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) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 612 | 617 | theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R)
(hI : Set.Pairwise (↑s) (IsCoprime on I)) :
(s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i ∈ s, I i} := by |
ext x
simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]
exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans 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, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 655 | 664 | theorem append_right_eq_snoc {α : Type*} {n : ℕ} (x : Fin n → α) (x₀ : Fin 1 → α) :
Fin.append x x₀ = Fin.snoc x (x₀ 0) := by |
ext i
refine Fin.addCases ?_ ?_ i <;> clear i
· intro i
rw [Fin.append_left]
exact (@snoc_castSucc _ (fun _ => α) _ _ i).symm
· intro i
rw [Subsingleton.elim i 0, Fin.append_right]
exact (@snoc_last _ (fun _ => α) _ _).symm
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,104 | 2,106 | theorem transpose_map {f : α → β} {M : Matrix m n α} : Mᵀ.map f = (M.map f)ᵀ := by |
ext
rfl
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 275 | 280 | theorem le_dist_coe (z w : ℍ) : w.im * (1 - Real.exp (-dist z w)) ≤ dist (z : ℂ) w :=
calc
w.im * (1 - Real.exp (-dist z w)) =
dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) := by |
rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]; ring
_ ≤ dist (z : ℂ) w := sub_le_iff_le_add.2 <| dist_triangle _ _ _
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 594 | 602 | theorem surj₂ (f : LocalizationMap S N) (z w : N) : ∃ z' w' : M, ∃ d : S,
(z * f.toMap d = f.toMap z') ∧ (w * f.toMap d = f.toMap w') := by |
let ⟨a, ha⟩ := surj f z
let ⟨b, hb⟩ := surj f w
refine ⟨a.1 * b.2, a.2 * b.1, a.2 * b.2, ?_, ?_⟩
· simp_rw [mul_def, map_mul, ← ha]
exact (mul_assoc z _ _).symm
· simp_rw [mul_def, map_mul, ← hb]
exact mul_left_comm w _ _
|
/-
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 | 111 | 114 | theorem gradedComm_symm : (gradedComm R 𝒜 ℬ).symm = gradedComm R ℬ 𝒜 := by |
rw [gradedComm, gradedComm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]
ext
rfl
|
/-
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 | 674 | 675 | theorem div_le_iff_of_neg' (hc : c < 0) : b / c ≤ a ↔ c * a ≤ b := by |
rw [mul_comm, div_le_iff_of_neg hc]
|
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 70 | 70 | theorem rightMoves_nim (o : Ordinal) : (nim o).RightMoves = o.out.α := by | rw [nim_def]; rfl
|
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_s... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 68 | 72 | theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n,
(⟪gramSchmidt 𝕜 f i, f n⟫ / (‖gramSchmidt 𝕜 f i‖ : 𝕜) ^ 2) • gramSchmidt 𝕜 f i := by |
convert gramSchmidt_def' 𝕜 f n
rw [orthogonalProjection_singleton, RCLike.ofReal_pow]
|
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 457 | 459 | theorem prod_union [DecidableEq α] (h : Disjoint s₁ s₂) :
∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := by |
rw [← prod_union_inter, disjoint_iff_inter_eq_empty.mp h]; exact (mul_one _).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.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.Interval.Set.UnorderedInterval
#align_import order.locally_finite from "... | Mathlib/Order/Interval/Finset/Defs.lean | 1,227 | 1,229 | theorem map_subtype_embedding_Iio : (Iio a).map (Embedding.subtype p) = (Iio a : Finset α) := by |
rw [subtype_Iio_eq]
exact Finset.subtype_map_of_mem fun x hx => hp (mem_Iio.1 hx).le a.prop
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 149 | 152 | theorem pairwise_filter (p : α → Prop) [DecidablePred p] {l : List α} :
Pairwise R (filter p l) ↔ Pairwise (fun x y => p x → p y → R x y) l := by |
rw [← filterMap_eq_filter, pairwise_filterMap]
simp
|
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,318 | 1,319 | theorem lTensor_id_apply (x : M ⊗[R] N) : (LinearMap.id : N →ₗ[R] N).lTensor M x = x := by |
rw [lTensor_id, id_coe, _root_.id]
|
/-
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 | 404 | 407 | theorem integral_zpow {n : ℤ} (h : 0 ≤ n ∨ n ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]]) :
∫ x in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1) := by |
replace h : -1 < (n : ℝ) ∨ (n : ℝ) ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]] := mod_cast h
exact mod_cast integral_rpow h
|
/-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_t... | Mathlib/GroupTheory/DoubleCoset.lean | 52 | 57 | theorem doset_eq_of_mem {H K : Subgroup G} {a b : G} (hb : b ∈ doset a H K) :
doset b H K = doset a H K := by |
obtain ⟨h, hh, k, hk, rfl⟩ := mem_doset.1 hb
rw [doset, doset, ← Set.singleton_mul_singleton, ← Set.singleton_mul_singleton, mul_assoc,
mul_assoc, Subgroup.singleton_mul_subgroup hk, ← mul_assoc, ← mul_assoc,
Subgroup.subgroup_mul_singleton hh]
|
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Integral
/-!
# Uniqueness of the conditional kernel
We prove that the conditional kernels `ProbabilityTheor... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 81 | 124 | theorem eq_condKernel_of_measure_eq_compProd (κ : kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by |
-- The idea is to transport the question to `ℝ` from `Ω` using `embeddingReal`
-- and then construct a measure on `α × ℝ`
let f := embeddingReal Ω
have hf := measurableEmbedding_embeddingReal Ω
set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def
have hρ' : ρ'.fst = ρ.fst := by
ext s hs
rw... |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 173 | 179 | theorem liftP_iff' {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (a : P.A) (f : P.B a ⟹ α) :
@LiftP.{u} _ P.Obj _ α p ⟨a, f⟩ ↔ ∀ i x, p (f i x) := by |
simp only [liftP_iff, Sigma.mk.inj_iff]; constructor
· rintro ⟨_, _, ⟨⟩, _⟩
assumption
· intro
repeat' first |constructor|assumption
|
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 562 | 571 | theorem oangle_sub_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
o.oangle (x - r • o.rotation (π / 2 : ℝ) x) x = Real.arctan r := by |
by_cases hr : r = 0; · simp [hr]
have hx : x = r⁻¹ • o.rotation (π / 2 : ℝ) (-(r • o.rotation (π / 2 : ℝ) x)) := by
simp [hr, ← Real.Angle.coe_add]
rw [sub_eq_add_neg, add_comm]
nth_rw 3 [hx]
nth_rw 2 [hx]
rw [o.oangle_add_left_smul_rotation_pi_div_two, inv_inv]
simpa [hr] using h
|
/-
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.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib... | Mathlib/Order/Chain.lean | 137 | 142 | theorem IsChain.exists3 (hchain : IsChain r s) [IsTrans α r] {a b c} (mem1 : a ∈ s) (mem2 : b ∈ s)
(mem3 : c ∈ s) : ∃ (z : _) (_ : z ∈ s), r a z ∧ r b z ∧ r c z := by |
rcases directedOn_iff_directed.mpr (IsChain.directed hchain) a mem1 b mem2 with ⟨z, mem4, H1, H2⟩
rcases directedOn_iff_directed.mpr (IsChain.directed hchain) z mem4 c mem3 with
⟨z', mem5, H3, H4⟩
exact ⟨z', mem5, _root_.trans H1 H3, _root_.trans H2 H3, H4⟩
|
/-
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 | 271 | 273 | theorem center_eq_infi' (S : Set G) (hS : closure S = ⊤) :
center G = ⨅ g : S, centralizer (zpowers (g : G)) := by |
rw [center_eq_iInf S hS, ← iInf_subtype'']
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 627 | 628 | theorem bit1_apply_ne (M : Matrix n n α) {i j : n} (h : i ≠ j) : (bit1 M) i j = bit0 (M i j) := by |
simp [bit1_apply, h]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Scott Morrison
-/
import Mathlib.CategoryTheory.Comma.Basic
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Category... | Mathlib/CategoryTheory/Comma/StructuredArrow.lean | 244 | 246 | theorem map_id {f : StructuredArrow S T} : (map (𝟙 S)).obj f = f := by |
rw [eq_mk f]
simp
|
/-
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 | 1,345 | 1,346 | theorem mul_mul_inv_cancel'_right (a b : G) : a * (b * a⁻¹) = b := by |
rw [← div_eq_mul_inv, mul_div_cancel a b]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 119 | 138 | theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
(hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 := by |
have H : IsLocalExtrOn T.reApplyInnerSelf {x : F | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀ := by
convert hextr
ext x
simp [dist_eq_norm]
-- find Lagrange multipliers for the function `T.re_apply_inner_self` and the
-- hypersurface-defining function `fun x ↦ ‖x‖ ^ 2`
obtain ⟨a, b, h₁, h₂⟩ :=
IsLocalExtrOn.exists... |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | Mathlib/Topology/Compactification/OnePoint.lean | 211 | 213 | theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) := by |
simp [isOpen_def, h]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 234 | 238 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by |
refine ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => ?_⟩⟩
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by omega), smul_smul, ← pow_succ',
tsub_add_eq_add_tsub hn]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 105 | 117 | theorem cycleType_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) :
cycleType l.attach.formPerm = {l.length} := by |
rw [← length_attach] at hn
rw [← nodup_attach] at hl
rw [cycleType_eq [l.attach.formPerm]]
· simp only [map, Function.comp_apply]
rw [support_formPerm_of_nodup _ hl, card_toFinset, dedup_eq_self.mpr hl]
· simp
· intro x h
simp [h, Nat.succ_le_succ_iff] at hn
· simp
· simpa using isCycle_f... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 918 | 920 | theorem preimage_inr_image_inl (s : Set α) : Sum.inr ⁻¹' (@Sum.inl α β '' s) = ∅ := by |
ext
simp
|
/-
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.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 102 | 105 | theorem mem_support_not_mem_vars_zero {f : MvPolynomial σ R} {x : σ →₀ ℕ} (H : x ∈ f.support)
{v : σ} (h : v ∉ vars f) : x v = 0 := by |
contrapose! h
exact (mem_vars v).mpr ⟨x, H, Finsupp.mem_support_iff.mpr h⟩
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 377 | 394 | theorem exists_linearIsometryEquiv_eq_of_det_pos {f : V ≃ₗᵢ[ℝ] V}
(hd : 0 < LinearMap.det (f.toLinearEquiv : V →ₗ[ℝ] V)) :
∃ θ : Real.Angle, f = o.rotation θ := by |
haveI : Nontrivial V :=
FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
use o.oangle x (f x)
apply LinearIsometryEquiv.toLinearEquiv_injective
apply LinearEquiv.toLinearMap_injective
apply (o.basisRightAngleRotation ... |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1d... | Mathlib/LinearAlgebra/Orientation.lean | 414 | 424 | theorem map_eq_neg_iff_det_neg (x : Orientation R M ι) (f : M ≃ₗ[R] M)
(h : Fintype.card ι = finrank R M) :
Orientation.map ι f x = -x ↔ LinearMap.det (f : M →ₗ[R] M) < 0 := by |
cases isEmpty_or_nonempty ι
· have H : finrank R M = 0 := h.symm.trans Fintype.card_eq_zero
simp [LinearMap.det_eq_one_of_finrank_eq_zero H, Module.Ray.ne_neg_self x]
have H : 0 < finrank R M := by
rw [← h]
exact Fintype.card_pos
haveI : FiniteDimensional R M := of_finrank_pos H
rw [map_eq_det_in... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ModularLattice
import Mathlib.Order.WellFounded
import Mathlib.Tactic.Nontriviality
#align_import order.atoms fr... | Mathlib/Order/Atoms.lean | 469 | 471 | theorem sSup_atoms_eq_top : sSup { a : α | IsAtom a } = ⊤ := by |
refine Eq.trans (congr rfl (Set.ext fun x => ?_)) (sSup_atoms_le_eq ⊤)
exact (and_iff_left le_top).symm
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.alg... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 127 | 129 | theorem mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f := by |
ext
rfl
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Complement
import Mathlib.GroupTheory.Sylow
import Mathlib.GroupTheory.Subgroup.Center
#align_import group_theory.transfer from "leanpro... | Mathlib/GroupTheory/Transfer.lean | 165 | 181 | theorem transfer_eq_pow [FiniteIndex H] (g : G)
(key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k) :
transfer ϕ g = ϕ ⟨g ^ H.index, transfer_eq_pow_aux g key⟩ := by |
classical
letI := H.fintypeQuotientOfFiniteIndex
change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key
rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_to_list]
refine (List.prod_map_hom _ _ _).trans ?_ -- Porting note: this used to be in the `r... |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 69 | 75 | theorem IsLindelof.induction_on (hs : IsLindelof s) {p : Set X → Prop}
(hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s)
(hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S))
(hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s := by |
let f : Filter X := ofCountableUnion p hcountable_union (fun t ht _ hsub ↦ hmono hsub ht)
have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using hnhds)
rwa [← compl_compl s]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
#align_import topology.sheaves.skyscrape... | Mathlib/Topology/Sheaves/Skyscraper.lean | 100 | 107 | theorem SkyscraperPresheafFunctor.map'_comp {a b c : C} (f : a ⟶ b) (g : b ⟶ c) :
SkyscraperPresheafFunctor.map' p₀ (f ≫ g) =
SkyscraperPresheafFunctor.map' p₀ f ≫ SkyscraperPresheafFunctor.map' p₀ g := by |
ext U
-- Porting note: change `simp` to `rw`
rw [NatTrans.comp_app]
simp only [SkyscraperPresheafFunctor.map'_app]
split_ifs with h <;> aesop_cat
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 870 | 871 | theorem leadingCoeff_C_mul_X (a : R) : leadingCoeff (C a * X) = a := by |
simpa only [pow_one] using leadingCoeff_C_mul_X_pow a 1
|
/-
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, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 942 | 944 | theorem insertNth_le_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} :
i.insertNth x p ≤ q ↔ x ≤ q i ∧ p ≤ fun j ↦ q (i.succAbove j) := by |
simp [Pi.le_def, forall_iff_succAbove i]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Operations
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,073 | 1,073 | theorem equivOfEq_rfl (S : Subalgebra R A) : equivOfEq S S rfl = AlgEquiv.refl := by | ext; rfl
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearF... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 142 | 156 | theorem isSymmetric_iff_inner_map_self_real (T : V →ₗ[ℂ] V) :
IsSymmetric T ↔ ∀ v : V, conj ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ := by |
constructor
· intro hT v
apply IsSymmetric.conj_inner_sym hT
· intro h x y
rw [← inner_conj_symm x (T y)]
rw [inner_map_polarization T x y]
simp only [starRingEnd_apply, star_div', star_sub, star_add, star_mul]
simp only [← starRingEnd_apply]
rw [h (x + y), h (x - y), h (x + Complex.I • y... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,652 | 1,653 | theorem map_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :
(p.append q).map f = (p.map f).append (q.map f) := by | induction p <;> simp [*]
|
/-
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.LinearAlgebra.QuadraticForm.IsometryEquiv
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.qua... | Mathlib/LinearAlgebra/QuadraticForm/Dual.lean | 52 | 72 | theorem separatingLeft_dualProd :
(dualProd R M).SeparatingLeft ↔ Function.Injective (Module.Dual.eval R M) := by |
classical
rw [separatingLeft_iff_ker_eq_bot, ker_eq_bot]
let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M
let h_d := e.symm.toLinearMap.comp (dualProd R M)
refine (Function.Injective.of_comp_iff e.symm.injective
(dualProd R M)).symm.trans ?_
rw [← LinearEquiv.c... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,764 | 1,766 | theorem map_dualCoannihilator (W : Subspace K (Dual K V)) [FiniteDimensional K V] :
W.dualCoannihilator.map (Dual.eval K V) = W.dualAnnihilator := by |
rw [← dualAnnihilator_dualAnnihilator_eq_map, dualCoannihilator_dualAnnihilator_eq]
|
/-
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.LinearAlgebra.PiTensorProduct
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.DirectSum.Algebra
#align_import linear_algebra.tensor_power from "leanpr... | Mathlib/LinearAlgebra/TensorPower.lean | 195 | 216 | theorem mul_assoc {na nb nc} (a : (⨂[R]^na) M) (b : (⨂[R]^nb) M) (c : (⨂[R]^nc) M) :
cast R M (add_assoc _ _ _) (a ₜ* b ₜ* c) = a ₜ* (b ₜ* c) := by |
let mul : ∀ n m : ℕ, ⨂[R]^n M →ₗ[R] (⨂[R]^m) M →ₗ[R] (⨂[R]^(n + m)) M := fun n m =>
(TensorProduct.mk R _ _).compr₂ ↑(mulEquiv : _ ≃ₗ[R] (⨂[R]^(n + m)) M)
-- replace `a`, `b`, `c` with `tprod R a`, `tprod R b`, `tprod R c`
let e : (⨂[R]^(na + nb + nc)) M ≃ₗ[R] (⨂[R]^(na + (nb + nc))) M := cast R M (add_assoc... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,551 | 1,554 | theorem image_injective : Injective (image f) ↔ Injective f := by |
refine ⟨fun h x x' hx => ?_, Injective.image_injective⟩
rw [← singleton_eq_singleton_iff]; apply h
rw [image_singleton, image_singleton, hx]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 26 | 27 | theorem then_eq_gt {o₁ o₂ : Ordering} : o₁.then o₂ = gt ↔ o₁ = gt ∨ o₁ = eq ∧ o₂ = gt := by |
cases o₁ <;> cases o₂ <;> decide
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,205 | 1,207 | theorem tendsto_mul_const_atTop_iff [NeBot l] :
Tendsto (fun x => f x * r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by |
simp only [mul_comm _ r, tendsto_const_mul_atTop_iff]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/m... | Mathlib/Topology/Order/LeftRightLim.lean | 201 | 208 | theorem continuousWithinAt_Iio_iff_leftLim_eq :
ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x := by |
rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
· simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h']
haveI : (𝓝[Iio x] x).NeBot := neBot_iff.2 h'
refine ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => ?_⟩
have := hf.tendsto_leftLim x
rwa [h] at this
|
/-
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, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 760 | 765 | theorem exists_not_mem_null_le_laverage (hμ : μ ≠ 0) (hf : AEMeasurable f μ) (hN : μ N = 0) :
∃ x, x ∉ N ∧ f x ≤ ⨍⁻ a, f a ∂μ := by |
have := measure_le_laverage_pos hμ hf
rw [← measure_diff_null hN] at this
obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
exact ⟨x, hxN, hx⟩
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
import Mathlib.RingTheory.Adjoin.... | Mathlib/RingTheory/IntegralClosure.lean | 166 | 173 | theorem isIntegral_iff_isIntegral_closure_finite {r : B} :
IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r := by |
constructor <;> intro hr
· rcases hr with ⟨p, hmp, hpr⟩
refine ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, ?_⟩
rw [← aeval_def, ← aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]
rcases hr with ⟨s, _, hsr⟩
exact hsr.of_subring _
|
/-
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.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,591 | 1,594 | theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c := by |
by_contra! h'
rcases lt_aleph0.1 h' with ⟨n, rfl⟩
exact not_isLimit_natCast n h
|
/-
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.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.NumberField.Discriminant
#align_import number_theory.cyclotomic.discriminant from... | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | 141 | 186 | theorem discr_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} K L] [hp : Fact (p : ℕ).Prime]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) (hirr : Irreducible (cyclotomic (↑(p ^ k) : ℕ) K)) :
discr K (hζ.powerBasis K).basis =
(-1) ^ ((p ^ k : ℕ).totient / 2) * p ^ ((p : ℕ) ^ (k - 1) * ((p - 1) * k - 1)) := by |
cases' k with k k
· simp only [coe_basis, _root_.pow_zero, powerBasis_gen _ hζ, totient_one, mul_zero, mul_one,
show 1 / 2 = 0 by rfl, discr, traceMatrix]
have hζone : ζ = 1 := by simpa using hζ
rw [hζ.powerBasis_dim _, hζone, ← (algebraMap K L).map_one,
minpoly.eq_X_sub_C_of_algebraMap_inj _ (... |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model... | Mathlib/ModelTheory/Encoding.lean | 67 | 98 | theorem listDecode_encode_list (l : List (L.Term α)) :
listDecode (l.bind listEncode) = l.map Option.some := by |
suffices h : ∀ (t : L.Term α) (l : List (Sum α (Σi, L.Functions i))),
listDecode (t.listEncode ++ l) = some t::listDecode l by
induction' l with t l lih
· rfl
· rw [cons_bind, h t (l.bind listEncode), lih, List.map]
intro t
induction' t with a n f ts ih <;> intro l
· rw [listEncode, singleton... |
/-
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 | 785 | 786 | theorem dirac_prod_dirac {x : α} {y : β} : (dirac x).prod (dirac y) = dirac (x, y) := by |
rw [prod_dirac, map_dirac measurable_prod_mk_right]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 59 | 61 | theorem antidiagonal_succ {n : ℕ} :
antidiagonal (n + 1) = (0, n + 1) ::ₘ (antidiagonal n).map (Prod.map Nat.succ id) := by |
simp only [antidiagonal, List.Nat.antidiagonal_succ, map_coe, cons_coe]
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 213 | 218 | theorem degree_le_degree (h : coeff q (natDegree p) ≠ 0) : degree p ≤ degree q := by |
by_cases hp : p = 0
· rw [hp, degree_zero]
exact bot_le
· rw [degree_eq_natDegree hp]
exact le_degree_of_ne_zero h
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 414 | 423 | theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)
(hw : w < b ^ u) : log b (b ^ u * v + w) = u := by |
have hne' := (opow_mul_add_pos (zero_lt_one.trans hb).ne' u hv w).ne'
by_contra! hne
cases' lt_or_gt_of_ne hne with h h
· rw [← lt_opow_iff_log_lt hb hne'] at h
exact h.not_le ((le_mul_left _ (Ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _))
· conv at h => change u < log b (b ^ u * v + w)
rw ... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 67 | 94 | theorem strictConcaveOn_log_Ioi : StrictConcaveOn ℝ (Ioi 0) log := by |
apply strictConcaveOn_of_slope_strict_anti_adjacent (convex_Ioi (0 : ℝ))
intro x y z (hx : 0 < x) (hz : 0 < z) hxy hyz
have hy : 0 < y := hx.trans hxy
trans y⁻¹
· have h : 0 < z - y := by linarith
rw [div_lt_iff h]
have hyz' : 0 < z / y := by positivity
have hyz'' : z / y ≠ 1 := by
contrapo... |
/-
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 | 183 | 184 | theorem preimage_coe_Ioi_bot : (some : α → WithBot α) ⁻¹' Ioi ⊥ = univ := by |
rw [← range_coe, preimage_range]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 58 | 61 | theorem finrank_eq_of_rank_eq {n : ℕ} (h : Module.rank R M = ↑n) : finrank R M = n := by |
apply_fun toNat at h
rw [toNat_natCast] at h
exact mod_cast h
|
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprov... | Mathlib/Algebra/CharP/MixedCharZero.lean | 344 | 352 | theorem split_by_characteristic (h_pos : ∀ p : ℕ, p ≠ 0 → CharP R p → P) (h_equal : Algebra ℚ R → P)
(h_mixed : ∀ p : ℕ, Nat.Prime p → MixedCharZero R p → P) : P := by |
cases CharP.exists R with
| intro p p_charP =>
by_cases h : p = 0
· rw [h] at p_charP
haveI h0 : CharZero R := CharP.charP_to_charZero R
exact split_equalCharZero_mixedCharZero R h_equal h_mixed
· exact h_pos p h p_charP
|
/-
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.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.Derivation.Basic
#align_import data.mv_polynomial.derivation from "leanprover-community/mat... | Mathlib/Algebra/MvPolynomial/Derivation.lean | 96 | 114 | theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
(∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
(monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by |
refine ⟨fun H p i => H _ _, fun H => ?_⟩
have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
intro p i
induction' p using MvPolynomial.induction_on' with s r p q hp hq
· rw [← mul_one r,... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 61 | 63 | theorem IsPrimitive.ne_zero [Nontrivial R] {p : R[X]} (hp : p.IsPrimitive) : p ≠ 0 := by |
rintro rfl
exact (hp 0 (dvd_zero (C 0))).ne_zero rfl
|
/-
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 | 221 | 224 | theorem exists_measurable_superset₂ (μ ν : Measure α) (s : Set α) :
∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s := by |
simpa only [Bool.forall_bool.trans and_comm] using
exists_measurable_superset_forall_eq (fun b => cond b μ ν) s
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,203 | 1,205 | theorem map_Ici (f : α ≃o β) (a : α) : map f (Ici a) = Ici (f a) := by |
ext
simp
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Basic
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 170 | 175 | theorem leftAdjointUniq_trans_app {F F' F'' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
(adj3 : F'' ⊣ G) (x : C) :
(leftAdjointUniq adj1 adj2).hom.app x ≫ (leftAdjointUniq adj2 adj3).hom.app x =
(leftAdjointUniq adj1 adj3).hom.app x := by |
rw [← leftAdjointUniq_trans adj1 adj2 adj3]
rfl
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Functor.Flat
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.cover_preserving fr... | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | 103 | 112 | theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
(x.functorPushforward G).Compatible := by |
rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
unfold FamilyOfElements.functorPushforward
rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
simp... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@... | Mathlib/Analysis/Calculus/Dslope.lean | 114 | 115 | theorem continuousAt_dslope_of_ne (h : b ≠ a) : ContinuousAt (dslope f a) b ↔ ContinuousAt f b := by |
simp only [← continuousWithinAt_univ, continuousWithinAt_dslope_of_ne h]
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 107 | 108 | theorem isBlock_singleton (a : X) : IsBlock G ({a} : Set X) := by |
simp [IsBlock.def, Classical.or_iff_not_imp_left]
|
/-
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 | 767 | 769 | theorem isLittleO_norm_right : (f =o[l] fun x => ‖g' x‖) ↔ f =o[l] g' := by |
simp only [IsLittleO_def]
exact forall₂_congr fun _ _ => isBigOWith_norm_right
|
/-
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 | 70 | 71 | theorem taylor_monomial (i : ℕ) (k : R) : taylor r (monomial i k) = C k * (X + C r) ^ i := by |
simp [taylor_apply]
|
/-
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.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 850 | 851 | theorem biSup_adjoin_simple : ⨆ x ∈ S, F⟮x⟯ = adjoin F S := by |
rw [← iSup_subtype'', ← gc.l_iSup, iSup_subtype'']; congr; exact S.biUnion_of_singleton
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Algebra.Module.Sub... | Mathlib/Algebra/Module/Submodule/Map.lean | 213 | 217 | theorem le_comap_pow_of_le_comap (p : Submodule R M) {f : M →ₗ[R] M} (h : p ≤ p.comap f) (k : ℕ) :
p ≤ p.comap (f ^ k) := by |
induction' k with k ih
· simp [LinearMap.one_eq_id]
· simp [LinearMap.iterate_succ, comap_comp, h.trans (comap_mono ih)]
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f... | Mathlib/Data/Multiset/Fintype.lean | 194 | 202 | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ := by |
ext ⟨y, i⟩
simp only [eq_comm, Finset.mem_filter, Multiset.mem_toEnumFinset, Finset.mem_map,
Finset.mem_range, Function.Embedding.coeFn_mk, Prod.mk.inj_iff, exists_prop,
exists_eq_right_right', and_congr_left_iff]
rintro rfl
rfl
|
/-
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.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 389 | 389 | theorem smul_eq_C_mul (a : R) : a • p = C a * p := by | simp [ext_iff]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 266 | 269 | theorem mapAccumr_eq_map_of_constant_state (f : α → σ → σ × β) (s : σ) (h : ∀ a, (f a s).fst = s) :
mapAccumr f xs s = (s, (map (fun x => (f x s).snd) xs)) := by |
clear ys
induction xs using revInductionOn <;> simp_all
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_th... | Mathlib/NumberTheory/SumTwoSquares.lean | 147 | 156 | theorem Nat.eq_sq_add_sq_of_isSquare_mod_neg_one {n : ℕ} (h : IsSquare (-1 : ZMod n)) :
∃ x y : ℕ, n = x ^ 2 + y ^ 2 := by |
induction' n using induction_on_primes with p n hpp ih
· exact ⟨0, 0, rfl⟩
· exact ⟨0, 1, rfl⟩
· haveI : Fact p.Prime := ⟨hpp⟩
have hp : IsSquare (-1 : ZMod p) := ZMod.isSquare_neg_one_of_dvd ⟨n, rfl⟩ h
obtain ⟨u, v, huv⟩ := Nat.Prime.sq_add_sq (ZMod.exists_sq_eq_neg_one_iff.mp hp)
obtain ⟨x, y, hx... |
/-
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.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
/-!
# Relations hol... | Mathlib/Data/Finset/Pairwise.lean | 44 | 48 | theorem PairwiseDisjoint.image_finset_of_le [DecidableEq ι] {s : Finset ι} {f : ι → α}
(hs : (s : Set ι).PairwiseDisjoint f) {g : ι → ι} (hf : ∀ a, f (g a) ≤ f a) :
(s.image g : Set ι).PairwiseDisjoint f := by |
rw [coe_image]
exact hs.image_of_le hf
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 565 | 566 | theorem coeff_one [Nontrivial S] (h : IsAdjoinRootMonic S f) : h.coeff 1 = Pi.single 0 1 := by |
rw [← h.coeff_root_pow h.deg_pos, pow_zero]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
import Mathlib.CategoryTheory.Limits.Preserves.Fil... | Mathlib/CategoryTheory/Functor/Flat.lean | 297 | 304 | theorem flat_iff_lan_flat (F : C ⥤ D) :
RepresentablyFlat F ↔ RepresentablyFlat (lan F.op : _ ⥤ Dᵒᵖ ⥤ Type u₁) :=
⟨fun H => inferInstance, fun H => by
haveI := preservesFiniteLimitsOfFlat (lan F.op : _ ⥤ Dᵒᵖ ⥤ Type u₁)
haveI : PreservesFiniteLimits F := by |
apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{u₁}
intros; apply preservesLimitOfLanPreservesLimit
apply flat_of_preservesFiniteLimits⟩
|
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functorial constructions
Lemmas on the interaction... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 218 | 220 | theorem Pi.hasProd {f : ι → ∀ x, π x} {g : ∀ x, π x} :
HasProd f g ↔ ∀ x, HasProd (fun i ↦ f i x) (g x) := by |
simp only [HasProd, tendsto_pi_nhds, prod_apply]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Spe... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 384 | 388 | theorem Complex.volume_ball (a : ℂ) (r : ℝ) :
volume (Metric.ball a r) = .ofReal r ^ 2 * NNReal.pi := by |
rw [InnerProductSpace.volume_ball a r, finrank_real_complex, Nat.cast_ofNat, div_self two_ne_zero,
one_add_one_eq_two, Real.Gamma_two, div_one, Real.sq_sqrt (by positivity),
← NNReal.coe_real_pi, ofReal_coe_nnreal]
|
/-
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.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 301 | 311 | theorem measurable_of_tendsto' {ι : Type*} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞} (u : Filter ι)
[NeBot u] [IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
Measurable g := by |
rcases u.exists_seq_tendsto with ⟨x, hx⟩
rw [tendsto_pi_nhds] at lim
have : (fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop) = g := by
ext1 y
exact ((lim y).comp hx).liminf_eq
rw [← this]
show Measurable fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop
exact measurable_liminf fun n => hf (x... |
/-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 308 | 311 | theorem sSup {ℐ : Set (Ideal A)} (h : ∀ I ∈ ℐ, Ideal.IsHomogeneous 𝒜 I) :
(sSup ℐ).IsHomogeneous 𝒜 := by |
rw [sSup_eq_iSup]
exact iSup₂ 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.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 183 | 185 | theorem matrix_vecEmpty_coeff {R} (i j) :
@coeff p R (Matrix.vecEmpty i) j = (Matrix.vecEmpty i : ℕ → R) j := by |
rcases i with ⟨_ | _ | _ | _ | i_val, ⟨⟩⟩
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Yury Kudryashov
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
#align_import geometry.manifold.diffeo... | Mathlib/Geometry/Manifold/Diffeomorph.lean | 468 | 470 | theorem uniqueDiffOn_image (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) (hn : 1 ≤ n) {s : Set E} :
UniqueDiffOn 𝕜 (h '' s) ↔ UniqueDiffOn 𝕜 s := by |
simp only [← uniqueMDiffOn_iff_uniqueDiffOn, uniqueMDiffOn_image, hn]
|
/-
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, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc... | Mathlib/Data/Real/Pi/Bounds.lean | 200 | 204 | theorem pi_lt_3141593 : π < 3.141593 := by |
pi_upper_bound
[27720 / 19601, 56935 / 30813, 49359 / 25163, 258754 / 130003, 113599 / 56868,
1101994 / 551163, 8671537 / 4336095, 3877807 / 1938940, 52483813 / 26242030,
56946167 / 28473117, 23798415 / 11899211]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,571 | 2,573 | theorem comap_neBot_iff_frequently {f : Filter β} {m : α → β} :
NeBot (comap m f) ↔ ∃ᶠ y in f, y ∈ range m := by |
simp only [comap_neBot_iff, frequently_iff, mem_range, @and_comm (_ ∈ _), exists_exists_eq_and]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/Algebra/Lie/Engel.lean | 192 | 216 | theorem LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer {K : LieSubalgebra R L}
(hK₁ : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R K) (hK₂ : K < K.normalizer) :
∃ (K' : LieSubalgebra R L), LieAlgebra.IsEngelian.{u₁, u₂, u₄} R K' ∧ K < K' := by |
obtain ⟨x, hx₁, hx₂⟩ := SetLike.exists_of_lt hK₂
let K' : LieSubalgebra R L :=
{ (R ∙ x) ⊔ (K : Submodule R L) with
lie_mem' := fun {y z} => LieSubalgebra.lie_mem_sup_of_mem_normalizer hx₁ }
have hxK' : x ∈ K' := Submodule.mem_sup_left (Submodule.subset_span (Set.mem_singleton _))
have hKK' : K ≤ K' ... |
/-
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.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.b... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 117 | 122 | theorem toMatrix_smul {R₁ S : Type*} [CommRing R₁] [Ring S] [Algebra R₁ S] [Fintype ι]
[DecidableEq ι] (x : S) (b : Basis ι R₁ S) (w : ι → S) :
(b.toMatrix (x • w)) = (Algebra.leftMulMatrix b x) * (b.toMatrix w) := by |
ext
rw [Basis.toMatrix_apply, Pi.smul_apply, smul_eq_mul, ← Algebra.leftMulMatrix_mulVec_repr]
rfl
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.LinearAlgebra.Dual
import Math... | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 91 | 97 | theorem dualMap_surjective_iff {W} [AddCommGroup W] [Module R W] [FiniteDimensional R W]
{f : W →ₗ[R] V} : Surjective (f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Injective f := by |
constructor <;> intro hf
· exact LinearMap.dualMap_surjective_iff.mp hf.of_comp
have := (separatingDual_iff_injective.mp ‹_›).comp hf
rw [← LinearMap.coe_comp] at this
exact LinearMap.flip_surjective_iff₁.mpr this
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 569 | 571 | theorem mem_rootSet {p : T[X]} {S : Type*} [CommRing S] [IsDomain S] [Algebra T S]
[NoZeroSMulDivisors T S] {a : S} : a ∈ p.rootSet S ↔ p ≠ 0 ∧ aeval a p = 0 := by |
rw [mem_rootSet', Polynomial.map_ne_zero_iff (NoZeroSMulDivisors.algebraMap_injective T S)]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 443 | 448 | theorem sq_add_sq (R : Type*) [CommRing R] [IsDomain R] (p : ℕ) [NeZero p] [CharP R p] (x : ℤ) :
∃ a b : ℕ, ((a : R) ^ 2 + (b : R) ^ 2) = x := by |
haveI := char_is_prime_of_pos R p
obtain ⟨a, b, hab⟩ := ZMod.sq_add_sq p x
refine ⟨a.val, b.val, ?_⟩
simpa using congr_arg (ZMod.castHom dvd_rfl R) hab
|
/-
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.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 411 | 411 | theorem neg_add {X Y : C} (a b : X ⟶ Y) : -(a + b) = -a - b := by | rw [add_def, neg_sub', add_neg]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying
-/
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathl... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 389 | 391 | theorem mem_pairSelfAdjointMatricesSubmodule' :
A ∈ pairSelfAdjointMatricesSubmodule J J₃ ↔ Matrix.IsAdjointPair J J₃ A A := by |
simp only [mem_pairSelfAdjointMatricesSubmodule]
|
/-
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.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.b... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 165 | 172 | theorem restrictScalars_toMatrix [Fintype ι] [DecidableEq ι] {S : Type*} [CommRing S] [Nontrivial S]
[Algebra R₂ S] [Module S M₂] [IsScalarTower R₂ S M₂] [NoZeroSMulDivisors R₂ S]
(b : Basis ι S M₂) (v : ι → span R₂ (Set.range b)) :
(algebraMap R₂ S).mapMatrix ((b.restrictScalars R₂).toMatrix v) =
b.t... |
ext
rw [RingHom.mapMatrix_apply, Matrix.map_apply, Basis.toMatrix_apply,
Basis.restrictScalars_repr_apply, Basis.toMatrix_apply]
|
/-
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 | 534 | 540 | theorem derivative_comp (p q : R[X]) :
derivative (p.comp q) = derivative q * p.derivative.comp q := by |
induction p using Polynomial.induction_on'
· simp [*, mul_add]
· simp only [derivative_pow, derivative_mul, monomial_comp, derivative_monomial, derivative_C,
zero_mul, C_eq_natCast, zero_add, RingHom.map_mul]
ring
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 388 | 389 | theorem IsCompl.bihimp_eq_bot {a b : α} (h : IsCompl a b) : a ⇔ b = ⊥ := by |
rw [h.eq_compl, compl_bihimp_self]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Algebra.Module.StrongTopology
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.strong_topology ... | Mathlib/Analysis/LocallyConvex/StrongTopology.lean | 47 | 54 | theorem locallyConvexSpace (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty)
(h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
LocallyConvexSpace R (UniformConvergenceCLM σ F 𝔖) := by |
apply LocallyConvexSpace.ofBasisZero _ _ _ _
(UniformConvergenceCLM.hasBasis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂
(LocallyConvexSpace.convex_basis_zero R F)) _
rintro ⟨S, V⟩ ⟨_, _, hVconvex⟩ f hf g hg a b ha hb hab x hx
exact hVconvex (hf x hx) (hg x hx) ha hb hab
|
/-
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.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 398 | 412 | theorem modByMonic_eq_zero_iff_dvd (hq : Monic q) : p %ₘ q = 0 ↔ q ∣ p :=
⟨fun h => by rw [← modByMonic_add_div p hq, h, zero_add]; exact dvd_mul_right _ _, fun h => by
nontriviality R
obtain ⟨r, hr⟩ := exists_eq_mul_right_of_dvd h
by_contra hpq0
have hmod : p %ₘ q = q * (r - p /ₘ q) := by | rw [modByMonic_eq_sub_mul_div _ hq, mul_sub, ← hr]
have : degree (q * (r - p /ₘ q)) < degree q := hmod ▸ degree_modByMonic_lt _ hq
have hrpq0 : leadingCoeff (r - p /ₘ q) ≠ 0 := fun h =>
hpq0 <|
leadingCoeff_eq_zero.1
(by rw [hmod, leadingCoeff_eq_zero.1 h, mul_zero, leadingCoeff_zero])
... |
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