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 |
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/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 971 | 974 | theorem nonpos_of_restrict_le_zero (hi₂ : v ≤[i] 0) : v i ≤ 0 := by |
by_cases hi₁ : MeasurableSet i
· exact (restrict_le_restrict_iff _ _ hi₁).1 hi₂ hi₁ Set.Subset.rfl
· rw [v.not_measurable hi₁]
|
/-
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.Extreme
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.Top... | Mathlib/Analysis/Convex/Exposed.lean | 176 | 178 | theorem inter_right (hC : IsExposed 𝕜 B C) (hCA : C ⊆ A) : IsExposed 𝕜 (A ∩ B) C := by |
rw [inter_comm]
exact hC.inter_left hCA
|
/-
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 Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 248 | 263 | theorem get?_inj
(h₀ : i < xs.length) (h₁ : Nodup xs) (h₂ : xs.get? i = xs.get? j) : i = j := by |
induction xs generalizing i j with
| nil => cases h₀
| cons x xs ih =>
match i, j with
| 0, 0 => rfl
| i+1, j+1 => simp; cases h₁ with
| cons ha h₁ => exact ih (Nat.lt_of_succ_lt_succ h₀) h₁ h₂
| i+1, 0 => ?_ | 0, j+1 => ?_
all_goals
simp at h₂
cases h₁; rename_i h' h
... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 277 | 281 | theorem mfderivWithin_fst {s : Set (M × M')} {x : M × M'}
(hxs : UniqueMDiffWithinAt (I.prod I') s x) :
mfderivWithin (I.prod I') I Prod.fst s x =
ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) := by |
rw [MDifferentiable.mfderivWithin (mdifferentiableAt_fst I I') hxs]; exact mfderiv_fst I I'
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Group.Hom.CompType... | Mathlib/GroupTheory/GroupAction/Hom.lean | 512 | 513 | theorem id_apply (x : A) : DistribMulActionHom.id M x = x := by |
rfl
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 130 | 154 | theorem mongePoint_eq_affineCombination_of_pointsWithCircumcenter {n : ℕ}
(s : Simplex ℝ P (n + 2)) :
s.mongePoint =
(univ : Finset (PointsWithCircumcenterIndex (n + 2))).affineCombination ℝ
s.pointsWithCircumcenter (mongePointWeightsWithCircumcenter n) := by |
rw [mongePoint_eq_smul_vsub_vadd_circumcenter,
centroid_eq_affineCombination_of_pointsWithCircumcenter,
circumcenter_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
← LinearMap.map_smul, weightedVSub_vadd_affineCombination]
congr with i
rw [Pi.add_apply, Pi.smul_apply, smul_eq... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,289 | 1,297 | theorem map_eq_singleton {f : α → β} {s : Multiset α} {b : β} :
map f s = {b} ↔ ∃ a : α, s = {a} ∧ f a = b := by |
constructor
· intro h
obtain ⟨a, ha⟩ : ∃ a, s = {a} := by rw [← card_eq_one, ← card_map, h, card_singleton]
refine ⟨a, ha, ?_⟩
rw [← mem_singleton, ← h, ha, map_singleton, mem_singleton]
· rintro ⟨a, rfl, rfl⟩
simp
|
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
/-... | Mathlib/Data/Real/Pi/Wallis.lean | 91 | 98 | theorem le_W (k : ℕ) : ((2 : ℝ) * k + 1) / (2 * k + 2) * (π / 2) ≤ W k := by |
rw [← le_div_iff pi_div_two_pos, div_eq_inv_mul (W k) _]
rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff (integral_sin_pow_pos _)]
convert integral_sin_pow_succ_le (2 * k + 1)
rw [integral_sin_pow (2 * k)]
simp only [sin_zero, ne_eq, add_eq_zero, and_false, not_false_eq_true, zero_pow, cos_zero,
... |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Submonoid.Basic
/-!
# Submonoid of opposite monoids
For every monoid `M`, we construct an... | Mathlib/Algebra/Group/Submonoid/MulOpposite.lean | 142 | 145 | theorem unop_closure (s : Set Mᵐᵒᵖ) : (closure s).unop = closure (MulOpposite.op ⁻¹' s) := by |
simp_rw [closure, unop_sInf, Set.preimage_setOf_eq, Submonoid.op_coe]
congr with a
exact MulOpposite.op_surjective.forall
|
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 350 | 355 | theorem mem_support_derivative [NoZeroSMulDivisors ℕ R] (p : R[X]) (n : ℕ) :
n ∈ (derivative p).support ↔ n + 1 ∈ p.support := by |
suffices ¬p.coeff (n + 1) * (n + 1 : ℕ) = 0 ↔ coeff p (n + 1) ≠ 0 by
simpa only [mem_support_iff, coeff_derivative, Ne, Nat.cast_succ]
rw [← nsmul_eq_mul', smul_eq_zero]
simp only [Nat.succ_ne_zero, false_or_iff]
|
/-
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.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 120 | 125 | theorem smeval_smul (r : R) : (r • p).smeval x = r • p.smeval x := by |
induction p using Polynomial.induction_on' with
| h_add p q ph qh =>
rw [smul_add, smeval_add, ph, qh, ← smul_add, smeval_add]
| h_monomial n a =>
rw [smul_monomial, smeval_monomial, smeval_monomial, smul_assoc]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geo... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 777 | 781 | theorem _root_.Wbtw.oangle_sign_eq_of_ne_left {p₁ p₂ p₃ : P} (p₄ : P) (h : Wbtw ℝ p₁ p₂ p₃)
(hne : p₁ ≠ p₂) : (∡ p₁ p₄ p₂).sign = (∡ p₁ p₄ p₃).sign :=
haveI hc : Collinear ℝ ({p₁, p₂, p₁, p₃} : Set P) := by |
simpa [Set.insert_comm p₂] using h.collinear
hc.oangle_sign_of_sameRay_vsub _ hne (h.left_ne_right_of_ne_left hne.symm) h.sameRay_vsub_left
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 270 | 273 | theorem imageSubobjectMap_comp_imageToKernel (p : α.right = β.left) :
imageToKernel f g w ≫ kernelSubobjectMap β = imageSubobjectMap α ≫ imageToKernel f' g' w' := by |
ext
simp [p]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 583 | 590 | theorem uniformity_lift_le_comp {f : Set (α × α) → Filter β} (h : Monotone f) :
((𝓤 α).lift fun s => f (s ○ s)) ≤ (𝓤 α).lift f :=
calc
((𝓤 α).lift fun s => f (s ○ s)) = ((𝓤 α).lift' fun s : Set (α × α) => s ○ s).lift f := by |
rw [lift_lift'_assoc]
· exact monotone_id.compRel monotone_id
· exact h
_ ≤ (𝓤 α).lift f := lift_mono comp_le_uniformity le_rfl
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 572 | 581 | theorem Filter.EventuallyEq.mdifferentiableWithinAt_iff (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
MDifferentiableWithinAt I I' f s x ↔ MDifferentiableWithinAt I I' f₁ s x := by |
constructor
· intro h
apply h.congr_of_eventuallyEq h₁ hx
· intro h
apply h.congr_of_eventuallyEq _ hx.symm
apply h₁.mono
intro y
apply Eq.symm
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,106 | 1,116 | theorem mk_perm_eq_self_power : #(Equiv.Perm α) = #α ^ #α :=
((mk_equiv_le_embedding α α).trans (mk_embedding_le_arrow α α)).antisymm <| by
suffices Nonempty ((α → Bool) ↪ Equiv.Perm (α × Bool)) by
obtain ⟨e⟩ : Nonempty (α ≃ α × Bool) := by |
erw [← Cardinal.eq, mk_prod, lift_uzero, mk_bool,
lift_natCast, mul_two, add_eq_self (aleph0_le_mk α)]
erw [← le_def, mk_arrow, lift_uzero, mk_bool, lift_natCast 2] at this
rwa [← power_def, power_self_eq (aleph0_le_mk α), e.permCongr.cardinal_eq]
refine ⟨⟨fun f ↦ Involutive.toPerm (f... |
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 782 | 786 | theorem finrank_vectorSpan_insert_le_set (s : Set P) (p : P) :
finrank k (vectorSpan k (insert p s)) ≤ finrank k (vectorSpan k s) + 1 := by |
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, direction_affineSpan]
refine (finrank_vectorSpan_insert_le _ _).trans (add_le_add_right ?_ _)
rw [direction_affineSpan]
|
/-
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 | 262 | 267 | theorem gcd_eq (h : a ≡ b [MOD m]) : gcd a m = gcd b m := by |
have h1 := gcd_dvd_right a m
have h2 := gcd_dvd_right b m
exact
dvd_antisymm (dvd_gcd ((h.dvd_iff h1).mp (gcd_dvd_left a m)) h1)
(dvd_gcd ((h.dvd_iff h2).mpr (gcd_dvd_left b m)) h2)
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | Mathlib/FieldTheory/SeparableClosure.lean | 100 | 103 | theorem separableClosure.comap_eq_of_algHom (i : E →ₐ[F] K) :
(separableClosure F K).comap i = separableClosure F E := by |
ext x
exact map_mem_separableClosure_iff i
|
/-
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 | 218 | 220 | theorem volume_Icc_pi_toReal {a b : ι → ℝ} (h : a ≤ b) :
(volume (Icc a b)).toReal = ∏ i, (b i - a i) := by |
simp only [volume_Icc_pi, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | Mathlib/Algebra/Divisibility/Units.lean | 187 | 188 | theorem IsRelPrime.of_dvd_left (h : IsRelPrime y z) (dvd : x ∣ y) : IsRelPrime x z := by |
obtain ⟨d, rfl⟩ := dvd; exact IsRelPrime.of_mul_left_left h
|
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Module.Pi
#align_import data.holor from "leanprover-community/mathlib"@"509de852e1de5... | Mathlib/Data/Holor.lean | 330 | 350 | theorem cprankMax_upper_bound [Ring α] : ∀ {ds}, ∀ x : Holor α ds, CPRankMax ds.prod x
| [], x => cprankMax_nil x
| d :: ds, x => by
have h_summands :
∀ i : { x // x ∈ Finset.range d },
CPRankMax ds.prod (unitVec d i.1 ⊗ slice x i.1 (mem_range.1 i.2)) :=
fun i => cprankMax_mul _ _ _ (cprankM... |
simp [Finset.card_range]
have :
CPRankMax (Finset.card (Finset.attach (Finset.range d)) * prod ds)
(∑ i ∈ Finset.attach (Finset.range d),
unitVec d i.val ⊗ slice x i.val (mem_range.1 i.2)) :=
cprankMax_sum (Finset.range d).attach _ fun i _ => h_summands i
have h_cprankMax_su... |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Martin Dvorak
-/
import Mathlib.Algebra.Order.Kleene
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.List.Join
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.... | Mathlib/Computability/Language.lean | 263 | 266 | theorem map_kstar (f : α → β) (l : Language α) : map f l∗ = (map f l)∗ := by |
rw [kstar_eq_iSup_pow, kstar_eq_iSup_pow]
simp_rw [← map_pow]
exact image_iUnion
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 659 | 661 | theorem spanSingleton_le_iff_mem {x : P} {I : FractionalIdeal S P} :
spanSingleton S x ≤ I ↔ x ∈ I := by |
rw [← coe_le_coe, coe_spanSingleton, Submodule.span_singleton_le_iff_mem, mem_coe]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 636 | 640 | theorem sup_sdiff_left (s : Finset ι) (f : ι → α) (a : α) :
(s.sup fun b => a \ f b) = a \ s.inf f := by |
induction s using Finset.cons_induction with
| empty => rw [sup_empty, inf_empty, sdiff_top]
| cons _ _ _ h => rw [sup_cons, inf_cons, h, sdiff_inf]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Reid Barton
-/
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types f... | Mathlib/CategoryTheory/Limits/Types.lean | 62 | 65 | theorem isLimit_iff_bijective_sectionOfCone (c : Cone F) :
Nonempty (IsLimit c) ↔ (Types.sectionOfCone c).Bijective := by |
simp_rw [isLimit_iff, Function.bijective_iff_existsUnique, Subtype.forall, F.sections_ext_iff,
sectionOfCone]
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.BilinearForm.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Al... | Mathlib/LinearAlgebra/BilinearForm/Hom.lean | 90 | 92 | theorem sum_apply {α} (t : Finset α) (B : α → BilinForm R M) (v w : M) :
(∑ i ∈ t, B i) v w = ∑ i ∈ t, B i v w := by |
simp only [coeFn_sum, Finset.sum_apply]
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 215 | 224 | theorem gradedMul_algebraMap (x : (⨁ i, 𝒜 i) ⊗[R] (⨁ i, ℬ i)) (r : R) :
gradedMul R 𝒜 ℬ x (algebraMap R _ r ⊗ₜ 1) = r • x := by |
suffices (gradedMul R 𝒜 ℬ).flip (algebraMap R _ r ⊗ₜ 1) = DistribMulAction.toLinearMap R _ r by
exact DFunLike.congr_fun this x
ext
dsimp
erw [tmul_of_gradedMul_of_tmul]
rw [mul_zero, uzpow_zero, one_smul, smul_tmul']
erw [mul_one, _root_.Algebra.smul_def, Algebra.commutes]
rfl
|
/-
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.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 171 | 172 | theorem singleton_mul_closedBall : {x} * closedBall y δ = closedBall (x * y) δ := by |
simp_rw [singleton_mul, ← smul_eq_mul, image_smul, smul_closedBall]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594b... | Mathlib/Algebra/Polynomial/BigOperators.lean | 66 | 77 | theorem degree_list_sum_le (l : List S[X]) : degree l.sum ≤ (l.map natDegree).maximum := by |
by_cases h : l.sum = 0
· simp [h]
· rw [degree_eq_natDegree h]
suffices (l.map natDegree).maximum = ((l.map natDegree).foldr max 0 : ℕ) by
rw [this]
simpa using natDegree_list_sum_le l
rw [← List.foldr_max_of_ne_nil]
· congr
contrapose! h
rw [List.map_eq_nil] at h
simp [h]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
Some proofs and docs came from `algebra/commute` (c) Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Units
#align_import a... | Mathlib/Algebra/Group/Semiconj/Units.lean | 48 | 51 | theorem units_inv_right {a : M} {x y : Mˣ} (h : SemiconjBy a x y) : SemiconjBy a ↑x⁻¹ ↑y⁻¹ :=
calc
a * ↑x⁻¹ = ↑y⁻¹ * (y * a) * ↑x⁻¹ := by | rw [Units.inv_mul_cancel_left]
_ = ↑y⁻¹ * a := by rw [← h.eq, mul_assoc, Units.mul_inv_cancel_right]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 727 | 734 | theorem zipWith_single_single (f : ∀ i, β₁ i → β₂ i → β i) (hf : ∀ i, f i 0 0 = 0)
{i} (b₁ : β₁ i) (b₂ : β₂ i) :
zipWith f hf (single i b₁) (single i b₂) = single i (f i b₁ b₂) := by |
ext j
rw [zipWith_apply]
obtain rfl | hij := Decidable.eq_or_ne i j
· rw [single_eq_same, single_eq_same, single_eq_same]
· rw [single_eq_of_ne hij, single_eq_of_ne hij, single_eq_of_ne hij, hf]
|
/-
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.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f1... | Mathlib/Data/List/NodupEquivFin.lean | 168 | 205 | theorem sublist_iff_exists_fin_orderEmbedding_get_eq {l l' : List α} :
l <+ l' ↔
∃ f : Fin l.length ↪o Fin l'.length,
∀ ix : Fin l.length, l.get ix = l'.get (f ix) := by |
rw [sublist_iff_exists_orderEmbedding_get?_eq]
constructor
· rintro ⟨f, hf⟩
have h : ∀ {i : ℕ}, i < l.length → f i < l'.length := by
intro i hi
specialize hf i
rw [get?_eq_get hi, eq_comm, get?_eq_some] at hf
obtain ⟨h, -⟩ := hf
exact h
refine ⟨OrderEmbedding.ofMapLEIff (fun... |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Mi... | Mathlib/FieldTheory/KummerExtension.lean | 230 | 235 | theorem X_pow_sub_C_irreducible_of_prime_pow
{p : ℕ} (hp : p.Prime) (hp' : p ≠ 2) (n : ℕ) {a : K} (ha : ∀ b : K, b ^ p ≠ a) :
Irreducible (X ^ (p ^ n) - C a) := by |
apply X_pow_sub_C_irreducible_of_odd (hp.odd_of_ne_two hp').pow
intros q hq hq'
simpa [(Nat.prime_dvd_prime_iff_eq hq hp).mp (hq.dvd_of_dvd_pow hq')] using ha
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Mario Carneiro, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 922 | 923 | theorem norm_le (C0 : (0 : ℝ) ≤ C) : ‖f‖ ≤ C ↔ ∀ x : α, ‖f x‖ ≤ C := by |
simpa using @dist_le _ _ _ _ f 0 _ C0
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 698 | 698 | theorem div_inv_eq_mul : a / b⁻¹ = a * b := by | simp
|
/-
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.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 294 | 300 | theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
bind₁ (xInTermsOfW p R) (W_ R k) = X k := by |
rw [wittPolynomial_eq_sum_C_mul_X_pow, AlgHom.map_sum]
simp only [Nat.cast_pow, AlgHom.map_pow, C_pow, AlgHom.map_mul, algHom_C]
rw [sum_range_succ_comm, tsub_self, pow_zero, pow_one, bind₁_X_right, mul_comm, ← C_pow,
xInTermsOfW_aux]
simp only [Nat.cast_pow, C_pow, bind₁_X_right, sub_add_cancel]
|
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Topology.Algebra.Order.Liminf... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 300 | 302 | theorem superpolynomialDecay_mul_param_pow_iff (hk : Tendsto k l atTop) (n : ℕ) :
SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f := by |
simpa [mul_comm f] using superpolynomialDecay_param_pow_mul_iff f hk n
|
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
#align_import algebra.grou... | Mathlib/Algebra/Order/Ring/Basic.lean | 177 | 178 | theorem pow_lt_self_of_lt_one (h₀ : 0 < a) (h₁ : a < 1) (hn : 1 < n) : a ^ n < a := by |
simpa only [pow_one] using pow_lt_pow_right_of_lt_one h₀ h₁ hn
|
/-
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.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
/-!
# Lexicographic order on... | Mathlib/Data/Sigma/Lex.lean | 151 | 162 | theorem lex_iff {a b : Σ' i, α i} :
Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by |
constructor
· rintro (⟨a, b, hij⟩ | ⟨i, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ h
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d... | Mathlib/Order/Interval/Set/Disjoint.lean | 60 | 61 | theorem Ici_disjoint_Iic : Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b := by |
rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff]
|
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,195 | 2,200 | theorem foldl_assoc : ∀ {l : List α} {a₁ a₂}, (l <*> a₁ ⋆ a₂) = a₁ ⋆ l <*> a₂
| [], a₁, a₂ => rfl
| a :: l, a₁, a₂ =>
calc
((a :: l) <*> a₁ ⋆ a₂) = l <*> a₁ ⋆ a₂ ⋆ a := by | simp only [foldl_cons, ha.assoc]
_ = a₁ ⋆ (a :: l) <*> a₂ := by rw [foldl_assoc, foldl_cons]
|
/-
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.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 119 | 160 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
(hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E) :
Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
(𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) := by |
let K := cthickening (‖v‖) (tsupport f)
have K_compact : IsCompact K := IsCompact.cthickening h'f
apply tendsto_integral_filter_of_dominated_convergence
(K.indicator (fun x ↦ (C * ‖v‖) * ‖g x‖))
· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeas... |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 897 | 899 | theorem support_C_mul_X_pow (n : ℕ) {c : R} (h : c ≠ 0) :
Polynomial.support (C c * X ^ n) = singleton n := by |
rw [C_mul_X_pow_eq_monomial, support_monomial n h]
|
/-
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 | 177 | 178 | theorem isLittleO_iff_forall_isBigOWith : f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g := by |
rw [IsLittleO_def]
|
/-
Copyright (c) 2022 Anand Rao, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anand Rao, Rémi Bottinelli
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_im... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 298 | 302 | theorem end_hom_mk_of_mk {s} (sec : s ∈ G.end) {K L : (Finset V)ᵒᵖ} (h : L ⟶ K) {v : V}
(vnL : v ∉ L.unop) (hs : s L = G.componentComplMk vnL) :
s K = G.componentComplMk (Set.not_mem_subset (le_of_op_hom h : _ ⊆ _) vnL) := by |
rw [← sec h, hs]
apply ComponentCompl.hom_mk _ (le_of_op_hom h : _ ⊆ _)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 599 | 617 | theorem sMod'_eq_sMod (p k : ℕ) (hp : 2 ≤ p) : (sMod' (2 ^ p - 1) k : ℤ) = sMod p k := by |
have h1 := calc
4 = 2 ^ 2 := by norm_num
_ ≤ 2 ^ p := Nat.pow_le_pow_of_le_right (by norm_num) hp
have h2 : 1 ≤ 2 ^ p := by omega
induction k with
| zero =>
rw [sMod', sMod, Int.ofNat_emod]
simp [h2]
| succ k ih =>
rw [sMod', sMod, ← ih]
have h3 : 2 ≤ 2 ^ p - 1 := by
zify [h2]
... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 378 | 379 | theorem isClosed_preimage (h : X ≃ₜ Y) {s : Set Y} : IsClosed (h ⁻¹' s) ↔ IsClosed s := by |
simp only [← isOpen_compl_iff, ← preimage_compl, isOpen_preimage]
|
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 300 | 304 | theorem eq_iSup_nat (m : Set X → ℝ≥0∞) : mkMetric' m = ⨆ n : ℕ, mkMetric'.pre m n⁻¹ := by |
ext1 s
rw [iSup_apply]
refine tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s)
(tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s)
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# The... | Mathlib/CategoryTheory/Comma/Arrow.lean | 162 | 163 | theorem hom.congr_left {f g : Arrow T} {φ₁ φ₂ : f ⟶ g} (h : φ₁ = φ₂) : φ₁.left = φ₂.left := by |
rw [h]
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | Mathlib/FieldTheory/SeparableClosure.lean | 309 | 310 | theorem sepDegree_self : sepDegree F F = 1 := by |
rw [sepDegree, Subsingleton.elim (separableClosure F F) ⊥, IntermediateField.rank_bot]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.t... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 187 | 191 | theorem thickenedIndicator_le_one {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) (x : α) :
thickenedIndicator δ_pos E x ≤ 1 := by |
rw [thickenedIndicator.coeFn_eq_comp]
simpa using (toNNReal_le_toNNReal thickenedIndicatorAux_lt_top.ne one_ne_top).mpr
(thickenedIndicatorAux_le_one δ E x)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 545 | 569 | theorem contMDiffOn_iff :
ContMDiffOn I I' n f s ↔
ContinuousOn f s ∧
∀ (x : M) (y : M'),
ContDiffOn 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
((extChartAt I x).target ∩
(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) := by |
constructor
· intro h
refine ⟨fun x hx => (h x hx).1, fun x y z hz => ?_⟩
simp only [mfld_simps] at hz
let w := (extChartAt I x).symm z
have : w ∈ s := by simp only [w, hz, mfld_simps]
specialize h w this
have w1 : w ∈ (chartAt H x).source := by simp only [w, hz, mfld_simps]
have w2 : f... |
/-
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.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 855 | 881 | theorem measure_toMeasurable_inter_of_sum {s : Set α} (hs : MeasurableSet s) {t : Set α}
{m : ℕ → Measure α} (hv : ∀ n, m n t ≠ ∞) (hμ : μ = sum m) :
μ (toMeasurable μ t ∩ s) = μ (t ∩ s) := by |
-- we show that there is a measurable superset of `t` satisfying the conclusion for any
-- measurable set `s`. It is built for each measure `mₙ` using `toMeasurable`
-- (which is well behaved for finite measure sets thanks to `measure_toMeasurable_inter`), and
-- then taking the intersection over `n`.
have A... |
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 65 | 69 | theorem Ico_add_bij : BijOn (· + d) (Ico a b) (Ico (a + d) (b + d)) := by |
rw [← Ici_inter_Iio, ← Ici_inter_Iio]
exact
(Ici_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx =>
lt_of_add_lt_add_right hx.2
|
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanpro... | Mathlib/ModelTheory/Satisfiability.lean | 366 | 380 | theorem models_iff_finset_models {φ : L.Sentence} :
T ⊨ᵇ φ ↔ ∃ T0 : Finset L.Sentence, (T0 : L.Theory) ⊆ T ∧ (T0 : L.Theory) ⊨ᵇ φ := by |
simp only [models_iff_not_satisfiable]
rw [← not_iff_not, not_not, isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]
push_neg
letI := Classical.decEq (Sentence L)
constructor
· intro h T0 hT0
simpa using h (T0 ∪ {Formula.not φ})
(by
simp only [Finset.coe_union, Finset.coe_si... |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Mario Carneiro, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 194 | 205 | theorem dist_lt_iff_of_compact [CompactSpace α] (C0 : (0 : ℝ) < C) :
dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by |
fconstructor
· intro w x
exact lt_of_le_of_lt (dist_coe_le_dist x) w
· by_cases h : Nonempty α
· exact dist_lt_of_nonempty_compact
· rintro -
convert C0
apply le_antisymm _ dist_nonneg'
rw [dist_eq]
exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempt... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 74 | 78 | theorem continuousAt_const_cpow' {a b : ℂ} (h : b ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b := by |
by_cases ha : a = 0
· rw [ha, continuousAt_congr (zero_cpow_eq_nhds h)]
exact continuousAt_const
· exact continuousAt_const_cpow ha
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 322 | 324 | theorem isLUB_le_iff (h : IsLUB s a) : a ≤ b ↔ b ∈ upperBounds s := by |
rw [h.upperBounds_eq]
rfl
|
/-
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
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 512 | 513 | theorem tanh_mul_I : tanh (x * I) = tan x * I := by |
rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.SetTheory.Car... | Mathlib/Combinatorics/Configuration.lean | 400 | 402 | theorem pointCount_eq_pointCount [Finite P] [Finite L] (l m : L) :
pointCount P l = pointCount P m := by |
apply lineCount_eq_lineCount (Dual P)
|
/-
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.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 325 | 326 | theorem coe_filter_univ (p : α → Prop) [DecidablePred p] :
(univ.filter p : Set α) = { x | p x } := by | simp
|
/-
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.BigOperators.Group.List
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.Group.Action.Defs
import Mathli... | Mathlib/Algebra/GradedMonoid.lean | 621 | 624 | theorem list_prod_ofFn_mem_graded {n} (i : Fin n → ι) (r : Fin n → R) (h : ∀ j, r j ∈ A (i j)) :
(List.ofFn r).prod ∈ A (List.ofFn i).sum := by |
rw [List.ofFn_eq_map, List.ofFn_eq_map]
exact list_prod_map_mem_graded _ _ _ fun _ _ => h _
|
/-
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.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.wi... | Mathlib/RingTheory/WittVector/IsPoly.lean | 384 | 409 | theorem ext [Fact p.Prime] {f g} (hf : IsPoly₂ p f) (hg : IsPoly₂ p g)
(h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
ghostComponent n (f x y) = ghostComponent n (g x y)) :
∀ (R) [_Rcr : CommRing R] (x y : 𝕎 R), f x y = g x y := by |
obtain ⟨φ, hf⟩ := hf
obtain ⟨ψ, hg⟩ := hg
intros
ext n
rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq' p φ ψ]
-- porting note: `clear x y` does not work, since `x, y` are now hygienic
intro k
apply MvPolynomial.funext
intro x
simp only [hom_bind₁]
specialize h (ULift ℤ) (mk p fun i => ⟨x (0, i)⟩) ... |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.Language
import Mathlib.Tactic.AdaptationNote
#align_import computability.regular_expressions from "leanprover-community/mathlib"@"369525b73f2... | Mathlib/Computability/RegularExpressions.lean | 254 | 297 | theorem mul_rmatch_iff (P Q : RegularExpression α) (x : List α) :
(P * Q).rmatch x ↔ ∃ t u : List α, x = t ++ u ∧ P.rmatch t ∧ Q.rmatch u := by |
induction' x with a x ih generalizing P Q
· rw [rmatch]; simp only [matchEpsilon]
constructor
· intro h
refine ⟨[], [], rfl, ?_⟩
rw [rmatch, rmatch]
rwa [Bool.and_eq_true_iff] at h
· rintro ⟨t, u, h₁, h₂⟩
cases' List.append_eq_nil.1 h₁.symm with ht hu
subst ht
subst ... |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 980 | 986 | theorem IsPathConnected.image' (hF : IsPathConnected F)
{f : X → Y} (hf : ContinuousOn f F) : IsPathConnected (f '' F) := by |
rcases hF with ⟨x, x_in, hx⟩
use f x, mem_image_of_mem f x_in
rintro _ ⟨y, y_in, rfl⟩
refine ⟨(hx y_in).somePath.map' ?_, fun t ↦ ⟨_, (hx y_in).somePath_mem t, rfl⟩⟩
exact hf.mono (range_subset_iff.2 (hx y_in).somePath_mem)
|
/-
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 | 830 | 833 | theorem integral_sin_pow_three :
∫ x in a..b, sin x ^ 3 = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3 := by |
have := @integral_sin_pow_odd_mul_cos_pow a b 1 0
norm_num at this; exact this
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 293 | 297 | theorem Submodule.exists_lieSubmodule_coe_eq_iff (p : Submodule R M) :
(∃ N : LieSubmodule R L M, ↑N = p) ↔ ∀ (x : L) (m : M), m ∈ p → ⁅x, m⁆ ∈ p := by |
constructor
· rintro ⟨N, rfl⟩ _ _; exact N.lie_mem
· intro h; use { p with lie_mem := @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.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 868 | 868 | theorem coe_eq_zero_iff : (φ : MvPowerSeries σ R) = 0 ↔ φ = 0 := by | rw [← coe_zero, coe_inj]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 926 | 933 | theorem isConnected_setOf_sOppSide {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s)
(h : (s : Set P).Nonempty) : IsConnected { y | s.SOppSide x y } := by |
obtain ⟨p, hp⟩ := h
haveI : Nonempty s := ⟨⟨p, hp⟩⟩
rw [setOf_sOppSide_eq_image2 hx hp, ← Set.image_prod]
refine (isConnected_Iio.prod (isConnected_iff_connectedSpace.2 ?_)).image _
((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn
convert AddTorsor.connectedSpace s.direction s
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 568 | 572 | theorem AlternatingMap.eq_smul_basis_det (f : M [⋀^ι]→ₗ[R] R) : f = f e • e.det := by |
refine Basis.ext_alternating e fun i h => ?_
let σ : Equiv.Perm ι := Equiv.ofBijective i (Finite.injective_iff_bijective.1 h)
change f (e ∘ σ) = (f e • e.det) (e ∘ σ)
simp [AlternatingMap.map_perm, Basis.det_self]
|
/-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
imp... | Mathlib/RingTheory/Noetherian.lean | 323 | 345 | theorem isNoetherian_iff_fg_wellFounded :
IsNoetherian R M ↔
WellFounded
((· > ·) : { N : Submodule R M // N.FG } → { N : Submodule R M // N.FG } → Prop) := by |
let α := { N : Submodule R M // N.FG }
constructor
· intro H
let f : α ↪o Submodule R M := OrderEmbedding.subtype _
exact OrderEmbedding.wellFounded f.dual (isNoetherian_iff_wellFounded.mp H)
· intro H
constructor
intro N
obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=
WellFounded.has_min H { N'... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 687 | 708 | theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg : ℂ → Real.Angle) x := by |
by_cases hs : x ∈ slitPlane
· exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
· rw [← Function.comp_id (((↑) : ℝ → Real.Angle) ∘ arg),
(Function.funext_iff.2 fun _ => (neg_neg _).symm : (id : ℂ → ℂ) = Neg.neg ∘ Neg.neg), ←
Function.comp.assoc]
refine ContinuousAt.comp ?_ c... |
/-
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 | 590 | 595 | theorem disjSups_assoc : ∀ s t u : Finset α, s ○ t ○ u = s ○ (t ○ u) := by |
refine associative_of_commutative_of_le disjSups_comm ?_
simp only [le_eq_subset, disjSups_subset_iff, mem_disjSups]
rintro s t u _ ⟨a, ha, b, hb, hab, rfl⟩ c hc habc
rw [disjoint_sup_left] at habc
exact ⟨a, ha, _, ⟨b, hb, c, hc, habc.2, rfl⟩, hab.sup_right habc.1, (sup_assoc ..).symm⟩
|
/-
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.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
/-!... | Mathlib/Order/Interval/Multiset.lean | 194 | 194 | theorem Ico_self : Ico a a = 0 := by | rw [Ico, Finset.Ico_self, Finset.empty_val]
|
/-
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.Topology.Algebra.Ring.Ideal
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed_space.units from "leanprover-community/mathl... | Mathlib/Analysis/NormedSpace/Units.lean | 200 | 202 | theorem inverse_add_norm_diff_first_order (x : Rˣ) :
(fun t : R => inverse (↑x + t) - ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ := by |
simpa using inverse_add_norm_diff_nth_order x 1
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.MeasureTheory.Integral.Asymptotics
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#al... | Mathlib/MeasureTheory/Integral/ExpDecay.lean | 30 | 36 | theorem exp_neg_integrableOn_Ioi (a : ℝ) {b : ℝ} (h : 0 < b) :
IntegrableOn (fun x : ℝ => exp (-b * x)) (Ioi a) := by |
have : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b)) := by
refine Tendsto.div_const (Tendsto.neg ?_) _
exact tendsto_exp_atBot.comp (tendsto_id.const_mul_atTop_of_neg (neg_neg_iff_pos.2 h))
refine integrableOn_Ioi_deriv_of_nonneg' (fun x _ => ?_) (fun x _ => (exp_pos _).le) this
simpa [h.ne'] ... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 241 | 244 | theorem cardinal_mk_eq_cardinal_mk_field_pow_rank (K V : Type u) [Ring K] [StrongRankCondition K]
[AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] :
#V = #K ^ Module.rank K V := by |
simpa using lift_cardinal_mk_eq_lift_cardinal_mk_field_pow_lift_rank K V
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.Fin... | Mathlib/FieldTheory/Finite/GaloisField.lean | 146 | 148 | theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n := by |
let b := IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)
rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,587 | 1,591 | theorem add_moveRight_inl {x : PGame} (y : PGame) (i) :
(x + y).moveRight (toRightMovesAdd (Sum.inl i)) = x.moveRight i + y := by |
cases x
cases y
rfl
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 606 | 618 | theorem LinearIndependent.group_smul {G : Type*} [hG : Group G] [DistribMulAction G R]
[DistribMulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : ι → M}
(hv : LinearIndependent R v) (w : ι → G) : LinearIndependent R (w • v) := by |
rw [linearIndependent_iff''] at hv ⊢
intro s g hgs hsum i
refine (smul_eq_zero_iff_eq (w i)).1 ?_
refine hv s (fun i => w i • g i) (fun i hi => ?_) ?_ i
· dsimp only
exact (hgs i hi).symm ▸ smul_zero _
· rw [← hsum, Finset.sum_congr rfl _]
intros
dsimp
rw [smul_assoc, smul_comm]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic.Abel
import Mathlib.Tactic.GCongr.Core
#align_import data.int.modeq from "leanprover-community/mathlib"@"47a1a73351de... | Mathlib/Data/Int/ModEq.lean | 237 | 238 | theorem of_div (h : a / c ≡ b / c [ZMOD m / c]) (ha : c ∣ a) (ha : c ∣ b) (ha : c ∣ m) :
a ≡ b [ZMOD m] := by | convert h.mul_left' <;> rwa [Int.mul_ediv_cancel']
|
/-
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.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.SpecialFunctions.... | Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | 95 | 97 | theorem HasStrictDerivAt.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : HasStrictDerivAt f f' x)
(h₂ : f x ∈ slitPlane) : HasStrictDerivAt (fun t => log (f t)) (f' / f x) x := by |
simpa only [div_eq_inv_mul] using (hasStrictFDerivAt_log_real h₂).comp_hasStrictDerivAt x h₁
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 904 | 908 | theorem eventually_filterAt_integrableOn (x : α) {f : α → E} (hf : LocallyIntegrable f μ) :
∀ᶠ a in v.filterAt x, IntegrableOn f a μ := by |
rcases hf x with ⟨w, w_nhds, hw⟩
filter_upwards [v.eventually_filterAt_subset_of_nhds w_nhds] with a ha
exact hw.mono_set ha
|
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 223 | 226 | theorem sizeUpTo_succ {i : ℕ} (h : i < c.length) :
c.sizeUpTo (i + 1) = c.sizeUpTo i + c.blocks.get ⟨i, h⟩ := by |
simp only [sizeUpTo]
rw [sum_take_succ _ _ h]
|
/-
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 | 246 | 253 | theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet)
(hv : v ∈ e) : v ∈ G'.verts := by |
revert hv
refine Sym2.ind (fun v w he ↦ ?_) e he
intro hv
rcases Sym2.mem_iff.mp hv with (rfl | rfl)
· exact G'.edge_vert he
· exact G'.edge_vert (G'.symm he)
|
/-
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.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theo... | Mathlib/MeasureTheory/Constructions/Pi.lean | 132 | 135 | theorem generateFrom_eq_pi [h : ∀ i, MeasurableSpace (α i)] {C : ∀ i, Set (Set (α i))}
(hC : ∀ i, generateFrom (C i) = h i) (h2C : ∀ i, IsCountablySpanning (C i)) :
generateFrom (pi univ '' pi univ C) = MeasurableSpace.pi := by |
simp only [← funext hC, generateFrom_pi_eq h2C]
|
/-
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.Opposite
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mat... | Mathlib/Algebra/Ring/Equiv.lean | 675 | 678 | theorem symm_toNonUnitalRingHom_comp_toNonUnitalRingHom (e : R ≃+* S) :
e.symm.toNonUnitalRingHom.comp e.toNonUnitalRingHom = NonUnitalRingHom.id _ := by |
ext
simp
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 373 | 377 | theorem expSeries_div_hasSum_exp_of_mem_ball [CompleteSpace 𝔸] (x : 𝔸)
(hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
HasSum (fun n => x ^ n / n !) (exp 𝕂 x) := by |
rw [← expSeries_apply_eq_div' (𝕂 := 𝕂) x]
exact expSeries_hasSum_exp_of_mem_ball x hx
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 1,248 | 1,251 | theorem natAbs_valMinAbs_neg {n : ℕ} (a : ZMod n) : (-a).valMinAbs.natAbs = a.valMinAbs.natAbs := by |
by_cases h2a : 2 * a.val = n
· rw [a.neg_eq_self_iff.2 (Or.inr h2a)]
· rw [valMinAbs_neg_of_ne_half h2a, Int.natAbs_neg]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.Algebra.CharP.Algebra
#align_import field_theory.splitting_field.construction from "leanprover-... | Mathlib/FieldTheory/SplittingField/Construction.lean | 69 | 72 | theorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by |
by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _
rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]
exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from... | Mathlib/Data/Finsupp/BigOperators.lean | 99 | 111 | theorem Multiset.support_sum_eq [AddCommMonoid M] (s : Multiset (ι →₀ M))
(hs : s.Pairwise (_root_.Disjoint on Finsupp.support)) :
s.sum.support = (s.map Finsupp.support).sup := by |
induction' s using Quot.inductionOn with a
obtain ⟨l, hl, hd⟩ := hs
suffices a.Pairwise (_root_.Disjoint on Finsupp.support) by
convert List.support_sum_eq a this
· simp only [Multiset.quot_mk_to_coe'', Multiset.sum_coe]
· dsimp only [Function.comp_def]
simp only [quot_mk_to_coe'', map_coe, sup... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 712 | 722 | theorem Field.span_map_pow_expChar_pow_eq_top_of_isSeparable [IsSeparable F E]
(h : Submodule.span F (Set.range v) = ⊤) :
Submodule.span F (Set.range (v · ^ q ^ n)) = ⊤ := by |
erw [← Algebra.top_toSubmodule, ← top_toSubalgebra, ← adjoin_univ,
adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable' F E _ q n,
adjoin_algebraic_toSubalgebra fun x _ ↦ Algebra.IsAlgebraic.isAlgebraic x,
Set.image_univ, Algebra.adjoin_eq_span, (powMonoidHom _).mrange.closure_eq]
refine (Submodule.span_mo... |
/-
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.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Order.Ring.Finset
import Mat... | Mathlib/Analysis/Normed/Field/Basic.lean | 1,102 | 1,103 | theorem nnnorm_norm [SeminormedAddCommGroup α] (a : α) : ‖‖a‖‖₊ = ‖a‖₊ := by |
rw [Real.nnnorm_of_nonneg (norm_nonneg a)]; rfl
|
/-
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.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#alig... | Mathlib/Analysis/PSeries.lean | 100 | 104 | theorem sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))) ≤ ∑ k ∈ Ico 2 (2 ^ n + 1), f k := by |
convert sum_schlomilch_le' hf (fun n => pow_pos zero_lt_two n)
(fun m n hm => pow_le_pow_right one_le_two hm) n using 2
simp [pow_succ, mul_two, two_mul]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span
import Mathlib.LinearAlgebra.BilinearMap
#align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"60... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 135 | 137 | theorem map₂_eq_span_image2 (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N) :
map₂ f p q = span R (Set.image2 (fun m n => f m n) (p : Set M) (q : Set N)) := by |
rw [← map₂_span_span, span_eq, span_eq]
|
/-
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.Logic.Equiv.List
import Mathlib.Logic.Function.Iterate
#align_import computability.primrec from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3b... | Mathlib/Computability/Primrec.lean | 1,530 | 1,532 | theorem prec' {n f g h} (hf : @Primrec' n f) (hg : @Primrec' n g) (hh : @Primrec' (n + 2) h) :
@Primrec' n fun v => (f v).rec (g v) fun y IH : ℕ => h (y ::ᵥ IH ::ᵥ v) := by |
simpa using comp' (prec hg hh) (hf.cons idv)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb537... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 271 | 278 | theorem restrictDvd_surjective (hpq : p ∣ q) (hq : q ≠ 0) :
Function.Surjective (restrictDvd hpq) := by |
classical
-- Porting note: was `simp only [restrictDvd_def, dif_neg hq, restrict_surjective]`
haveI := Fact.mk <|
splits_of_splits_of_dvd (algebraMap F q.SplittingField) hq (SplittingField.splits q) hpq
simp only [restrictDvd_def, dif_neg hq]
exact restrict_surjective _ _
|
/-
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.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 222 | 227 | theorem cycleType_le_of_mem_cycleFactorsFinset {f g : Perm α} (hf : f ∈ g.cycleFactorsFinset) :
f.cycleType ≤ g.cycleType := by |
have hf' := mem_cycleFactorsFinset_iff.1 hf
rw [cycleType_def, cycleType_def, hf'.left.cycleFactorsFinset_eq_singleton]
refine map_le_map ?_
simpa only [Finset.singleton_val, singleton_le, Finset.mem_val] using hf
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.... | Mathlib/GroupTheory/Finiteness.lean | 408 | 415 | theorem Group.rank_le_of_surjective [Group.FG G] [Group.FG G'] (f : G →* G')
(hf : Function.Surjective f) : Group.rank G' ≤ Group.rank G := by |
classical
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
trans (S.image f).card
· apply Group.rank_le
rw [Finset.coe_image, ← MonoidHom.map_closure, hS2, Subgroup.map_top_of_surjective f hf]
· exact Finset.card_image_le.trans_eq hS1
|
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