Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 835 | 843 | |
/-
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... |
@[simp]
theorem Ico_filter_lt (a b c : α) : {x ∈ Ico a b | x < c} = Ico a (min b c) := by
cases le_total b c with
| inl h => rw [Ico_filter_lt_of_right_le h, min_eq_left h]
| inr h => rw [Ico_filter_lt_of_le_right h, min_eq_right h]
| Mathlib/Order/Interval/Finset/Basic.lean | 849 | 855 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Fintype.Sigma
/-!
# Darts in graphs
A `Dart` or half-edge or bond in a graph is an ordered pair of adja... | theorem dart_edge_eq_mk'_iff' :
∀ {d : G.Dart} {u v : V},
d.edge = s(u, v) ↔ d.fst = u ∧ d.snd = v ∨ d.fst = v ∧ d.snd = u := by
| Mathlib/Combinatorics/SimpleGraph/Dart.lean | 107 | 109 |
/-
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.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.RingTheory.AlgebraTower
import Mathli... | /-- Two finite and free modules are isomorphic if they have the same (finite) rank. -/
noncomputable def LinearEquiv.ofFinrankEq [Module.Finite R M] [Module.Finite R M']
(cond : finrank R M = finrank R M') : M ≃ₗ[R] M' :=
Classical.choice <| FiniteDimensional.nonempty_linearEquiv_of_finrank_eq cond
| Mathlib/LinearAlgebra/Dimension/Free.lean | 158 | 161 |
/-
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.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | theorem coe_one : (↑(1 : FractionalIdeal S P) : Submodule R P) = 1 := by
rw [coe_one_eq_coeSubmodule_top, coeSubmodule_top]
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 371 | 373 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Nat.Cast.Order.Ring
/-! # Bernoulli's inequality -/
variable {R : Type*}
section OrderedSemiring
variabl... | /-- **Bernoulli's inequality**. This version works for semirings but requires
additional hypotheses `0 ≤ a * a` and `0 ≤ (1 + a) * (1 + a)`. -/
lemma one_add_mul_le_pow' (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a)) (H : 0 ≤ 2 + a) :
∀ n : ℕ, 1 + n * a ≤ (1 + a) ^ n
| 0 => by simp
| 1 => by simp
| n + 2 =>... | Mathlib/Algebra/Order/Ring/Pow.lean | 16 | 35 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | contrapose! hφ
cases mk_eq_iff.mp (mk_embedding (mk φ)) with
| inl h => rwa [h] at hφ
| inr h => rwa [← ComplexEmbedding.isReal_conjugate_iff, h] at hφ
@[simp]
theorem conjugate_embedding_eq_of_isReal {w : InfinitePlace K} (h : IsReal w) :
ComplexEmbedding.conjugate (embedding w) = embedding w :=
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 385 | 392 |
/-
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.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... |
section LT
| Mathlib/Order/SuccPred/Limit.lean | 336 | 338 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... | IsLocallyInjective_iff_isOpenEmbedding.trans
⟨IsOpenEmbedding.isOpenMap, .of_continuous_injective_isOpenMap
(IsEmbedding.toPullbackDiag f).continuous (injective_toPullbackDiag f)⟩
theorem discreteTopology_iff_locallyInjective (y : Y) :
DiscreteTopology X ↔ IsLocallyInjective fun _ : X ↦ y := by
| Mathlib/Topology/SeparatedMap.lean | 155 | 160 |
/-
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.Induced
import Mathlib.MeasureTheory.OuterMeasure.AE
import Mathlib.Order.Filter.CountableInter
/-!
# Measu... | ⟨by rintro rfl s; rfl, fun h ↦ Measure.ext (fun s _ ↦ h s)⟩
| Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 145 | 146 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | theorem pullback_comp {f : Y ⟶ X} {g : Z ⟶ Y} (S : Sieve X) :
S.pullback (g ≫ f) = (S.pullback f).pullback g := by simp [Sieve.ext_iff]
| Mathlib/CategoryTheory/Sites/Sieves.lean | 535 | 536 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,155 | 1,156 | |
/-
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` ... | ContinuousWithinAt g s x :=
h.congr_of_eventuallyEq h₁ (mem_of_mem_nhdsWithin hx h₁ :)
| Mathlib/Topology/ContinuousOn.lean | 857 | 859 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,321 | 3,322 | |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | constructor
· intro h
rcases lt_trichotomy x 0 with (x_lt_zero | rfl | x_gt_zero)
· refine Or.inr (Or.inr (neg_eq_iff_eq_neg.mp ?_))
rw [← log_neg_eq_log x] at h
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 256 | 260 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | /-- If `2` is invertible in `R`, then applying the inverse of `2` in `End R M` to an element
of `M` is the same as multiplying by the inverse of `2` in `R`. -/
@[simp]
lemma half_moduleEnd_apply_eq_half_smul [Invertible (2 : R)] (x : M) :
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 858 | 861 |
/-
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 | 957 | 979 | |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... |
lemma trace_eq_trace_restrict_of_le_idealizer
(hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N := fun m _ ↦ N.lie_mem (N.mem_idealizer.mp (h hy) m)) :
trace R M (φ x ∘ₗ φ y) = trace R N ((φ x ∘ₗ φ y).restrict hy') := by
suffices ∀ m, ⁅x, ⁅y, m⁆⁆ ∈ N by
have : (trace R { x // x ∈ N }) ((φ x ∘ₗ φ y).restrict _) = (trace... | Mathlib/Algebra/Lie/TraceForm.lean | 302 | 307 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | exact ⟨Int.sub_floor_div_mul_nonneg _ hp, Int.sub_floor_div_mul_lt _ hp⟩
theorem toIocDiv_eq_neg_floor (a b : α) : toIocDiv hp a b = -⌊(a + p - b) / p⌋ := by
refine toIocDiv_eq_of_sub_zsmul_mem_Ioc hp ?_
rw [Set.mem_Ioc, zsmul_eq_mul, Int.cast_neg, neg_mul, sub_neg_eq_add, ← sub_nonneg,
sub_add_eq_sub_sub]
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 864 | 883 |
/-
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` ... | nhds_discrete, prod_pure, map_map]; rfl
/-!
### Pi
-/
| Mathlib/Topology/ContinuousOn.lean | 1,162 | 1,167 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... |
/-- A non-empty reduced word determines an element of the free product, given by multiplication. -/
def prod {i j} (w : NeWord M i j) :=
w.toWord.prod
@[simp]
| Mathlib/GroupTheory/CoprodI.lean | 710 | 715 |
/-
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... | theorem bsup_lt_ord {o : Ordinal} {f : ∀ a < o, Ordinal} {c : Ordinal} (ho : o.card < c.cof) :
(∀ i hi, f i hi < c) → bsup.{u, u} o f < c :=
bsup_lt_ord_lift (by rwa [o.card.lift_id])
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 358 | 360 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | end Bisim
theorem bisim_simple (s₁ s₂ : Stream' α) :
head s₁ = head s₂ → s₁ = tail s₁ → s₂ = tail s₂ → s₁ = s₂ := fun hh ht₁ ht₂ =>
eq_of_bisim (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂)
(fun s₁ s₂ ⟨h₁, h₂, h₃⟩ => by
| Mathlib/Data/Stream/Init.lean | 274 | 279 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | theorem HasFDerivWithinAt.congr_of_eventuallyEq (h : HasFDerivWithinAt f f' s x)
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : HasFDerivWithinAt f₁ f' s x :=
HasFDerivAtFilter.congr_of_eventuallyEq h h₁ hx
theorem HasFDerivAt.congr_of_eventuallyEq (h : HasFDerivAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f) :
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 900 | 904 |
/-
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... | simp only [sublists_singleton, map_cons, bind_eq_flatMap, nil_append, cons_append, map_nil]
@[simp]
theorem sublists_concat (l : List α) (a : α) :
sublists (l ++ [a]) = sublists l ++ map (fun x => x ++ [a]) (sublists l) := by
rw [sublists_append, sublists_singleton, bind_eq_flatMap, flatMap_cons, flatMap_cons,... | Mathlib/Data/List/Sublists.lean | 140 | 147 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | "Given a finite set `s`, the sum of `f i + g i` over `i ∈ s` equals the sum of `f i`
over `i ∈ s` plus the sum of `g i` over `i ∈ s`."]
| Mathlib/Algebra/BigOperators/Finprod.lean | 585 | 586 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | Wbtw R (f x) (f y) (f z) ↔ Wbtw R x y z := by
have : Function.Injective f.toAffineMap := f.injective
-- `refine` or `exact` are very slow, `apply` is fast. Please check before golfing.
apply this.wbtw_map_iff
| Mathlib/Analysis/Convex/Between.lean | 174 | 177 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Group.Measure
/-!
# Bochner Integration on Groups
We develop properties of inte... | simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_right_eq_self]
@[to_additive]
theorem Integrable.comp_mul_left {f : G → F} [IsMulLeftInvariant μ] (hf : Integrable f μ) (g : G) :
| Mathlib/MeasureTheory/Group/Integral.lean | 124 | 127 |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.DenselyOrdered
import Mathlib.Data.Real.Archimedean
import Mathlib.Order.LiminfLimsu... | [AddLeftMono R] (f : ι → R) (c : R)
(cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F := by
rcases F.eq_or_neBot with rfl | _
· simp only [liminf, limsInf, limsup, limsSup, map_bot, eventually_bot, Set.setOf_true]
... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 568 | 578 |
/-
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.Arctan
import Mathlib.Analysis.SpecialFunctions.... | open Real
section deriv
variable {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ}
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | 108 | 112 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... |
section Ring
variable [Ring R]
@[simp]
theorem trailingDegree_neg (p : R[X]) : trailingDegree (-p) = trailingDegree p := by
unfold trailingDegree
rw [support_neg]
@[simp]
theorem natTrailingDegree_neg (p : R[X]) : natTrailingDegree (-p) = natTrailingDegree p := by
simp [natTrailingDegree]
@[simp]
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 370 | 384 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,156 | 1,158 | |
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
/-!
# Vieta's Formula
The main result is `Multiset.prod_X_add_C_eq_sum_... |
section MvPolynomial
open Finset Polynomial Fintype
variable (R σ : Type*) [CommSemiring R] [Fintype σ]
| Mathlib/RingTheory/Polynomial/Vieta.lean | 139 | 145 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | /-! We now show that a continuous function on `[a, a + p]` satisfying `f a = f (a + p)` is the
pullback of a continuous function on `AddCircle p`. -/
variable {p a}
theorem liftIco_eq_lift_Icc {f : 𝕜 → B} (h : f a = f (a + p)) :
liftIco p a f =
Quot.lift (restrict (Icc a <| a + p) f)
| Mathlib/Topology/Instances/AddCircle.lean | 624 | 632 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... |
theorem lowerBounds_Ioc {a b : γ} (hab : a < b) : lowerBounds (Ioc a b) = Iic a :=
(isGLB_Ioc hab).lowerBounds_eq
end
section
variable [SemilatticeInf γ] [DenselyOrdered γ]
| Mathlib/Order/Bounds/Basic.lean | 586 | 595 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.RingTheory.MatrixPolynomialAlgebra
/-!
# Characteristic polynomials... | @[simp]
lemma charmatrix_blockTriangular_iff {α : Type*} [Preorder α] {M : Matrix n n R} {b : n → α} :
M.charmatrix.BlockTriangular b ↔ M.BlockTriangular b := by
rw [charmatrix, scalar_apply, RingHom.mapMatrix_apply, (blockTriangular_diagonal _).sub_iff_right]
simp [BlockTriangular]
| Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 91 | 95 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | theorem det_updateCol_add (M : Matrix n n R) (j : n) (u v : n → R) :
det (updateCol M j <| u + v) = det (updateCol M j u) + det (updateCol M j v) := by
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 384 | 385 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.Algebra.Homology.TotalComplex
/-!
# Behaviour of the total complex with respect to shifts
There are two ... | by_cases h : (up ℤ).Rel n₀ n₁
· apply total.hom_ext
intro p q hpq
dsimp at h hpq
dsimp [totalShift₁XIso]
rw [ι_D₁_assoc, Linear.comp_units_smul, ι_totalDesc_assoc, ι_D₁,
((shiftFunctor₁ C x).obj K).d₁_eq _ rfl _ _ (by dsimp; omega),
K.d₁_eq _ (show p + x + 1 = p + 1 + x by omega) _ _ (by... | Mathlib/Algebra/Homology/TotalComplexShift.lean | 141 | 159 |
/-
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
... |
variable (R)
-- Porting note: Added 2 new theorems for convenience
private theorem two_mul_C_half_eq_one [Invertible (2 : R)] : 2 * C (⅟ 2 : R) = 1 := by
rw [two_mul, ← C_add, invOf_two_add_invOf_two, C_1]
private theorem C_half_mul_two_eq_one [Invertible (2 : R)] : C (⅟ 2 : R) * 2 = 1 := by
rw [mul_comm, two_mu... | Mathlib/RingTheory/Polynomial/Dickson.lean | 128 | 137 |
/-
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.Data.Finsupp.Lex
import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
This file establishes man... |
lemma leadingCoeff_toLex_C (r : R) : leadingCoeff toLex (C (σ := σ) r) = r :=
leadingCoeff_single toLex.injective _ r
| Mathlib/Algebra/MvPolynomial/Variables.lean | 335 | 338 |
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
/-!
# Hermite polynomials
This file defines `Polynomial.hermite n`, the `n`... |
@[simp]
theorem natDegree_hermite {n : ℕ} : (hermite n).natDegree = n :=
natDegree_eq_of_degree_eq_some (degree_hermite n)
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 103 | 107 |
/-
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, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.MonoidLocalization.Away
import Mathlib.Algebra.Algebra.Pi
import Mathlib.RingTheory.I... | convert IsLocalization.commutes S₁ S₂ T (Submonoid.powers x) (Submonoid.powers y)
ext x
| Mathlib/RingTheory/Localization/Away/Basic.lean | 278 | 279 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | Scheme.comp_app, Scheme.Opens.topIso_hom, Scheme.ΓSpecIso_inv_naturality,
Scheme.inv_appTop, -- `check_compositions` reports defeq problems starting after this step.
IsIso.inv_comp_eq]
rw [Scheme.toSpecΓ_appTop]
-- We need `erw` here because the goal has
| Mathlib/AlgebraicGeometry/AffineScheme.lean | 343 | 347 |
/-
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.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... |
@[simp]
| Mathlib/Algebra/Polynomial/Derivative.lean | 103 | 104 |
/-
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.Partition.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | splitMany I (insert p s) = splitMany I s ⊓ split I p.1 p.2 := by
rw [splitMany, Finset.inf_insert, inf_comm, splitMany]
theorem splitMany_le_split (I : Box ι) {s : Finset (ι × ℝ)} {p : ι × ℝ} (hp : p ∈ s) :
splitMany I s ≤ split I p.1 p.2 :=
Finset.inf_le hp
| Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 229 | 234 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
/-! # Lemmas about coprimality with big products.
These lemmas ... | Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
| Mathlib/Data/Nat/GCD/BigOperators.lean | 20 | 22 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
/-!
# Perpendicular bisector of a segment
We def... |
@[simp] theorem perpBisector_ne_bot : perpBisector p₁ p₂ ≠ ⊥ := by
rw [← nonempty_iff_ne_bot]; exact perpBisector_nonempty
| Mathlib/Geometry/Euclidean/PerpBisector.lean | 112 | 115 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 310 | 311 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | theorem dite_eq_or_eq : (∃ h, dite P A B = A h) ∨ ∃ h, dite P A B = B h :=
if h : _ then .inl ⟨h, dif_pos h⟩ else .inr ⟨h, dif_neg h⟩
| Mathlib/Logic/Basic.lean | 873 | 874 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... |
@[deprecated (since := "2025-01-24")]
alias aestronglyMeasurable'_condexpL1 := aestronglyMeasurable_condexpL1
theorem condExpL1_congr_ae (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (h : f =ᵐ[μ] g) :
condExpL1 hm μ f = condExpL1 hm μ g :=
setToFun_congr_ae _ h
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 603 | 609 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,353 | 2,370 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | @[simp]
theorem oangle_sub_left {p p₁ p₂ p₃ : P} (hp₁ : p₁ ≠ p) (hp₂ : p₂ ≠ p) (hp₃ : p₃ ≠ p) :
∡ p₁ p p₃ - ∡ p₁ p p₂ = ∡ p₂ p p₃ :=
o.oangle_sub_left (vsub_ne_zero.2 hp₁) (vsub_ne_zero.2 hp₂) (vsub_ne_zero.2 hp₃)
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 250 | 253 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | end LinearOrderedAddCommGroup
| Mathlib/Algebra/Order/Group/Abs.lean | 255 | 258 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by
apply ae_of_forall_measure_lt_top_ae_restrict
intro t t_meas t_lt_top
apply ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter (hf_int_finite t t_meas t_lt_top)
intro s s_meas _
exact
hf_zero _ (s_meas.inter t_meas)... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 165 | 222 |
/-
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.Algebra.Polynomial.Taylor
import Mathlib.FieldTheory.RatFunc.AsPolynomial
/-!
# Laurent expansions of rational functions
## Main declarations
* `R... | -- Porting note: added `by exact taylor_mem_nonZeroDivisors r`
map_apply_div _ (by exact taylor_mem_nonZeroDivisors r) _ _
@[simp]
theorem laurentAux_algebraMap : laurentAux r (algebraMap _ _ p) = algebraMap _ _ (taylor r p) := by
| Mathlib/FieldTheory/Laurent.lean | 66 | 70 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | open MeasureTheory
| Mathlib/MeasureTheory/Measure/Stieltjes.lean | 195 | 196 |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | have h₆ : ¬p ∣ b * (b ^ 2 - 1) := Nat.not_dvd_of_pos_of_lt h₄ h₅
have h₇ : b ≤ b * (b ^ 2 - 1) := Nat.le_mul_of_pos_right _ h₃
have h₈ : 2 ≤ b * (b ^ 2 - 1) := le_trans b_ge_two h₇
have h₉ : 2 < p := gt_of_gt_of_ge h₅ h₈
have h₁₀ := psp_from_prime_gt_p b_ge_two hp₂ h₉
use psp_from_prime b p
... | Mathlib/NumberTheory/FermatPsp.lean | 335 | 368 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.P... | /-- The Fubini theorem for a functor `G : J × K ⥤ C`,
showing that the limit of `G` can be computed as
the limit of the limits of the functors `G.obj (j, _)`.
-/
noncomputable def limitIsoLimitCurryCompLim : limit G ≅ limit (curry.obj G ⋙ lim) := by
have i : G ≅ uncurry.obj ((@curry J _ K _ C _).obj G) := currying.sy... | Mathlib/CategoryTheory/Limits/Fubini.lean | 584 | 597 |
/-
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.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functor... | intro bs hbs
simp only [Set.mem_preimage, Finset.le_iff_subset] at hu
have : Tendsto (fun t : Finset (Σ b, γ b) ↦ ∏ p ∈ t with p.1 ∈ bs, f p) atTop
| Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 126 | 128 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,142 | 1,145 | |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... | lt_of_pow_lt_pow_left₀ n hb h
@[deprecated le_of_pow_le_pow_left₀ (since := "2024-11-12")]
| Mathlib/Algebra/Order/Ring/Basic.lean | 164 | 166 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Cayley-Hamilton theorem for f.g. modules.
Given a fixed finite spannin... | delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, map_mul]
ext
dsimp [PiToModule.fromEnd]
rw [← h'.congr_fun, ← h.congr_fun]
rfl
theorem Matrix.Represents.one : (1 : Matrix ι ι R).Represents b 1 := by
delta Matrix.Represents PiToModule.fromMatrix
rw [Li... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 100 | 111 |
/-
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.... | theorem Convex.openSegment_self_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
(hx : x ∈ s) (hy : y ∈ interior s) : openSegment 𝕜 x y ⊆ interior s :=
hs.openSegment_closure_interior_subset_interior (subset_closure hx) hy
| Mathlib/Analysis/Convex/Topology.lean | 200 | 203 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | have hle := encard_le_encard_of_injOn hf f_inj
to_encard_tac; rwa [ht.cast_ncard_eq, (ht.finite_of_encard_le hle).cast_ncard_eq]
theorem exists_ne_map_eq_of_ncard_lt_of_maps_to {t : Set β} (hc : t.ncard < s.ncard) {f : α → β}
(hf : ∀ a ∈ s, f a ∈ t) (ht : t.Finite := by toFinite_tac) :
∃ x ∈ s, ∃ y ∈ s, x ... | Mathlib/Data/Set/Card.lean | 779 | 784 |
/-
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.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | theorem card_interedges_div_card (s t : Finset α) :
(#(G.interedges s t) : ℚ) / (#s * #t) = G.edgeDensity s t :=
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 287 | 288 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... |
@[to_additive zsmul_sub]
| Mathlib/Algebra/Group/Basic.lean | 617 | 618 |
/-
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.MeasureTheory.Measure.AEMeasurable
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` etc and prove d... | @[to_additive]
class MeasurableConstSMul (M α : Type*) [SMul M α] [MeasurableSpace α] : Prop where
measurable_const_smul : ∀ c : M, Measurable (c • · : α → α)
| Mathlib/MeasureTheory/Group/Arithmetic.lean | 478 | 480 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Range
/-! ### Linear equivalences involvi... |
/-- Given `p` a submodule of the module `M` and `q` a submodule of `p`, `p.equivSubtypeMap q`
is the natural `LinearEquiv` between `q` and `q.map p.subtype`. -/
| Mathlib/Algebra/Module/Submodule/Equiv.lean | 213 | 215 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | rw [exp_log_eq_abs hx.ne']
exact abs_of_pos hx
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 55 | 56 |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | simp? [rTensor.inverse_of_rightInverse] says
simp only [inverse_of_rightInverse, coe_comp, Function.comp_apply, rTensor_tmul,
lift.tmul, coe_mk, AddHom.coe_mk, mk_apply, Submodule.mkQ_apply]
rw [Submodule.Quotient.eq, ← TensorProduct.sub_tmul]
apply le_comap_range_rTensor f
rw [← hfg, mem_ker, map_sub... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 331 | 338 |
/-
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... | simpa only [not_forall₂, not_le, bex_def] using not_congr (@limit_le _ h a)
@[simp]
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 233 | 235 |
/-
Copyright (c) 2023 Jason Yuen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jason Yuen
-/
import Mathlib.Data.Real.ConjExponents
import Mathlib.Data.Real.Irrational
/-!
# Rayleigh's theorem on Beatty sequences
This file proves Rayleigh's theorem on Beatty sequen... |
theorem beattySeq'_symmDiff_beattySeq_pos {r s : ℝ} (hrs : r.HolderConjugate s) :
{beattySeq' r k | k > 0} ∆ {beattySeq s k | k > 0} = {n | 0 < n} := by
| Mathlib/NumberTheory/Rayleigh.lean | 157 | 159 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... |
protected theorem measurableSet_eq_of_countable' [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[hτ.measurableSpace] {ω | τ ω = i} :=
hτ.measurableSet_eq_of_countable_range' (Set.to_countable _) i
protected theorem measurableSet_ge_of_countable_range' (hτ : IsStoppingTime f τ)
(h_countable :... | Mathlib/Probability/Process/Stopping.lean | 468 | 474 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
MeasurableSet {b | BddAbove (range (fun i ↦ f i b))} := by
rcases isEmpty_or_nonempty α with hα|hα
· have : ∀ b, range (fun i ↦ f i b) = ∅ := fun b ↦ eq_empty_of_isEmpty _
simp [this]
have A : ∀ (i : ι) (c ... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 695 | 716 |
/-
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... | _ ≤ f ⊓ Filter.cocompact X := inf_le_inf_left f (Filter.comap_cocompact_le hg)
_ = ⊥ := disjoint_iff.mp hf
| Mathlib/Topology/Compactness/Compact.lean | 801 | 803 |
/-
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.Divisibility.Hom
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.A... | simp only [Equiv.neg_symm, Equiv.neg_apply, mul_neg, neg_inj, Dvd.dvd]
/-- The negation of an element `a` of a semigroup with a distributive negation divides another
| Mathlib/Algebra/Ring/Divisibility/Basic.lean | 95 | 97 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Jordan.Basic
import Mathlib.Algebra.Module.Defs
/-!
# Symmetrized algebra
A commutative multiplication on a real or complex space can... | end SymAlg
| Mathlib/Algebra/Symmetrized.lean | 326 | 327 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,620 | 2,621 | |
/-
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... | rw [← Ico_succ_singleton, List.drop_length, ofDigits] at this
have h₁ : 1 ≤ tl.length := List.length_pos_iff.mpr h'
rw [← sum_range_add_sum_Ico _ <| h₁, ← add_zero (∑ x ∈ Ico _ _, ofDigits p (tl.drop x)),
← this, sum_Ico_consecutive _ h₁ <| (le_add_right tl.length 1),
← ... | Mathlib/Data/Nat/Digits.lean | 571 | 604 |
/-
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.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... | mem_nhdsLE_iff_exists_Ioc_subset' hl'
@[deprecated (since := "2024-12-22")]
alias mem_nhdsWithin_Iic_iff_exists_Ioc_subset := mem_nhdsLE_iff_exists_Ioc_subset
/-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]`
| Mathlib/Topology/Order/LeftRightNhds.lean | 337 | 342 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | refine IntegrableOn.mul_continuousOn hf.norm ?_ hK
exact continuous_norm.comp_continuousOn hg
· exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet)
end SMul
namespace LocallyIntegrableOn
| Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 652 | 659 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... | have he' : ∀ x, HasDerivAt e ((fun _ ↦ 1) x) x := fun _ ↦ (hasDerivAt_id _).sub_const y
change (gaussianReal μ v).map e.symm = gaussianReal (μ + y) v
ext s' hs'
rw [MeasurableEquiv.gaussianReal_map_symm_apply hv e he' hs']
simp only [abs_neg, abs_one, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk, one_mul, ne_eq]
... | Mathlib/Probability/Distributions/Gaussian.lean | 275 | 306 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Tactic.Group
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
... |
/-- In a space with an inner regular measure, the everywhere positive subset of a measurable set
is itself everywhere positive. This is not obvious as `μ.everywherePosSubset s` is defined as
the points whose neighborhoods intersect `s` along positive measure subsets, but this does not
say they also intersect `μ.everyw... | Mathlib/MeasureTheory/Measure/EverywherePos.lean | 130 | 145 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
/-!
## `ℤ` forms a conditionally complete linear order
The integer... | example : Lattice ℤ := inferInstance
| Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 94 | 96 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
theorem Set.Finite.isClosed_biUnion {s : Set α} {f : α → Set X} (hs : s.Finite)
(h : ∀ i ∈ s, IsClosed (f i)) :
| Mathlib/Topology/Basic.lean | 164 | 167 |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne... | theorem lt_of_eq_of_lt {a b : α} (ha : a = 0) (hb : b < 0) : a + b < 0 := by
simp [*]
| Mathlib/Tactic/Linarith/Lemmas.lean | 39 | 40 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Separation.Hausdorff
/-!
# Order-closed topologies
In this file we ... | continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg
theorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y))
(hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x :=
| Mathlib/Topology/Order/OrderClosed.lean | 886 | 889 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Yongle Hu
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.RingTheory.Ideal.Pointwise
import Mathlib.RingTheory.Ideal.Quot... | Mathlib/RingTheory/Ideal/Over.lean | 449 | 461 | |
/-
Copyright (c) 2021 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.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | have hg_eq : ∀ x ∈ s, ∀ n, g_seq_s n x = hf.approx n x := by
intro x hx n
rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_mem hx]
have hg_zero : ∀ x ∉ s, ∀ n, g_seq_s n x = 0 := by
intro x hx n
rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_not_mem hx]
refine ⟨g_seq_s, fun x => ?_, hg_zero⟩... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 927 | 944 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... | fun _S hS ↦ isOpen_iSup_iff.2
fun i ↦ @isOpen_sInter _ (t i) _ _
| Mathlib/Topology/AlexandrovDiscrete.lean | 138 | 139 |
/-
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 coe_rpow_def (x : ℝ≥0) (y : ℝ) :
(x : ℝ≥0∞) ^ y = if x = 0 ∧ y < 0 then ⊤ else ↑(x ^ y) :=
rfl
theorem rpow_ofNNReal {M : ℝ≥0} {P : ℝ} (hP : 0 ≤ P) : (M : ℝ≥0∞) ^ P = ↑(M ^ P) := by
rw [ENNReal.coe_rpow_of_nonneg _ hP, ← ENNReal.rpow_eq_pow]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 502 | 509 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | ext ⟨a, b⟩
rw [Finsupp.uncurry_apply_pair, curry_apply]
/-- `finsuppProdEquiv` defines the `Equiv` between `((α × β) →₀ M)` and `(α →₀ (β →₀ M))` given by
currying and uncurrying. -/
| Mathlib/Data/Finsupp/Basic.lean | 1,111 | 1,115 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | section ToSubring
variable (p : R[X]) (T : Subring R)
/-- Given a polynomial `p` and a subring `T` that contains the coefficients of `p`,
| Mathlib/RingTheory/Polynomial/Basic.lean | 368 | 372 |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | reindex (el.prodCongr en) (em.prodCongr ep) (kroneckerMap f M N) := by
ext ⟨i, i'⟩ ⟨j, j'⟩
rfl
| Mathlib/Data/Matrix/Kronecker.lean | 149 | 152 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | @[to_additive (attr := simp)]
theorem Step.cons_not_rev {x b} : Red.Step ((x, !b) :: (x, b) :: L) L :=
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 109 | 110 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.VectorBundle.Basic
/-! # Tangent bundles
This file defines the tangent bundle as a `C^n` vector bundle.
Let `... | apply (tangentBundleCore I M).coordChange_self
rw [tangentBundleCore_baseSet, coe_achart, ← extChartAt_source I]
exact h
lemma tangentCoordChange_comp {w x y z : M} {v : E}
| Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean | 143 | 147 |
/-
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.Algebra.Order.Group.Pointwise.Interval
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
import Ma... |
@[simp]
theorem lineMap_apply_one (p₀ p₁ : P1) : lineMap p₀ p₁ (1 : k) = p₁ := by
simp [lineMap_apply]
@[simp]
theorem lineMap_eq_lineMap_iff [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c₁ c₂ : k} :
| Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 480 | 486 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | nth_rw 2 [← hf.lt_iff_lt]
convert Iff.rfl using 2
exact ih m' (Nat.lt_add_one_iff.mp hm) fun hfi ↦ hm.trans (h hfi)
| Mathlib/Data/Nat/Nth.lean | 300 | 302 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
exact CommRingCat.hom_ext_iff.mp (StructureSheaf.toOpen_res _ _ _ _)
instance _root_.AlgebraicGeometry.isLocalization_away_of_isAffine
[IsAffine X] (r : Γ(X, ⊤)) :
IsLocalization.Away r Γ(X, X.basicOpen r) :=
isLocalization_basicOpen (isAffineOpen... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 621 | 627 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | declared as an instance because there are several natural choices for defining the norm of a
matrix. -/
@[local instance]
def frobeniusSeminormedAddCommGroup [SeminormedAddCommGroup α] :
SeminormedAddCommGroup (Matrix m n α) :=
inferInstanceAs (SeminormedAddCommGroup (PiLp 2 fun _i : m => PiLp 2 fun _j : n => α))... | Mathlib/Analysis/Matrix.lean | 464 | 481 |
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