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 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Data.Nat.Find
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Basic properties of holors
Holors... | ∀ t : HolorIndex (d :: ds),
(∀ i is, ∀ h : t.1 = i :: is, p ⟨i :: is, by rw [← h]; exact t.2⟩) → p t
| ⟨[], hforall₂⟩, _ => absurd (forall₂_nil_left_iff.1 hforall₂) (cons_ne_nil d ds)
| ⟨i :: is, _⟩, hp => hp i is rfl
| Mathlib/Data/Holor.lean | 178 | 181 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Nat.Cast.Defs
/-!
# Lemmas about nonzero elements of an `AddMonoidWithOne`
-/
open Nat
namespace NeZero
theorem one_le {n : ℕ} ... | lemma natCast_ne (n : ℕ) (R) [AddMonoidWithOne R] [h : NeZero (n : R)] : (n : R) ≠ 0 := h.out
| Mathlib/Data/Nat/Cast/NeZero.lean | 18 | 19 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | lemma dualSeminorms_bounded : BddAbove {p | ∃ (G : Type (max uι u𝕜 uE))
(_ : SeminormedAddCommGroup G) (_ : NormedSpace 𝕜 G),
p = Seminorm.comp (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G))
(toDualContinuousMultilinearMap G (𝕜 := 𝕜) (E := E))} := by
existsi projectiveSeminorm
rw [mem_... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 131 | 142 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | (x * y).Numeric := by
have ihxy := ih1 ih
have ihyx := ih1_swap ih
have ihxyn := ih1_neg_left (ih1_neg_right ihxy)
have ihyxn := ih1_neg_left (ih1_neg_right ihyx)
refine numeric_def.mpr ⟨?_, ?_, ?_⟩
· simp_rw [lt_iff_game_lt]
intro i
rw [rightMoves_mul_iff]
constructor <;> (intro j l; revert... | Mathlib/SetTheory/Surreal/Multiplication.lean | 259 | 281 |
/-
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.Data.NNReal.Basic
import Mathlib.Order.Fin.Tuple
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 519 | 527 | |
/-
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.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
/-!
# Simplicial complexes
In this file, we define ... | have h := K.inter_subset_convexHull hx hs ⟨by simp, h⟩
by_contra H
rwa [← coe_inter, Finset.disjoint_iff_inter_eq_empty.1 (Finset.disjoint_singleton_right.2 H).symm,
coe_empty, convexHull_empty] at h
| Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 158 | 162 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 684 | 685 | |
/-
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.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | @[simp]
theorem convexJoin_union_right (s t₁ t₂ : Set E) :
| Mathlib/Analysis/Convex/Join.lean | 70 | 71 |
/-
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 | 1,230 | 1,243 | |
/-
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.BigOperators.Field
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Math... |
variable {α : Type*} [DecidableEq α] [Fintype α] {P : Finpartition (univ : Finset α)}
| Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 58 | 59 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
import Mathlib.RingTheory.Norm.Transitivity
/-!
# No... | ⟨(univ \ {AlgEquiv.refl}).prod fun σ : L ≃ₐ[K] L => σ x,
prod_mem fun σ _ => x.2.map (σ : L →+* L).toIntAlgHom⟩ _ ?_
convert hx using 1
ext
convert_to ((univ \ {AlgEquiv.refl}).prod fun σ : L ≃ₐ[K] L => σ x) *
∏ σ ∈ {(AlgEquiv.refl : L ≃ₐ[K] L)}, σ x = _
· rw [prod_singleton, AlgEquiv.coe_refl, _r... | Mathlib/NumberTheory/NumberField/Norm.lean | 72 | 85 |
/-
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... |
theorem zneg_toZNumNeg (n : Num) : -n.toZNumNeg = n.toZNum := by cases n <;> rfl
| Mathlib/Data/Num/Lemmas.lean | 666 | 667 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | simp only [Polynomial.map_multiset_prod, Multiset.map_map]
congr; ext1
simp only [Function.comp_apply, Polynomial.map_sub, map_X, map_C]
theorem count_map_roots_of_injective [IsDomain A] [DecidableEq B] (p : A[X]) {f : A →+* B}
| Mathlib/Algebra/Polynomial/Roots.lean | 752 | 756 |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... |
@[simp]
theorem toOuterMeasure_bind_apply :
(p.bind f).toOuterMeasure s = ∑' a, p a * (f a).toOuterMeasure s := by
| Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 152 | 155 |
/-
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... |
theorem log_le_iff_le_exp (hx : 0 < x) : log x ≤ y ↔ x ≤ exp y := by rw [← exp_le_exp, exp_log hx]
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 151 | 152 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | Nat.cast_smul_eq_nsmul]
exact genWeightSpace_chainTopCoeff_add_one_nsmul_add (-α) β (Weight.IsNonZero.neg hα)
| Mathlib/Algebra/Lie/Weights/RootSystem.lean | 164 | 166 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 601 | 610 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,218 | 1,220 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | @[deprecated (since := "2025-01-21")] alias condexp_bot_ae_eq := condExp_bot_ae_eq
theorem condExp_bot [IsProbabilityMeasure μ] (f : α → E) : μ[f|⊥] = fun _ => ∫ x, f x ∂μ := by
refine (condExp_bot' f).trans ?_
rw [measureReal_univ_eq_one, inv_one, one_smul]
@[deprecated (since := "2025-01-21")] alias condexp_bot... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 314 | 324 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 945 | 953 | |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | theorem sup_prod (f₁ f₂ : Filter α) (g : Filter β) : (f₁ ⊔ f₂) ×ˢ g = (f₁ ×ˢ g) ⊔ (f₂ ×ˢ g) := by
simp only [prod_eq_inf, comap_sup, inf_sup_right]
| Mathlib/Order/Filter/Prod.lean | 107 | 109 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitel... | (Multiset.mapAddMonoidHom g).map_nsmul]
rfl
@[to_additive (attr := simp)]
theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
| Mathlib/Data/Finsupp/Multiset.lean | 71 | 79 |
/-
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 | 1,860 | 1,861 | |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | simp
theorem Monic.prime_of_degree_eq_one (hp1 : degree p = 1) (hm : Monic p) : Prime p :=
have : p = X - C (-p.coeff 0) := by simpa [hm.leadingCoeff] using eq_X_add_C_of_degree_eq_one hp1
this.symm ▸ prime_X_sub_C _
| Mathlib/Algebra/Polynomial/RingDivision.lean | 198 | 203 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... |
lemma card_ne_zero : Nat.card α ≠ 0 ↔ Nonempty α ∧ Finite α := by simp [card_eq_zero, not_or]
| Mathlib/SetTheory/Cardinal/Finite.lean | 72 | 73 |
/-
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.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | @subset_closure E _ _
end
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 541 | 543 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.Analysis.NormedSpace.Multiline... |
protected theorem t2Space [T2Space R] : T2Space V := by
apply (t2Space_iff _).2 (fun {x} {y} hxy ↦ ?_)
rcases exists_separating_of_ne (R := R) hxy with ⟨f, hf⟩
| Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 67 | 70 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex
import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex
import Mathlib.AlgebraicTopology.CechNerve
import Mathlib.A... |
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11119): @[simp] removed as the linter complains the LHS is not in normal form
theorem ExtraDegeneracy.s_comp_π_succ (n : ℕ) (i : Fin (n + 1)) :
ExtraDegeneracy.s f S n ≫ WidePullback.π _ i.succ =
@WidePullback.π _ _ _ f.right (fun _ : F... | Mathlib/AlgebraicTopology/ExtraDegeneracy.lean | 288 | 294 |
/-
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.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... | refine (dvd_add_left ?_).mp dvd_eval
apply Finset.dvd_sum
intro j hj
exact dvd_terms j (Finset.ne_of_mem_erase hj)
· convert dvd_zero p
rw [not_mem_support_iff] at hi
simp [hi]
theorem dvd_term_of_isRoot_of_dvd_terms {r p : S} {f : S[X]} (i : ℕ) (hr : f.IsRoot r)
(h : ∀ j ≠ i, p ∣ f.coeff... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 521 | 537 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.List.Pairwise
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Pairwise relations on a list
This file provides basic results ... | R (l.head <| ne_nil_of_mem ha) a :=
h₁.rel_head_of_rel_head_head ha (refl_of ..)
| Mathlib/Data/List/Pairwise.lean | 76 | 78 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h
| Mathlib/Topology/Order/DenselyOrdered.lean | 25 | 29 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | apply Iso.completeGraph
simpa using (Fintype.equivFin h.choose).symm
rw [← ha] at this
convert (Embedding.induce ↑h.choose.toSet).comp this.toEmbedding
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 339 | 342 |
/-
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.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | section LinearOrderedCommGroup
variable [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] {a : α}
| Mathlib/Algebra/Order/Group/Defs.lean | 98 | 100 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | (h3 : IsRelPrime f f.mirror) : Irreducible f := by
constructor
| Mathlib/Algebra/Polynomial/Mirror.lean | 193 | 194 |
/-
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.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... | @[simp]
| Mathlib/Order/Interval/Set/ProjIcc.lean | 99 | 99 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | rw [← inv_le_inv₀ s_pos p_pos] at this
apply le_trans this
have p_pos₂ : 0 < (∏ i ∈ s, (z i) ^ w i)⁻¹ :=
inv_pos.2 (prod_pos fun i hi => rpow_pos_of_pos ((hz i hi)) _ )
rw [← inv_inv (∏ i ∈ s, z i ^ w i), inv_le_inv₀ p_pos p_pos₂, ← Finset.prod_inv_distrib]
gcongr
· exact fun i hi ↦ inv_no... | Mathlib/Analysis/MeanInequalities.lean | 368 | 380 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,509 | 1,512 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | variable {R}
theorem of_irreducible_expand {p : ℕ} (hp : p ≠ 0) {f : R[X]} (hf : Irreducible (expand R p f)) :
Irreducible f :=
let _ := isLocalHom_expand R hp.bot_lt
hf.of_map
| Mathlib/Algebra/Polynomial/Expand.lean | 307 | 312 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite.Range
import Mathlib.Order.Partition.Finpartition
/-!
# Equivalenc... | (H y).elim fun _ hc _ => eq_eqv_class_of_mem H hc.1 hc.2 ▸ hc.1
theorem eqv_class_mem' {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {x} :
{ y : α | mkClasses c H x y } ∈ c := by
convert @Setoid.eqv_class_mem _ _ H x using 3
rw [Setoid.comm']
/-- Distinct elements of a set of sets partitioning α are disjoint... | Mathlib/Data/Setoid/Partition.lean | 122 | 130 |
/-
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... |
lemma nonsingular_iff (x y : R) : W'.Nonsingular x y ↔ W'.Equation x y ∧
(W'.a₁ * y ≠ 3 * x ^ 2 + 2 * W'.a₂ * x + W'.a₄ ∨ y ≠ -y - W'.a₁ * x - W'.a₃) := by
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 261 | 263 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | natCast := fun n => monomial R 0 n
natCast_zero := by simp [Nat.cast]
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 209 | 210 |
/-
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.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathli... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 573 | 575 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
/-!
# Wide pullbacks
We define the category `WidePullbackShape`, (re... | /-- The obvious functor `(WidePushoutShape J)ᵒᵖ ⥤ WidePullbackShape J` -/
@[simps!]
| Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 440 | 441 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,414 | 1,417 | |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies... | Mathlib/Topology/Category/Profinite/Limits.lean | 123 | 126 | |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.Order.LowerUpperTopology
import Mathlib.Topology.Order.ScottTopology
/-!
# Lawson topology
This file introduces the Lawson topology... | protected theorem isTopologicalBasis : TopologicalSpace.IsTopologicalBasis (lawsonBasis α) := by
have lawsonBasis_image2 : lawsonBasis α =
(image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1)
(IsLower.lowerBasis (WithLower α)) {U | IsOpen[scott α univ] U}) := by
rw [lawsonBasi... | Mathlib/Topology/Order/LawsonTopology.lean | 86 | 101 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | (HasDerivAt.scomp x hg.hasDerivAt hh.hasDerivAt).deriv
theorem deriv.scomp_of_eq
(hg : DifferentiableAt 𝕜' g₁ y) (hh : DifferentiableAt 𝕜 h x) (hy : y = h x) :
deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x) := by
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 144 | 148 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Finset.Max
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
/-!
# Conditionally complete la... |
theorem Set.Finite.ciInf_mem_image {s : Set ι} (hs : s.Finite) (h : ∃ x ∈ s, f x ≤ sInf ∅) :
| Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | 96 | 97 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Riccardo Brasca, Adam Topaz, Jujian Zhang, Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Projective.Resolution
/-!
# Left-der... | (φ : P.complex ⟶ Q.complex) (comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f)
(F : C ⥤ D) [F.Additive] :
(P.isoLeftDerivedToHomotopyCategoryObj F).inv ≫ F.leftDerivedToHomotopyCategory.map f =
(F.mapHomologicalComplex _ ⋙ HomotopyCategory.quotient _ _).map φ ≫
(Q.isoLeftDerivedToHomotopyCategoryObj F).in... | Mathlib/CategoryTheory/Abelian/LeftDerived.lean | 78 | 89 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... |
lemma bi_min_div_two_pos : 0 < b (min_bi b) / 2 := div_pos (R.b_pos _) (by norm_num)
lemma exists_eventually_const_mul_le_r :
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 198 | 201 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | { DynkinSystem.instLEDynkinSystem with
le_refl := fun _ _ => le_rfl
le_trans := fun _ _ _ hab hbc => le_def.mpr (le_trans hab hbc)
le_antisymm := fun _ _ h₁ h₂ => ext fun s => ⟨h₁ s, h₂ s⟩ }
| Mathlib/MeasureTheory/PiSystem.lean | 555 | 559 |
/-
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.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... |
namespace LieModule
section IsNilpotent
variable [LieRing.IsNilpotent L] (χ₁ χ₂ : L → R) (p q : ℤ)
section
variable [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M] (hχ₁ : χ₁ ≠ 0)
include hχ₁
| Mathlib/Algebra/Lie/Weights/Chain.lean | 59 | 69 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Algebra.Order.Monoid.Defs
/-!
# The integers form a linear ordered group
This file contains the instance necessar... | Mathlib/Algebra/Order/Group/Int.lean | 64 | 67 | |
/-
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 | 1,917 | 1,922 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Int.Parity
im... | Mathlib/Algebra/Order/Ring/Int.lean | 73 | 74 | |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | rw [ZMod.natCast_zmod_eq_zero_iff_dvd] at h
apply eq_zero_of_dvd_of_lt h (appr_lt _ _)
· intro h
rw [← sub_zero x] at h
dsimp [toZModPow, toZModHom]
rw [zmod_congr_of_sub_mem_span n x _ 0 _ h, cast_zero]
apply appr_spec
-- This is not a simp lemma; simp can't match the LHS.
theorem zmod_cast_... | Mathlib/NumberTheory/Padics/RingHoms.lean | 405 | 421 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | disjoint Borel measurable subsets of diameter at most ε that cover the whole space. -/
lemma SeparableSpace.exists_measurable_partition_diam_le {ε : ℝ} (ε_pos : 0 < ε) :
∃ (As : ℕ → Set Ω), (∀ n, MeasurableSet (As n)) ∧ (∀ n, Bornology.IsBounded (As n)) ∧
(∀ n, diam (As n) ≤ ε) ∧ (⋃ n, As n = univ) ∧
... | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | 516 | 611 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | refine ⟨v, ?_⟩
rw [maxDegree, ht]
rfl
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 346 | 348 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.W.Basic
import Mathlib.SetTheory.Cardinal.Arithmetic
/-!
# Cardinality of W-types
This file proves some theorems about the cardinality of W-types. T... |
@[deprecated (since := "2024-11-10")] alias cardinal_mk_eq_sum' := cardinalMk_eq_sum_lift
| Mathlib/Data/W/Cardinal.lean | 40 | 42 |
/-
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.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | ha.tsub_eq_of_eq_add <| add_comm a b
protected theorem lt_add_of_tsub_lt_left (hb : AddLECancellable b) (h : a - b < c) : a < b + c := by
rw [lt_iff_le_and_ne, ← tsub_le_iff_left]
| Mathlib/Algebra/Order/Sub/Defs.lean | 285 | 288 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,703 | 1,705 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | divisors_injective.eq_iff
theorem eq_properDivisors_of_subset_of_sum_eq_sum {s : Finset ℕ} (hsub : s ⊆ n.properDivisors) :
((∑ x ∈ s, x) = ∑ x ∈ n.properDivisors, x) → s = n.properDivisors := by
cases n
· rw [properDivisors_zero, subset_empty] at hsub
simp [hsub]
classical
rw [← sum_sdiff hsub]
... | Mathlib/NumberTheory/Divisors.lean | 417 | 430 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | fun h => by rw [h, mul_inv_cancel_right_of_invertible]⟩
lemma inv_mulVec_eq_vec {A : Matrix n n α} [Invertible A]
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 271 | 273 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | rw [segment_eq_image₂, segment_eq_image₂, image_image]
refine EqOn.image_eq fun a ha ↦ ?_
simp only [← update_smul, ← update_add, Convex.combo_self ha.2.2]
| Mathlib/Analysis/Convex/Segment.lean | 617 | 619 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | section TorsionPolynomial
/-! ### 2-torsion polynomials -/
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 291 | 294 |
/-
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... | ext x
rw [mem_filter, mem_Ico, mem_singleton, and_right_comm, ← le_antisymm_iff, eq_comm]
| Mathlib/Order/Interval/Finset/Basic.lean | 635 | 636 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 991 | 1,030 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import... | apply_fun fun x => x * mulShift ψ (-b) at h
simp only [mulShift_mul, mulShift_zero, add_neg_cancel, mulShift_apply] at h
simpa [← sub_eq_add_neg, sub_eq_zero] using (hψ · h)
-- `AddCommGroup.equiv_direct_sum_zmod_of_fintype`
-- gives the structure theorem for finite abelian groups.
| Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 71 | 76 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... | Balanced 𝕜 (insert 0 (interior A)) := by
intro a ha
obtain rfl | h := eq_or_ne a 0
| Mathlib/Analysis/LocallyConvex/Basic.lean | 244 | 246 |
/-
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... | rw [← mul_succ]
apply mul_le_mul_left'
rw [succ_le_iff]
exact lt_mul_succ_div b hd
· rw [le_div H, mul_assoc]
exact (mul_le_mul_left' (mul_div_le b d) a).trans (le_add_right _ c)
theorem mul_div_mul_cancel {a : Ordinal} (ha : a ≠ 0) (b c) : a * b / (a * c) = b / c := by
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 901 | 908 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | end FiniteKonig
namespace CategoryTheory
namespace Functor
variable {J : Type u} [Category J] (F : J ⥤ Type v) {i j k : J} (s : Set (F.obj i))
| Mathlib/CategoryTheory/CofilteredSystem.lean | 114 | 120 |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | @[simp] theorem derivative_inv {R} [CommRing R] (f : R⟦X⟧ˣ) :
d⁄dX R ↑f⁻¹ = -(↑f⁻¹ : R⟦X⟧) ^ 2 * d⁄dX R f := by
apply Derivation.leibniz_of_mul_eq_one
simp
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 153 | 156 |
/-
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.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | theorem one_toPGame_moveLeft (x) : (toPGame 1).moveLeft x = toPGame 0 := by simp
| Mathlib/SetTheory/Game/Ordinal.lean | 96 | 97 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.Group.Prod
/-!
# Typeclasses for power-associative structures
In this... | theorem npow_mul (x : M) (m n : ℕ) : x ^ (m * n) = (x ^ m) ^ n := by
induction n with
| zero => rw [npow_zero, Nat.mul_zero, npow_zero]
| succ n ih => rw [mul_add, npow_add, ih, mul_one, npow_add, npow_one]
| Mathlib/Algebra/Group/NatPowAssoc.lean | 72 | 75 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
/-!
# Inverse functio... | f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
variable (f)
variable [CompleteSpace E]
/-- Given a function with an invertible strict derivative at `a`, returns a `PartialHomeomorph`
with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
The other par... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 101 | 108 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.GroupWithZero.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
im... | theorem toRingHom_comp_symm_toRingHom (e : R ≃+* S) :
e.toRingHom.comp e.symm.toRingHom = RingHom.id _ := by
ext
simp
| Mathlib/Algebra/Ring/Equiv.lean | 780 | 783 |
/-
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... | @[simp]
theorem log_one : log 1 = 0 :=
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 99 | 100 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Pigeonhole
... | exact isMaximal_inf_right_of_isMaximal_sup hxb hyb
theorem second_iso_of_eq {x y a b : X} (hm : IsMaximal x a) (ha : x ⊔ y = a) (hb : x ⊓ y = b) :
Iso (x, a) (b, y) := by substs a b; exact second_iso hm
| Mathlib/Order/JordanHolder.lean | 109 | 113 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
import Mathlib.Topology.Metrizable.Real
im... | choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
filter_upwards [this]
intro x hx
rw [tendsto_iff_dist_tendsto_zero]
exact squeeze_zero (fun n => dist_... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 87 | 101 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Li... | theorem cRank_one [StrongRankCondition R] [DecidableEq m] :
(cRank (1 : Matrix m m R)) = lift.{uR} #m := by
have := nontrivial_of_invariantBasisNumber R
have h : LinearIndependent R (1 : Matrix m m R)ᵀ := by
convert Pi.linearIndependent_single_one m R
simp [funext_iff, Matrix.one_eq_pi_single]
rw [cRa... | Mathlib/Data/Matrix/Rank.lean | 113 | 122 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | theorem mk_sum_compl {α} (s : Set α) : #s + #(sᶜ : Set α) = #α := by
classical
exact mk_congr (Equiv.Set.sumCompl s)
theorem mk_le_mk_of_subset {α} {s t : Set α} (h : s ⊆ t) : #s ≤ #t :=
⟨Set.embeddingOfSubset s t h⟩
theorem mk_le_iff_forall_finset_subset_card_le {α : Type u} {n : ℕ} {t : Set α} :
| Mathlib/SetTheory/Cardinal/Basic.lean | 800 | 807 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Equiv
/-! # Infinite sums in top... | `[GroupWithZero γ]` if there was such a thing as `DistribMulActionWithZero`. -/
lemma tsum_const_smul'' {γ : Type*} [DivisionSemiring γ] [Module γ α] [ContinuousConstSMul γ α]
[T2Space α] (g : γ) : ∑' (i : β), g • f i = g • ∑' (i : β), f i := by
rcases eq_or_ne g 0 with rfl | hg
· simp
· exact tsum_const_sm... | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 51 | 58 |
/-
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.Data.Finsupp.Basic
import Mathlib.Data.List.AList
/-!
# Connections between `Finsupp` and `AList`
## Main definitions
* `Fin... | · suffices f.toAList.lookup a = some (f a) by simp [h, this]
apply mem_lookup_iff.2
simpa using h
theorem lookupFinsupp_surjective : Function.Surjective (@lookupFinsupp α M _) := fun f =>
| Mathlib/Data/Finsupp/AList.lean | 116 | 120 |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
import Mathlib.Control.Basic
/-!
# `applicative` instances
This file provides `Applicative` instances for... | seq_assoc := Comp.seq_assoc
| Mathlib/Control/Applicative.lean | 102 | 103 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
/-!
# Derivatives of aff... | theorem hasStrictDerivAt : HasStrictDerivAt f (f.linear 1) x := by
rw [f.decomp]
exact f.linear.hasStrictDerivAt.add_const (f 0)
| Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 32 | 34 |
/-
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.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.MetricSpace.ProperSpace.Lemmas
/-!
# Eucl... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 131 | 136 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | simp only [h2, rpow_zero, ne_self_iff_false] at h1
rw [← sum_filter_of_ne h, ← prod_filter_of_ne h', geom_mean_eq_arith_mean_weighted_iff']
· simp
· simp +contextual [(hw _ _).gt_iff_ne]
| Mathlib/Analysis/MeanInequalities.lean | 243 | 246 |
/-
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... | (hs.finite_of_encard_le hts).eq_of_subset_of_encard_le' hst hts
| Mathlib/Data/Set/Card.lean | 225 | 226 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | theorem rowLens_length_ofRowLens {w : List ℕ} {hw : w.Sorted (· ≥ ·)} (hpos : ∀ x ∈ w, 0 < x) :
(ofRowLens w hw).rowLens.length = w.length := by
simp only [length_rowLens, colLen, Nat.find_eq_iff, mem_cells, mem_ofRowLens,
lt_self_iff_false, IsEmpty.exists_iff, Classical.not_not]
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 428 | 431 |
/-
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... | noncomputable
def lTensor.inverse :
Q ⊗[R] P →ₗ[R] Q ⊗[R] N ⧸ LinearMap.range (lTensor Q f) :=
lTensor.inverse_of_rightInverse Q hfg (Function.rightInverse_surjInv hg)
| Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 241 | 244 |
/-
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`... | theorem hermite_zero : hermite 0 = C 1 :=
rfl
theorem hermite_one : hermite 1 = X := by
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 59 | 62 |
/-
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.Complex
import Mathlib.Analysis.SpecialFunctions... | simp [deriv_zero_of_not_differentiableAt this, h, sq]
else (hasDerivAt_tan h).deriv
@[simp]
theorem contDiffAt_tan {x : ℂ} {n : WithTop ℕ∞} : ContDiffAt ℂ n tan x ↔ cos x ≠ 0 :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 65 | 69 |
/-
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 | 3,028 | 3,030 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.KanExtension.Basic
import Mathlib.CategoryTheory.Localization.Predicate
/-!
# Right derived functors
In this file, given a functor `F : ... |
variable (F)
| Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean | 175 | 177 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | @[simp]
| Mathlib/Order/SymmDiff.lean | 248 | 248 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | have := (hax.trans_eq hax').le.trans_eq hf
norm_num at this
· exact hy.symm
· -- case 2: `a ≥ a₁`
have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
have hxx₁ := h.mul_inv_x_pos hx₁ hy
have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
have hyy := h.mul_inv_y_nonneg hx₁ hy
| Mathlib/NumberTheory/Pell.lean | 606 | 613 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | Nat.prime_two
instance {p : ℕ+} [h : Fact p.Prime] : Fact (p : ℕ).Prime := h
| Mathlib/Data/PNat/Prime.lean | 103 | 106 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... |
/-- If a continuous linear map is constructed from a linear map via the constructor `mkContinuous`,
then its norm is bounded by the bound given to the constructor if it is nonnegative. -/
theorem mkContinuous_norm_le (f : E →ₛₗ[σ₁₂] F) {C : ℝ} (hC : 0 ≤ C) (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mkContinuous C h‖ ≤ C :=
| Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 397 | 401 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | ‖f z - f x - (fderiv 𝕜 f x) (z - x) - (f y - f x - (fderiv 𝕜 f x) (y - x))‖ := by
simp only [map_sub]; abel_nf
_ ≤ ‖f z - f x - (fderiv 𝕜 f x) (z - x)‖ + ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ :=
norm_sub_le _ _
_ ≤ δ * ‖z - x‖ + δ * ‖y - x‖ :=
add_le_add (hR _ (ball_subset_ball hr.2... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 163 | 182 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | LeftFractionRel φ ψ := by
constructor
· intro h
rw [← MorphismProperty.LeftFraction.Localization.map_eq_iff]
apply map_eq_of_map_eq _ _ _ _ h
· intro h
simp only [← Localization.Hom.map_mk _ L (Localization.inverts _ _)]
congr 1
exact Quot.sound h
lemma MorphismProperty.map_eq_iff_postc... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 739 | 750 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.