | 1,106 | 1,112 | |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... |
/-- A left transversal can be viewed as a function mapping each element of the group
to the chosen representative from that left coset. -/
| Mathlib/GroupTheory/Complement.lean | 616 | 618 |
/-
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... | normUnit p :=
⟨C ↑(normUnit p.leadingCoeff), C ↑(normUnit p.leadingCoeff)⁻¹, by
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 197 | 198 |
/-
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.NNReal
/-!
# Limits and a... | /-- The function `x ^ s * exp (-b * x)` tends to `0` at `+∞`, for any real `s` and `b > 0`. -/
theorem tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (s : ℝ) (b : ℝ) (hb : 0 < b) :
Tendsto (fun x : ℝ => x ^ s * exp (-b * x)) atTop (𝓝 0) := by
refine (tendsto_exp_mul_div_rpow_atTop s b hb).inv_tendsto_atTop.congr' ... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 132 | 137 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | theorem thickening_subset_cthickening_of_le {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
thickening δ₁ E ⊆ cthickening δ₂ E :=
(thickening_subset_cthickening δ₁ E).trans (cthickening_mono hle E)
theorem _root_.Bornology.IsBounded.cthickening {α : Type*} [PseudoMetricSpace α] {δ : ℝ} {E : Set α}
| Mathlib/Topology/MetricSpace/Thickening.lean | 270 | 274 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | fun ⟨v₀, hv₀, H, h⟩ ↦ ⟨⟨v₀, hv₀⟩,
fun h' ↦ H (by rwa [AddSubmonoid.mk_eq_zero] at h'), fun ⟨v, hv⟩ ↦ ?_⟩⟩
· obtain ⟨r, hr⟩ := h ⟨v, hv⟩
exact ⟨r, by rwa [Subtype.ext_iff, coe_smul] at hr⟩
· obtain ⟨r, hr⟩ := h v hv
exact ⟨r, by rwa [Subtype.ext_iff, coe_smul]⟩
/-- A submodule has dimension at mos... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 145 | 153 |
/-
Copyright (c) 2020 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.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | @[nontriviality]
theorem contMDiffOn_of_subsingleton [Subsingleton M'] : ContMDiffOn I I' n f s :=
| Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 252 | 253 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.GaussianInt
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Facts about the gaussian... | Prime (p : ℤ[i]) ↔ p % 4 = 3 :=
⟨mod_four_eq_three_of_nat_prime_of_prime p, prime_of_nat_prime_of_mod_four_eq_three p⟩
end GaussianInt
| Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | 86 | 93 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | ext
· rw [← cancel_epi f]
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 816 | 817 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | -- `areaForm_map`, `rightAngleRotation_map` and `kahler_map` in the three theorems below,
-- so it has to be provided by unification (i.e. by naming the instance-implicit argument where
-- it belongs and using `(hF := _)`).
/-- The area form on an oriented real inner product space of dimension 2 can be evaluated in te... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 550 | 554 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... |
variable [IsDomain R₁]
theorem one_div_spanSingleton (x : K) : 1 / spanSingleton R₁⁰ x = spanSingleton R₁⁰ x⁻¹ := by
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 715 | 718 |
/-
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, Yury Kudryashov
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.Order.Conditionall... | · filter_upwards [hf, hI.compl_mem_cofinite] with i hsurj (hiI : i ∉ I)
simp [hiI, hsurj.range_eq]
| Mathlib/Order/Filter/Cofinite.lean | 173 | 175 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | simp [principal_iff_of_monotone h₁ h₂]
theorem principal_zero : Principal op 0 := fun a _ h =>
(Ordinal.not_lt_zero a h).elim
@[simp]
| Mathlib/SetTheory/Ordinal/Principal.lean | 69 | 74 |
/-
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.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | iInf_le_of_le j <| iInf_le_of_le hj <| iInf_le_of_le i <| iInf_le _ hi)
(le_iInf fun i => le_iInf fun hi => le_iInf fun j => le_iInf fun hj =>
| Mathlib/Order/Filter/Lift.lean | 117 | 118 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... | fac := by
rintro E (a | b)
· apply fac
· rw [← E.w (WalkingMulticospan.Hom.fst b), ← K.w (WalkingMulticospan.Hom.fst b), ←
Category.assoc]
congr 1
apply fac
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 393 | 399 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
/-!
# Association Lists
This file defines association lists. An association list is a list where every element consists o... | (insert a b s).entries = Sigma.mk a b :: kerase a s.entries :=
rfl
| Mathlib/Data/List/AList.lean | 241 | 242 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
/-!
# Connections between `TensorProduct` a... | toMatrix bM bM' f ⊗ₖ toMatrix bN bN' g := by
ext ⟨i, j⟩ ⟨i', j'⟩
simp_rw [Matrix.kroneckerMap_apply, toMatrix_apply, Basis.tensorProduct_apply,
TensorProduct.map_tmul, Basis.tensorProduct_repr_tmul_apply]
exact mul_comm _ _
| Mathlib/LinearAlgebra/TensorProduct/Matrix.lean | 39 | 44 |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | theorem map_iInf_comap {ι β} [Nonempty ι] {f : α → β} (m : ι → OuterMeasure β) :
map f (⨅ i, comap f (m i)) = ⨅ i, map f (comap f (m i)) := by
refine (map_iInf_le _ _).antisymm fun s => ?_
simp only [map_apply, comap_apply, iInf_apply, le_iInf_iff]
refine fun t ht => iInf_le_of_le (fun n => f '' t n ∪ (range ... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 421 | 428 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kyle Miller
-/
import Mathlib.Tactic.NormNum
/-! # `norm_num` extension for `Nat.sqrt`
This module defines a `norm_num` extension for `Nat.sqrt`.
-/
namespace Tactic... | Nat.sqrt x = y := by
rw [← hr, ← pow_two]
rw [two_mul] at hle
exact Nat.sqrt_add_eq' _ (Nat.le_of_ble_eq_true hle)
| Mathlib/Tactic/NormNum/NatSqrt.lean | 20 | 24 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 752 | 764 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Algebra.BigOperators.G... | rfl
theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) :
trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) := by
rcases s.eq_empty_or_nonempty with (rfl | h)
| Mathlib/Algebra/Tropical/BigOperators.lean | 85 | 89 |
/-
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.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.Pointwise
import Mat... |
@[to_additive lowerClosure_interior_subset] lemma lowerClosure_interior_subset' (s : Set α) :
(lowerClosure (interior s) : Set α) ⊆ interior (lowerClosure s) :=
lowerClosure_min (interior_mono subset_lowerClosure) (lowerClosure s).lower.interior
| Mathlib/Analysis/Normed/Order/UpperLower.lean | 66 | 69 |
/-
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.CategoryTheory.Limits.Shapes.Countable
import Mathlib.Topology.Category.Profinite.AsLimit
import Mathlib.Topology.Category.Profinite.CofilteredLimi... | Category.comp_id f])
functor_unitIso_comp _ := by simpa using lightDiagramToProfinite.preimage_id
instance : lightProfiniteToLightDiagram.IsEquivalence :=
show LightProfinite.equivDiagram.functor.IsEquivalence from inferInstance
instance : lightDiagramToLightProfinite.IsEquivalence :=
show LightProfinit... | Mathlib/Topology/Category/LightProfinite/Basic.lean | 322 | 335 |
/-
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.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | /-- `trim` sends the supremum of two outer measures to the supremum of the trimmed measures. -/
theorem trim_sup (m₁ m₂ : OuterMeasure α) : (m₁ ⊔ m₂).trim = m₁.trim ⊔ m₂.trim :=
ext fun s => (trim_binop (sup_apply m₁ m₂) s).trans (sup_apply _ _ _).symm
/-- `trim` sends the supremum of a countable family of outer mea... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 434 | 440 |
/-
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 Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | rfl
theorem mk_num_ne_zero_of_ne_zero {q : ℚ} {n d : ℤ} (hq : q ≠ 0) (hqnd : q = n /. d) : n ≠ 0 :=
fun this => hq <| by simpa [this] using hqnd
| Mathlib/Data/Rat/Defs.lean | 373 | 376 |
/-
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... | ∀ (p : R[X]) {q : R[X]} (_hq : Monic q), p %ₘ q = p - q * (p /ₘ q)
| p, q, hq =>
| Mathlib/Algebra/Polynomial/Div.lean | 237 | 238 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Sym.Sym2
import Mathlib.Logic.Relation
/-!
# Game addition relation
This file defines, given relations `rα : α → α → Prop` and `rβ : β → β → Prop`, a rela... | (Prod.GameAdd.fst h).to_sym2
theorem GameAdd.snd {a b₁ b₂ : α} (h : rα b₁ b₂) : GameAdd rα s(a, b₁) s(a, b₂) :=
(Prod.GameAdd.snd h).to_sym2
| Mathlib/Order/GameAdd.lean | 161 | 164 |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | (superpolynomialDecay_iff_norm_tendsto_zero l k f).trans (by simp [SuperpolynomialDecay])
variable {l k}
| Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 287 | 289 |
/-
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, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | Mathlib/Order/Monotone/Basic.lean | 1,175 | 1,178 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | lemma tilted_eq_withDensity_nnreal (μ : Measure α) (f : α → ℝ) :
μ.tilted f = μ.withDensity (fun x ↦ ((↑) : ℝ≥0 → ℝ≥0∞)
(⟨exp (f x) / ∫ x, exp (f x) ∂μ, by positivity⟩ : ℝ≥0)) := by
rw [Measure.tilted]
congr with x
rw [ENNReal.ofReal_eq_coe_nnreal]
| Mathlib/MeasureTheory/Measure/Tilted.lean | 90 | 95 |
/-
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.Data.Finite.Sum
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutations on... | · simp only [Set.mem_image, toEmbedding_apply, exists_eq_right, support_conj, coe_map,
apply_eq_iff_eq]
· intro x hx
simp only [trans_apply, symm_trans_apply, Equiv.setCongr_apply, Equiv.setCongr_symm_apply,
Equiv.sumCongr_apply]
rw [hd1', Set.mem_union] at hx
rcases hx wit... | Mathlib/GroupTheory/Perm/Finite.lean | 187 | 231 |
/-
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.LinearMap
import Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions
import Mathlib.MeasureTheory.Function.StronglyMeasur... | theorem eLpNorm_rpow_two_norm_lt_top (f : Lp F 2 μ) :
eLpNorm (fun x => ‖f x‖ ^ (2 : ℝ)) 1 μ < ∞ := by
have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp [zero_le_one]
rw [eLpNorm_norm_rpow f zero_lt_two, one_mul, h_two]
| Mathlib/MeasureTheory/Function/L2Space.lean | 118 | 121 |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Algebra.Order.Monoid.Defs... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 133 | 134 | |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... | lemma domain_eq (f : R ⟶ S) (φ : a ⟶ b) [p.IsHomLift f φ] : p.obj a = R := by
subst_hom_lift p f φ; rfl
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 76 | 77 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sets.OpenCover
/-!
# Sober spaces
A quasi-sober space is a topological space where every irreducible closed s... | /-- A generic point of the closure of an irreducible space. -/
noncomputable def IsIrreducible.genericPoint [QuasiSober α] {S : Set α} (hS : IsIrreducible S) :
α :=
(QuasiSober.sober hS.closure isClosed_closure).choose
| Mathlib/Topology/Sober.lean | 107 | 111 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 871 | 873 | |
/-
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... | fderiv_neg
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 454 | 455 |
/-
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 | 1,384 | 1,385 | |
/-
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.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual.Defs
/-!
# Characters of Lie algebras
A character of a Lie algebr... | theorem lieCharacter_apply_lie' (χ : LieCharacter R L) (x y : L) : ⁅χ x, χ y⁆ = 0 := by
rw [LieRing.of_associative_ring_bracket, mul_comm, sub_self]
| Mathlib/Algebra/Lie/Character.lean | 44 | 45 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | rw [forall_update_iff (p := fun x s => y x ∈ s)]
simp [eq_comm]
theorem update_preimage_univ_pi [DecidableEq ι] {f : ∀ i, α i} (hf : ∀ j ≠ i, f j ∈ t j) :
update f i ⁻¹' pi univ t = t i :=
update_preimage_pi (mem_univ i) fun j _ => hf j
theorem subset_pi_eval_image (s : Set ι) (u : Set (∀ i, α i)) : u ⊆ pi ... | Mathlib/Data/Set/Prod.lean | 879 | 886 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
/-!
# Homomorphisms of semirings and... | theorem codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 := by rw [map_one, eq_comm]
| Mathlib/Algebra/Ring/Hom/Defs.lean | 467 | 468 |
/-
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 | 537 | 540 | |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... |
end Point
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 942 | 944 |
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e' (f y)).2) s x := by
rw [Trivialization.mem_source] at he he'
refine (hp.coordChange hf he he').congr_of_eventuallyEq ?_ ?_
· filter_upwards [hp.continuousWithinAt (e.open_baseSet.mem_nhds he)] with y hy
rw [Function.comp_apply, e.coordChange_apply_snd _ hy]
... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 429 | 442 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
/-!
# Recursive computation rules for the Clifford algebra
This file provides API for a special case `CliffordAlg... | theorem foldr'_ι (f : M →ₗ[R] CliffordAlgebra Q × N →ₗ[R] N)
(hf : ∀ m x fx, f m (ι Q m * x, f m (x, fx)) = Q m • fx) (n m) :
foldr' Q f hf n (ι Q m) = f m (1, n) :=
congr_arg Prod.snd (foldr_ι _ _ _ _ _)
theorem foldr'_ι_mul (f : M →ₗ[R] CliffordAlgebra Q × N →ₗ[R] N)
| Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean | 195 | 200 |
/-
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.Real
/-!
# Power function... |
theorem rpow_le_self_of_le_one {x : ℝ≥0} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x := by
rcases eq_bot_or_bot_lt x with (rfl | (h : 0 < x))
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 362 | 364 |
/-
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... | @Filter.nonempty_of_mem α f hf s hs
| Mathlib/Order/Filter/Basic.lean | 348 | 348 |
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