Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
/-!
#... | Mathlib/Order/Filter/CardinalInter.lean | 123 | 127 | theorem EventuallyLE.cardinal_bUnion {S : Set ι} (hS : #S < c)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_› := by |
simp only [biUnion_eq_iUnion]
exact EventuallyLE.cardinal_iUnion hS fun i => h i i.2
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 111 | 117 | theorem ClassGroup.Quot_mk_eq_mk (I : (FractionalIdeal R⁰ (FractionRing R))ˣ) :
Quot.mk _ I = ClassGroup.mk I := by |
rw [ClassGroup.mk, canonicalEquiv_self, RingEquiv.coe_monoidHom_refl, Units.map_id]
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [MonoidHom.comp_apply]
rw [MonoidHom.id_apply, QuotientGroup.mk'_apply]
rfl
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 149 | 153 | theorem trace_comp_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]
[Finite κ] (b : Basis ι R S) (c : Basis κ S T) :
(trace R S).comp ((trace S T).restrictScalars R) = trace R T := by |
ext
rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace_of_basis b c]
|
/-
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.RingTheory.FiniteType
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.finite... | Mathlib/RingTheory/FinitePresentation.lean | 163 | 183 | theorem iff_quotient_mvPolynomial' :
FinitePresentation R A ↔
∃ (ι : Type*) (_ : Fintype ι) (f : MvPolynomial ι R →ₐ[R] A),
Surjective f ∧ f.toRingHom.ker.FG := by |
constructor
· rintro ⟨n, f, hfs, hfk⟩
set ulift_var := MvPolynomial.renameEquiv R Equiv.ulift
refine
⟨ULift (Fin n), inferInstance, f.comp ulift_var.toAlgHom, hfs.comp ulift_var.surjective,
Ideal.fg_ker_comp _ _ ?_ hfk ulift_var.surjective⟩
erw [RingHom.ker_coe_equiv ulift_var.toRingEquiv... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 507 | 513 | theorem lift_unique (g : S →+* 𝕎 R) (g_compat : ∀ k, (WittVector.truncate k).comp g = f k) :
lift _ f_compat = g := by |
ext1 x
rw [← sub_eq_zero, ← Ideal.mem_bot, ← iInf_ker_truncate, Ideal.mem_iInf]
intro i
simp only [RingHom.mem_ker, g_compat, ← RingHom.comp_apply, truncate_comp_lift, RingHom.map_sub,
sub_self]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a4... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 428 | 431 | theorem ContinuousMultilinearMap.apply_zero_curry0 (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
ContinuousMultilinearMap.curry0 𝕜 G (f x) = f := by |
ext m
simp [Subsingleton.elim x m]
|
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 369 | 370 | theorem mul_nonneg_of_nonpos_of_nonpos (ha : a ≤ 0) (hb : b ≤ 0) : 0 ≤ a * b := by |
simpa only [zero_mul] using mul_le_mul_of_nonpos_right ha hb
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Function.Iterate
import Mathlib.GroupTheory.Perm.Basic
#align_import dynamics.fixed_points.basic from "leanpr... | Mathlib/Dynamics/FixedPoints/Basic.lean | 185 | 187 | theorem Commute.invOn_fixedPoints_comp (h : Commute f g) :
Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) := by |
simpa only [h.comp_eq] using Function.invOn_fixedPoints_comp f g
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 782 | 786 | theorem liftOn'_div {P : Sort v} (p q : K[X]) (f : K[X] → K[X] → P) (f0 : ∀ p, f p 0 = f 0 1)
(H) :
(RatFunc.liftOn' (algebraMap _ (RatFunc K) p / algebraMap _ _ q)) f @H = f p q := by |
rw [RatFunc.liftOn', liftOn_div _ _ _ f0]
apply liftOn_condition_of_liftOn'_condition H -- Porting note: `exact` did not work. Also,
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorS... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 68 | 72 | theorem mapRange_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R)
(f : R →+* S) : Finsupp.mapRange f f.map_zero p = map f p := by |
rw [p.as_sum, Finsupp.mapRange_finset_sum, map_sum (map f)]
refine Finset.sum_congr rfl fun n _ => ?_
rw [map_monomial, ← single_eq_monomial, Finsupp.mapRange_single, single_eq_monomial]
|
/-
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.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 81 | 85 | theorem IsCountablySpanning.prod {C : Set (Set α)} {D : Set (Set β)} (hC : IsCountablySpanning C)
(hD : IsCountablySpanning D) : IsCountablySpanning (image2 (· ×ˢ ·) C D) := by |
rcases hC, hD with ⟨⟨s, h1s, h2s⟩, t, h1t, h2t⟩
refine ⟨fun n => s n.unpair.1 ×ˢ t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), ?_⟩
rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]
|
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 75 | 88 | theorem irreducible_of_span_eq_maximalIdeal {R : Type*} [CommRing R] [LocalRing R] [IsDomain R]
(ϖ : R) (hϖ : ϖ ≠ 0) (h : maximalIdeal R = Ideal.span {ϖ}) : Irreducible ϖ := by |
have h2 : ¬IsUnit ϖ := show ϖ ∈ maximalIdeal R from h.symm ▸ Submodule.mem_span_singleton_self ϖ
refine ⟨h2, ?_⟩
intro a b hab
by_contra! h
obtain ⟨ha : a ∈ maximalIdeal R, hb : b ∈ maximalIdeal R⟩ := h
rw [h, mem_span_singleton'] at ha hb
rcases ha with ⟨a, rfl⟩
rcases hb with ⟨b, rfl⟩
rw [show a * ... |
/-
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.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 114 | 116 | theorem mul_charP_coeff_zero [CharP R p] (x : 𝕎 R) : (x * p).coeff 0 = 0 := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_zero,
zero_pow hp.out.ne_zero]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 212 | 216 | theorem rootD_parent (self : UnionFind) (x : Nat) : self.rootD (self.parent x) = self.rootD x := by |
simp [rootD, parent_lt]; split <;> simp [parentD, parentD_of_not_lt, *, -Array.get_eq_getElem]
(conv => rhs; rw [root]); split
· rw [root, dif_pos] <;> simp [*, -Array.get_eq_getElem]
· simp
|
/-
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.Logic.Equiv.TransferInstance
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb... | Mathlib/Algebra/Group/AddChar.lean | 329 | 330 | theorem mulShift_zero (ψ : AddChar R M) : mulShift ψ 0 = 1 := by |
ext; rw [mulShift_apply, zero_mul, map_zero_eq_one, one_apply]
|
/-
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.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 399 | 401 | theorem affineCombination_apply_const (w : ι → k) (p : P) (h : ∑ i ∈ s, w i = 1) :
s.affineCombination k (fun _ => p) w = p := by |
rw [affineCombination_apply, s.weightedVSubOfPoint_apply_const, h, one_smul, vsub_vadd]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprove... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 141 | 150 | theorem pow_natDegree_le_of_aeval_zero_of_monic_mem_map {x : S} (hx : aeval x f = 0)
(hmo : f.Monic) :
∀ i, (f.map (algebraMap R S)).natDegree ≤ i → x ^ i ∈ 𝓟.map (algebraMap R S) := by |
suffices x ^ (f.map (algebraMap R S)).natDegree ∈ 𝓟.map (algebraMap R S) by
intro i hi
obtain ⟨k, hk⟩ := exists_add_of_le hi
rw [hk, pow_add]
exact mul_mem_right _ _ this
rw [aeval_def, eval₂_eq_eval_map, ← IsRoot.def] at hx
exact pow_natDegree_le_of_root_of_monic_mem (hf.map _) hx (hmo.map _) _... |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.AbstractCompletion
#align_import topology.uniform_space.completion from "leanprover-community/mathlib"@"dc6c365e... | Mathlib/Topology/UniformSpace/Completion.lean | 162 | 171 | theorem uniformInducing_pureCauchy : UniformInducing (pureCauchy : α → CauchyFilter α) :=
⟨have : (preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen = id :=
funext fun s =>
Set.ext fun ⟨a₁, a₂⟩ => by simp [preimage, gen, pureCauchy, prod_principal_principal]
calc
comap (fun... | simp [this]
⟩
|
/-
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
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 878 | 885 | theorem ncard_le_ncard_diff_add_ncard (s t : Set α) (ht : t.Finite := by | toFinite_tac) :
s.ncard ≤ (s \ t).ncard + t.ncard := by
cases' s.finite_or_infinite with hs hs
· to_encard_tac
rw [ht.cast_ncard_eq, hs.cast_ncard_eq, (hs.diff _).cast_ncard_eq]
apply encard_le_encard_diff_add_encard
convert Nat.zero_le _
rw [hs.ncard]
|
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Chain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Set.Subsingleton
#align_... | Mathlib/Order/Ideal.lean | 268 | 270 | theorem top_of_top_mem (h : ⊤ ∈ I) : I = ⊤ := by |
ext
exact iff_of_true (I.lower le_top h) trivial
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Subsingleton
import Mathlib.Data.Int... | Mathlib/Algebra/Group/Subgroup/ZPowers.lean | 222 | 223 | theorem zpowers_le {g : G} {H : Subgroup G} : zpowers g ≤ H ↔ g ∈ H := by |
rw [zpowers_eq_closure, closure_le, Set.singleton_subset_iff, SetLike.mem_coe]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 616 | 619 | theorem mul_floor_map_zero_le_floor_iterate_zero (n : ℕ) : ↑n * ⌊f 0⌋ ≤ ⌊f^[n] 0⌋ := by |
rw [le_floor, Int.cast_mul, Int.cast_natCast, ← zero_add ((n : ℝ) * _)]
apply le_iterate_of_add_int_le_map
simp [floor_le]
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 95 | 95 | theorem eval₂_one : (1 : R[X]).eval₂ f x = 1 := by | rw [← C_1, eval₂_C, f.map_one]
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Star.Prod
#align_import alg... | Mathlib/Algebra/Star/StarAlgHom.lean | 580 | 581 | theorem snd_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : (snd R B C).comp (prod f g) = g := by |
ext; rfl
|
/-
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.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 101 | 117 | theorem hasDerivAt_polynomial_eval_inv_mul (p : ℝ[X]) (x : ℝ) :
HasDerivAt (fun x ↦ p.eval x⁻¹ * expNegInvGlue x)
((X ^ 2 * (p - derivative (R := ℝ) p)).eval x⁻¹ * expNegInvGlue x) x := by |
rcases lt_trichotomy x 0 with hx | rfl | hx
· rw [zero_of_nonpos hx.le, mul_zero]
refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_
filter_upwards [gt_mem_nhds hx] with y hy
rw [zero_of_nonpos hy.le, mul_zero]
· rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
refine ((tendsto... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Support
#align_import algebra.monoid_algebra.degree from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 270 | 272 | theorem apply_eq_zero_of_not_le_supDegree {a : A} (hlt : ¬ D a ≤ p.supDegree D) : p a = 0 := by |
contrapose! hlt
exact Finset.le_sup (Finsupp.mem_support_iff.2 hlt)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 2,033 | 2,035 | theorem lift_apply (F : Multiplicative G →* A) (f : MonoidAlgebra k G) :
lift k G A F f = f.sum fun a b => b • F (Multiplicative.ofAdd a) := by |
simp only [lift_apply', Algebra.smul_def]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matri... | Mathlib/Data/Matrix/Notation.lean | 393 | 396 | theorem submatrix_cons_row (A : Matrix m' n' α) (i : m') (row : Fin m → m') (col : o' → n') :
submatrix A (vecCons i row) col = vecCons (fun j => A i (col j)) (submatrix A row col) := by |
ext i j
refine Fin.cases ?_ ?_ i <;> simp [submatrix]
|
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 52 | 60 | theorem hasStrictDerivAt_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) :
HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y := by |
rcases em (x = 0) with (rfl | hx)
· replace h := h.neg_resolve_left rfl
rw [log_zero, mul_zero]
refine (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq ?_
exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm
· simpa only [cpow_def_of_ne_zero hx, mul_one] using
((hasStrictDe... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 180 | 192 | theorem weightedHomogeneousSubmodule_mul (w : σ → M) (m n : M) :
weightedHomogeneousSubmodule R w m * weightedHomogeneousSubmodule R w n ≤
weightedHomogeneousSubmodule R w (m + n) := by |
classical
rw [Submodule.mul_le]
intro φ hφ ψ hψ c hc
rw [coeff_mul] at hc
obtain ⟨⟨d, e⟩, hde, H⟩ := Finset.exists_ne_zero_of_sum_ne_zero hc
have aux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0 := by
contrapose! H
by_cases h : coeff d φ = 0 <;>
simp_all only [Ne, not_false_iff, zero_mul, mul_zero]
rw [... |
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
/-!
# `L^p` distance on products of two metric space... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 664 | 666 | theorem prod_edist_eq_of_L2 (x y : WithLp 2 (α × β)) :
edist x y = (edist x.fst y.fst ^ 2 + edist x.snd y.snd ^ 2) ^ (1 / 2 : ℝ) := by |
simp [prod_edist_eq_add]
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 241 | 243 | theorem logb_neg_iff (h : 0 < x) : logb b x < 0 ↔ x < 1 := by |
rw [← logb_one]
exact logb_lt_logb_iff hb h zero_lt_one
|
/-
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
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 497 | 499 | theorem ncard_eq_toFinset_card' (s : Set α) [Fintype s] :
s.ncard = s.toFinset.card := by |
simp [← Nat.card_coe_set_eq, Nat.card_eq_fintype_card]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 764 | 775 | theorem integral_sin_pow_mul_cos_pow_odd (m n : ℕ) :
(∫ x in a..b, sin x ^ m * cos x ^ (2 * n + 1)) = ∫ u in sin a..sin b, u^m * (↑1 - u ^ 2) ^ n :=
have hc : Continuous fun u : ℝ => u ^ m * (↑1 - u ^ 2) ^ n := by | continuity
calc
(∫ x in a..b, sin x ^ m * cos x ^ (2 * n + 1)) =
∫ x in a..b, sin x ^ m * (↑1 - sin x ^ 2) ^ n * cos x := by
simp only [_root_.pow_zero, _root_.pow_succ, mul_assoc, pow_mul, one_mul]
congr! 5
rw [← sq, ← sq, cos_sq']
_ = ∫ u in sin a..sin b, u ^ m * (1 - u ^ 2) ^ n :... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#a... | Mathlib/Geometry/Euclidean/Triangle.lean | 109 | 143 | theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by |
by_cases hxy : x = y
· rw [hxy, angle_self hy]
simp
· rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
app... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,634 | 1,639 | theorem IsBigO.mul_isLittleO {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (h₁ : f₁ =O[l] g₁) (h₂ : f₂ =o[l] g₂) :
(fun x => f₁ x * f₂ x) =o[l] fun x => g₁ x * g₂ x := by |
simp only [IsLittleO_def] at *
intro c cpos
rcases h₁.exists_pos with ⟨c', c'pos, hc'⟩
exact (hc'.mul (h₂ (div_pos cpos c'pos))).congr_const (mul_div_cancel₀ _ (ne_of_gt c'pos))
|
/-
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... | Mathlib/NumberTheory/Rayleigh.lean | 126 | 128 | theorem compl_beattySeq' {r s : ℝ} (hrs : r.IsConjExponent s) :
{beattySeq' r k | k}ᶜ = {beattySeq s k | k} := by |
rw [← compl_beattySeq hrs.symm, compl_compl]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 389 | 391 | theorem closedBall_sub_closedBall [ProperSpace E] (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (a b : E) :
closedBall a ε - closedBall b δ = closedBall (a - b) (ε + δ) := by |
rw [sub_eq_add_neg, neg_closedBall, closedBall_add_closedBall hε hδ, sub_eq_add_neg]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 298 | 301 | theorem StarConvex.affinity (hs : StarConvex 𝕜 x s) (z : E) (c : 𝕜) :
StarConvex 𝕜 (z + c • x) ((fun x => z + c • x) '' s) := by |
have h := (hs.smul c).add_left z
rwa [← image_smul, image_image] at h
|
/-
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.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
/-!
# Formal (multivariate) power series - Truncation
`MvPowerSeries... | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 71 | 73 | theorem coeff_trunc (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
(trunc R n φ).coeff m = if m < n then coeff R m φ else 0 := by |
classical simp [trunc, coeff_truncFun]
|
/-
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
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 389 | 393 | theorem add_tsub_add_eq_tsub_right (a c b : α) : a + c - (b + c) = a - b := by |
refine add_tsub_add_le_tsub_right.antisymm (tsub_le_iff_right.2 <| ?_)
apply le_of_add_le_add_right
rw [add_assoc]
exact le_tsub_add
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.NormedSpace.Completion
#align_import ... | Mathlib/Analysis/Complex/Liouville.lean | 111 | 117 | theorem apply_eq_apply_of_bounded {f : E → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
(z w : E) : f z = f w := by |
set g : ℂ → F := f ∘ fun t : ℂ => t • (w - z) + z
suffices g 0 = g 1 by simpa [g]
apply liouville_theorem_aux
exacts [hf.comp ((differentiable_id.smul_const (w - z)).add_const z),
hb.subset (range_comp_subset_range _ _)]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 726 | 732 | theorem inf_sup_left (a b c : α) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c :=
calc
a ⊓ (b ⊔ c) = a ⊓ (a ⊔ c) ⊓ (b ⊔ c) := by | rw [inf_sup_self]
_ = a ⊓ (a ⊓ b ⊔ c) := by simp only [inf_assoc, sup_inf_right, eq_self_iff_true]
_ = (a ⊔ a ⊓ b) ⊓ (a ⊓ b ⊔ c) := by rw [sup_inf_self]
_ = (a ⊓ b ⊔ a) ⊓ (a ⊓ b ⊔ c) := by rw [sup_comm]
_ = a ⊓ b ⊔ a ⊓ c := by rw [sup_inf_left]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 201 | 202 | theorem image_coe_Iic : (some : α → WithBot α) '' Iic a = Ioc (⊥ : WithBot α) a := by |
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe, inter_comm, Ioi_inter_Iic]
|
/-
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.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 673 | 676 | theorem modEq_iff_toIcoDiv_eq_toIocDiv_add_one :
a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1 := by |
rw [modEq_iff_toIcoMod_add_period_eq_toIocMod hp, toIcoMod, toIocMod, ← eq_sub_iff_add_eq,
sub_sub, sub_right_inj, ← add_one_zsmul, (zsmul_strictMono_left hp).injective.eq_iff]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 376 | 378 | theorem PointedMap.headI_map {Γ Γ'} [Inhabited Γ] [Inhabited Γ'] (f : PointedMap Γ Γ')
(l : List Γ) : (l.map f).headI = f l.headI := by |
cases l <;> [exact (PointedMap.map_pt f).symm; rfl]
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 213 | 215 | theorem adjugate_apply (A : Matrix n n α) (i j : n) :
adjugate A i j = (A.updateRow j (Pi.single i 1)).det := by |
rw [adjugate_def, of_apply, cramer_apply, updateColumn_transpose, det_transpose]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
/-!
# Power ... | Mathlib/RingTheory/PowerBasis.lean | 265 | 267 | theorem constr_pow_gen (pb : PowerBasis A S) {y : S'} (hy : aeval y (minpoly A pb.gen) = 0) :
pb.basis.constr A (fun i => y ^ (i : ℕ)) pb.gen = y := by |
convert pb.constr_pow_aeval hy X <;> rw [aeval_X]
|
/-
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.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-com... | Mathlib/SetTheory/Game/Birthday.lean | 64 | 78 | theorem lt_birthday_iff {x : PGame} {o : Ordinal} :
o < x.birthday ↔
(∃ i : x.LeftMoves, o ≤ (x.moveLeft i).birthday) ∨
∃ i : x.RightMoves, o ≤ (x.moveRight i).birthday := by |
constructor
· rw [birthday_def]
intro h
cases' lt_max_iff.1 h with h' h'
· left
rwa [lt_lsub_iff] at h'
· right
rwa [lt_lsub_iff] at h'
· rintro (⟨i, hi⟩ | ⟨i, hi⟩)
· exact hi.trans_lt (birthday_moveLeft_lt i)
· exact hi.trans_lt (birthday_moveRight_lt i)
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over fro... | Mathlib/RingTheory/Ideal/Over.lean | 139 | 149 | theorem comap_eq_of_scalar_tower_quotient [Algebra R S] [Algebra (R ⧸ p) (S ⧸ P)]
[IsScalarTower R (R ⧸ p) (S ⧸ P)] (h : Function.Injective (algebraMap (R ⧸ p) (S ⧸ P))) :
comap (algebraMap R S) P = p := by |
ext x
rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,
IsScalarTower.algebraMap_apply R (R ⧸ p) (S ⧸ P), Quotient.algebraMap_eq]
constructor
· intro hx
exact (injective_iff_map_eq_zero (algebraMap (R ⧸ p) (S ⧸ P))).mp h _ hx
· intro hx
rw [hx, RingHom... |
/-
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.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 426 | 428 | theorem affineCombination_vsub (w₁ w₂ : ι → k) (p : ι → P) :
s.affineCombination k p w₁ -ᵥ s.affineCombination k p w₂ = s.weightedVSub p (w₁ - w₂) := by |
rw [← AffineMap.linearMap_vsub, affineCombination_linear, vsub_eq_sub]
|
/-
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.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 299 | 303 | theorem bound_aux (hd : d ≠ 0) (hn : 2 ≤ n) :
(d ^ (n - 2 :) / n : ℝ) ≤ rothNumberNat ((2 * d - 1) ^ n) := by |
convert bound_aux' n d using 1
rw [cast_mul, cast_pow, mul_comm, ← div_div, pow_sub₀ _ _ hn, ← div_eq_mul_inv, cast_pow]
rwa [cast_ne_zero]
|
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 405 | 407 | theorem map_one_fst_of_isTwoCocycle {f : G × G → A} (hf : IsTwoCocycle f) (g : G) :
f (1, g) = f (1, 1) := by |
simpa only [one_smul, one_mul, mul_one, add_right_inj] using (hf 1 1 g).symm
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
#align_import number_theory.dioph from ... | Mathlib/NumberTheory/Dioph.lean | 338 | 372 | theorem DiophList.forall (l : List (Set <| α → ℕ)) (d : l.Forall Dioph) :
Dioph {v | l.Forall fun S : Set (α → ℕ) => v ∈ S} := by |
suffices ∃ (β : _) (pl : List (Poly (Sum α β))), ∀ v, List.Forall (fun S : Set _ => S v) l ↔
∃ t, List.Forall (fun p : Poly (Sum α β) => p (v ⊗ t) = 0) pl
from
let ⟨β, pl, h⟩ := this
⟨β, Poly.sumsq pl, fun v => (h v).trans <| exists_congr fun t => (Poly.sumsq_eq_zero _ _).symm⟩
induction' l w... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 404 | 406 | theorem multichoose_one_right (n : ℕ) : multichoose n 1 = n := by |
induction' n with n IH; · simp
simp [multichoose_succ_succ n 0, IH]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.modeq... | Mathlib/Data/Nat/ModEq.lean | 256 | 259 | theorem dvd_iff (h : a ≡ b [MOD m]) (hdm : d ∣ m) : d ∣ a ↔ d ∣ b := by |
simp only [← modEq_zero_iff_dvd]
replace h := h.of_dvd hdm
exact ⟨h.symm.trans, h.trans⟩
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 677 | 678 | theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by | simp
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connect... | Mathlib/Order/Interval/Set/OrdConnected.lean | 265 | 268 | theorem ordConnected_range {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) :
OrdConnected (range e) := by |
simp_rw [← image_univ]
exact ordConnected_image (e : α ≃o β)
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 786 | 789 | theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) :
G'.verts.toFinset.card = Fintype.card V := by |
simp only [isSpanning_iff.1 h, Set.toFinset_univ]
congr
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Algebra.Hom
#align_import algebra.hom.non_unital_alg from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
/-!
# Morphisms... | Mathlib/Algebra/Algebra/NonUnitalHom.lean | 227 | 228 | theorem mk_coe (f : A →ₛₙₐ[φ] B) (h₁ h₂ h₃ h₄) : (⟨⟨⟨f, h₁⟩, h₂, h₃⟩, h₄⟩ : A →ₛₙₐ[φ] B) = f := by |
rfl
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 495 | 496 | theorem dimH_preimage (e : E ≃L[𝕜] F) (s : Set F) : dimH (e ⁻¹' s) = dimH s := by |
rw [← e.image_symm_eq_preimage, e.symm.dimH_image]
|
/-
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.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,408 | 1,410 | theorem coe_sInf {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) :
↑(sInf s) = (⨅ a ∈ s, ↑a : WithTop α) := by |
rw [coe_sInf' hs h's, sInf_image]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 1,100 | 1,106 | theorem update_eq_single_add_erase (f : α →₀ M) (a : α) (b : M) :
f.update a b = single a b + f.erase a := by |
classical
ext j
rcases eq_or_ne a j with (rfl | h)
· simp
· simp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
#align_import data.finset.sups from "leanprover-community/mathlib"@"8818fde... | Mathlib/Data/Finset/Sups.lean | 542 | 542 | theorem disjSups_empty_left : ∅ ○ t = ∅ := by | simp [disjSups]
|
/-
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.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 904 | 907 | theorem wbtw_pointReflection (x y : P) : Wbtw R y x (pointReflection R x y) := by |
refine ⟨2⁻¹, ⟨by norm_num, by norm_num⟩, ?_⟩
rw [lineMap_apply, pointReflection_apply, vadd_vsub_assoc, ← two_smul R (x -ᵥ y)]
simp
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 117 | 120 | theorem restrict_extend_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f).restrict (extend f g g') = fun x => g x.coe_prop.choose := by |
classical
exact restrict_dite _ _
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"... | Mathlib/Analysis/NormedSpace/Ray.lean | 68 | 69 | theorem norm_injOn_ray_right (hy : y ≠ 0) : { x | SameRay ℝ x y }.InjOn norm := by |
simpa only [SameRay.sameRay_comm] using norm_injOn_ray_left hy
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Range
#align_import data.fin.vec_notation from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b... | Mathlib/Data/Fin/VecNotation.lean | 146 | 148 | theorem cons_val_succ' {i : ℕ} (h : i.succ < m.succ) (x : α) (u : Fin m → α) :
vecCons x u ⟨i.succ, h⟩ = u ⟨i, Nat.lt_of_succ_lt_succ h⟩ := by |
simp only [vecCons, Fin.cons, Fin.cases_succ']
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 857 | 862 | theorem map_singletonSubgraph (f : G →g G') {v : V} :
Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by |
ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply,
exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts,
singletonSubgraph_verts, Set.image_singleton]
exact False.elim
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92... | Mathlib/LinearAlgebra/LinearPMap.lean | 847 | 854 | theorem mem_graph_snd_inj (f : E →ₗ.[R] F) {x y : E} {x' y' : F} (hx : (x, x') ∈ f.graph)
(hy : (y, y') ∈ f.graph) (hxy : x = y) : x' = y' := by |
rw [mem_graph_iff] at hx hy
rcases hx with ⟨x'', hx1, hx2⟩
rcases hy with ⟨y'', hy1, hy2⟩
simp only at hx1 hx2 hy1 hy2
rw [← hx1, ← hy1, SetLike.coe_eq_coe] at hxy
rw [← hx2, ← hy2, hxy]
|
/-
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.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,215 | 1,216 | theorem mk_intCast (m : ℤ) : (mk m 1 : Localization M) = m := by |
simpa using mk_algebraMap (R := R) (A := ℤ) _
|
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
/-!
# Associ... | Mathlib/Data/List/AList.lean | 337 | 340 | theorem insert_insert_of_ne {a a'} {b : β a} {b' : β a'} (s : AList β) (h : a ≠ a') :
((s.insert a b).insert a' b').entries ~ ((s.insert a' b').insert a b).entries := by |
simp only [insert_entries]; rw [kerase_cons_ne, kerase_cons_ne, kerase_comm] <;>
[apply Perm.swap; exact h; exact h.symm]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "lean... | Mathlib/GroupTheory/SemidirectProduct.lean | 185 | 185 | theorem rightHom_comp_inl : (rightHom : N ⋊[φ] G →* G).comp inl = 1 := by | ext; simp [rightHom]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 103 | 104 | theorem coe_injective : @Function.Injective (NormedAddGroupHom V₁ V₂) (V₁ → V₂) toFun := by |
apply coe_inj
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomi... | Mathlib/Algebra/Polynomial/Taylor.lean | 56 | 59 | theorem taylor_zero' : taylor (0 : R) = LinearMap.id := by |
ext
simp only [taylor_apply, add_zero, comp_X, _root_.map_zero, LinearMap.id_comp,
Function.comp_apply, LinearMap.coe_comp]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 834 | 836 | theorem indicatorConstLp_empty :
indicatorConstLp p MeasurableSet.empty (by simp : μ ∅ ≠ ∞) c = 0 := by |
simp only [indicatorConstLp, Set.indicator_empty', Memℒp.toLp_zero]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264... | Mathlib/Algebra/MvPolynomial/Rename.lean | 272 | 279 | theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by |
obtain ⟨s, q, rfl⟩ := exists_finset_rename p
let n := Fintype.card { x // x ∈ s }
let e := Fintype.equivFin { x // x ∈ s }
refine ⟨n, (↑) ∘ e.symm, Subtype.val_injective.comp e.symm.injective, rename e q, ?_⟩
rw [← rename_rename, rename_rename e]
simp only [Function.comp, Equiv.symm_apply_apply, rename_ren... |
/-
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.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,123 | 1,124 | theorem iUnion_Ioc_intCast : ⋃ n : ℤ, Ioc (n : α) (n + 1) = Set.univ := by |
simpa only [zero_add] using iUnion_Ioc_add_intCast (0 : α)
|
/-
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.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 715 | 716 | theorem cast_succ' [AddMonoidWithOne α] (n) : (succ' n : α) = n + 1 := by |
rw [← PosNum.cast_to_nat, succ'_to_nat, Nat.cast_add_one, cast_to_nat]
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | Mathlib/Analysis/NormedSpace/Dual.lean | 82 | 84 | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by |
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
|
/-
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.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,218 | 1,231 | theorem swapTrue_mem_C1 (f : π (C1 C ho) (ord I · < o)) :
SwapTrue o f.val ∈ C1 C ho := by |
obtain ⟨f, g, hg, rfl⟩ := f
convert hg
dsimp (config := { unfoldPartialApp := true }) [SwapTrue]
ext i
split_ifs with h
· rw [ord_term ho] at h
simpa only [← h] using hg.2.symm
· simp only [Proj, ite_eq_left_iff, not_lt, @eq_comm _ false, ← Bool.not_eq_true]
specialize hsC g hg.1 i
intro h'
... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 57 | 66 | theorem infEdist_smul₀ {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E) :
EMetric.infEdist (c • x) (c • s) = ‖c‖₊ • EMetric.infEdist x s := by |
simp_rw [EMetric.infEdist]
have : Function.Surjective ((c • ·) : E → E) :=
Function.RightInverse.surjective (smul_inv_smul₀ hc)
trans ⨅ (y) (_ : y ∈ s), ‖c‖₊ • edist x y
· refine (this.iInf_congr _ fun y => ?_).symm
simp_rw [smul_mem_smul_set_iff₀ hc, edist_smul₀]
· have : (‖c‖₊ : ENNReal) ≠ 0 := by ... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.Nat.Choose.Sum
#align_import algebra.char_p.basic from... | Mathlib/Algebra/CharP/Basic.lean | 110 | 116 | theorem sub_pow_char_pow_of_commute [Ring R] {p : ℕ} [Fact p.Prime] [CharP R p] {n : ℕ} (x y : R)
(h : Commute x y) : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n := by |
induction n with
| zero => simp
| succ n n_ih =>
rw [pow_succ, pow_mul, pow_mul, pow_mul, n_ih]
apply sub_pow_char_of_commute; apply Commute.pow_pow h
|
/-
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.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFound... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 165 | 170 | theorem closure_inv (s : Set G) : closure s⁻¹ = (closure s)⁻¹ := by |
apply le_antisymm
· rw [closure_le, coe_inv, ← Set.inv_subset, inv_inv]
exact subset_closure
· rw [inv_le, closure_le, coe_inv, ← Set.inv_subset]
exact subset_closure
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 127 | 132 | theorem volume_emetric_closedBall (a : ℝ) (r : ℝ≥0∞) : volume (EMetric.closedBall a r) = 2 * r := by |
rcases eq_or_ne r ∞ with (rfl | hr)
· rw [EMetric.closedBall_top, volume_univ, two_mul, _root_.top_add]
· lift r to ℝ≥0 using hr
rw [Metric.emetric_closedBall_nnreal, volume_closedBall, two_mul, ← NNReal.coe_add,
ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 1,018 | 1,019 | theorem mk'_add (m₁ m₂ : M) (s : S) : mk' f (m₁ + m₂) s = mk' f m₁ s + mk' f m₂ s := by |
rw [mk'_add_mk', ← smul_add, mk'_cancel_left]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 468 | 486 | theorem traceMatrix_of_matrix_vecMul [Fintype κ] (b : κ → B) (P : Matrix κ κ A) :
traceMatrix A (b ᵥ* P.map (algebraMap A B)) = Pᵀ * traceMatrix A b * P := by |
ext (α β)
rw [traceMatrix_apply, vecMul, dotProduct, vecMul, dotProduct, Matrix.mul_apply,
BilinForm.sum_left,
Fintype.sum_congr _ _ fun i : κ =>
BilinForm.sum_right _ _ (b i * P.map (algebraMap A B) i α) fun y : κ =>
b y * P.map (algebraMap A B) y β,
sum_comm]
congr; ext x
rw [Matrix... |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.Probability.Kernel.Basic
/-!
# Independence with respect to a kernel and a measure
A family of sets of sets... | Mathlib/Probability/Independence/Kernel.lean | 242 | 248 | theorem IndepSets.iUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
{κ : kernel α Ω} {μ : Measure α} (hyp : ∀ n, IndepSets (s n) s' κ μ) :
IndepSets (⋃ n, s n) s' κ μ := by |
intro t1 t2 ht1 ht2
rw [Set.mem_iUnion] at ht1
cases' ht1 with n ht1
exact hyp n t1 t2 ht1 ht2
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,368 | 1,370 | theorem isBigO_one_iff : f =O[l] (fun _x => 1 : α → F) ↔
IsBoundedUnder (· ≤ ·) l fun x => ‖f x‖ := by |
simp only [isBigO_iff, norm_one, mul_one, IsBoundedUnder, IsBounded, eventually_map]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 571 | 574 | theorem one_div_pow_lt_one_div_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) :
1 / a ^ n < 1 / a ^ m := by |
refine (one_div_lt_one_div ?_ ?_).2 (pow_lt_pow_right a1 mn) <;>
exact pow_pos (zero_lt_one.trans a1) _
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 834 | 835 | theorem nonempty_of_card_le [Fintype α] [Fintype β] (h : Fintype.card α ≤ Fintype.card β) :
Nonempty (α ↪ β) := by | classical exact (truncOfCardLE h).nonempty
|
/-
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
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 303 | 307 | theorem nhdsWithin_prod {α : Type*} [TopologicalSpace α] {β : Type*} [TopologicalSpace β]
{s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by |
rw [nhdsWithin_prod_eq]
exact prod_mem_prod hu hv
|
/-
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... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 244 | 247 | theorem HasDerivWithinAt.comp_of_eq (hh₂ : HasDerivWithinAt h₂ h₂' s' y)
(hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s s') (hy : y = h x) :
HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x := by |
rw [hy] at hh₂; exact hh₂.comp x hh hst
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 2,098 | 2,100 | theorem lebesgue_number_lemma_of_metric_sUnion {s : Set α} {c : Set (Set α)} (hs : IsCompact s)
(hc₁ : ∀ t ∈ c, IsOpen t) (hc₂ : s ⊆ ⋃₀ c) : ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t := by |
rw [sUnion_eq_iUnion] at hc₂; simpa using lebesgue_number_lemma_of_metric hs (by simpa) hc₂
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 51 | 51 | theorem rdrop_zero : rdrop l 0 = l := by | simp [rdrop]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 247 | 247 | theorem iteratedDeriv_zero : iteratedDeriv 0 f = f := by | ext x; simp [iteratedDeriv]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
-/
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db8... | Mathlib/CategoryTheory/Monoidal/End.lean | 200 | 206 | theorem obj_ε_app (n : M) (X : C) :
(F.obj n).map (F.ε.app X) = (F.map (λ_ n).inv).app X ≫ (F.μIso (𝟙_ M) n).inv.app X := by |
refine Eq.trans ?_ (Category.id_comp _)
rw [← Category.assoc, ← IsIso.comp_inv_eq, ← IsIso.comp_inv_eq, Category.assoc]
convert left_unitality_app F n X
· simp
· simp
|
/-
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
#align_impo... | Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | 215 | 218 | theorem exists_measurable_superset_forall_eq [Countable ι] (μ : ι → Measure α) (s : Set α) :
∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = μ i s := by |
simpa only [← measure_eq_trim] using
OuterMeasure.exists_measurable_superset_forall_eq_trim (fun i => (μ i).toOuterMeasure) s
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 114 | 117 | theorem toReal_le_add' (hle : a ≤ b + c) (hb : b = ∞ → a = ∞) (hc : c = ∞ → a = ∞) :
a.toReal ≤ b.toReal + c.toReal := by |
refine le_trans (toReal_mono' hle ?_) toReal_add_le
simpa only [add_eq_top, or_imp] using And.intro hb hc
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | Mathlib/Order/CompactlyGenerated/Basic.lean | 264 | 275 | theorem wellFounded_characterisations : List.TFAE
[WellFounded ((· > ·) : α → α → Prop),
IsSupFiniteCompact α, IsSupClosedCompact α, ∀ k : α, IsCompactElement k] := by |
tfae_have 1 → 2
· exact WellFounded.isSupFiniteCompact α
tfae_have 2 → 3
· exact IsSupFiniteCompact.isSupClosedCompact α
tfae_have 3 → 1
· exact IsSupClosedCompact.wellFounded α
tfae_have 2 ↔ 4
· exact isSupFiniteCompact_iff_all_elements_compact α
tfae_finish
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.