Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 106 | 106 | theorem singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by | simp [h]
|
/-
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.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 72 | 76 | theorem measure_biUnion_le {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α) :
μ (⋃ i ∈ I, s i) ≤ ∑' i : I, μ (s i) := by |
have := hI.to_subtype
rw [biUnion_eq_iUnion]
apply measure_iUnion_le
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 124 | 126 | theorem parent'_lt (self : UnionFind) (i : Fin self.size) :
(self.arr.get i).parent < self.size := by |
simp only [← parentD_eq, parentD_lt, Fin.is_lt, Array.data_length]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprove... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 127 | 140 | theorem eqId_iff_eq : A.EqId ↔ A.1 = Δ := by |
constructor
· intro h
dsimp at h
rw [h]
rfl
· intro h
rcases A with ⟨_, ⟨f, hf⟩⟩
simp only at h
subst h
refine ext _ _ rfl ?_
haveI := hf
simp only [eqToHom_refl, comp_id]
exact eq_id_of_epi f
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 99 | 107 | theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by |
refine le_antisymm ?_ ?_
· refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw [adjointAux_apply, LinearIsometryEquiv.norm_map]
exact toSesqForm_apply_norm_le
· nth_rw 1 [← adjointAux_adjointAux A]
refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw... |
/-
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.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
/-!
# Sober spaces
A quasi... | Mathlib/Topology/Sober.lean | 96 | 97 | theorem mem_closed_set_iff (h : IsGenericPoint x S) (hZ : IsClosed Z) : x ∈ Z ↔ S ⊆ Z := by |
rw [← h.def, hZ.closure_subset_iff, singleton_subset_iff]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connect... | Mathlib/Order/Interval/Set/OrdConnected.lean | 93 | 96 | theorem image_Ico (e : α ↪o β) (he : OrdConnected (range e)) (x y : α) :
e '' Ico x y = Ico (e x) (e y) := by |
rw [← e.preimage_Ico, image_preimage_eq_inter_range,
inter_eq_left.2 <| Ico_subset_Icc_self.trans <| he.out ⟨_, rfl⟩ ⟨_, rfl⟩]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.le... | Mathlib/Data/Int/Lemmas.lean | 64 | 67 | theorem natAbs_inj_of_nonpos_of_nonpos {a b : ℤ} (ha : a ≤ 0) (hb : b ≤ 0) :
natAbs a = natAbs b ↔ a = b := by |
simpa only [Int.natAbs_neg, neg_inj] using
natAbs_inj_of_nonneg_of_nonneg (neg_nonneg_of_nonpos ha) (neg_nonneg_of_nonpos hb)
|
/-
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.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
... | Mathlib/GroupTheory/Perm/Option.lean | 47 | 58 | theorem map_equiv_removeNone {α : Type*} [DecidableEq α] (σ : Perm (Option α)) :
(removeNone σ).optionCongr = swap none (σ none) * σ := by |
ext1 x
have : Option.map (⇑(removeNone σ)) x = (swap none (σ none)) (σ x) := by
cases' x with x
· simp
· cases h : σ (some _)
· simp [removeNone_none _ h]
· have hn : σ (some x) ≠ none := by simp [h]
have hσn : σ (some x) ≠ σ none := σ.injective.ne (by simp)
simp [removeNone... |
/-
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.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 46 | 48 | theorem Icc_mul_Icc_subset' (a b c d : α) : Icc a b * Icc c d ⊆ Icc (a * c) (b * d) := by |
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_le_mul' hya hzc, mul_le_mul' hyb hzd⟩
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 134 | 134 | theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by | simp only [mem_Icc, true_and_iff, le_rfl]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.Nat.Choose.Sum
#align_import algebra.char_p.basic from... | Mathlib/Algebra/CharP/Basic.lean | 98 | 101 | theorem add_pow_char_pow_of_commute [Semiring R] {p n : ℕ} [hp : Fact p.Prime] [CharP R p]
(x y : R) (h : Commute x y) : (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n := by |
let ⟨r, hr⟩ := h.exists_add_pow_prime_pow_eq hp.out n
simp [hr]
|
/-
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 | 39 | 58 | theorem strictConvexOn_exp : StrictConvexOn ℝ univ exp := by |
apply strictConvexOn_of_slope_strict_mono_adjacent convex_univ
rintro x y z - - hxy hyz
trans exp y
· have h1 : 0 < y - x := by linarith
have h2 : x - y < 0 := by linarith
rw [div_lt_iff h1]
calc
exp y - exp x = exp y - exp y * exp (x - y) := by rw [← exp_add]; ring_nf
_ = exp y * (1 - ... |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "lea... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 51 | 54 | theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :
∃ c : K, f.HasEigenvalue c := by |
simp_rw [hasEigenvalue_iff_mem_spectrum]
exact spectrum.nonempty_of_isAlgClosed_of_finiteDimensional K f
|
/-
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 | 118 | 123 | theorem volume_emetric_ball (a : ℝ) (r : ℝ≥0∞) : volume (EMetric.ball a r) = 2 * r := by |
rcases eq_or_ne r ∞ with (rfl | hr)
· rw [Metric.emetric_ball_top, volume_univ, two_mul, _root_.top_add]
· lift r to ℝ≥0 using hr
rw [Metric.emetric_ball_nnreal, volume_ball, two_mul, ← NNReal.coe_add,
ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 69 | 73 | theorem lift_comp_preservesLimitsIso_hom (t : Cone F) :
G.map (limit.lift _ t) ≫ (preservesLimitIso G F).hom =
limit.lift (F ⋙ G) (G.mapCone _) := by |
ext
simp [← G.map_comp]
|
/-
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.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.monoidal.center from "leanprover... | Mathlib/CategoryTheory/Monoidal/Center.lean | 97 | 98 | theorem ext {X Y : Center C} (f g : X ⟶ Y) (w : f.f = g.f) : f = g := by |
cases f; cases g; congr
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#a... | Mathlib/Data/Rat/Floor.lean | 86 | 87 | theorem cast_fract (x : ℚ) : (↑(fract x) : α) = fract (x : α) := by |
simp only [fract, cast_sub, cast_intCast, floor_cast]
|
/-
Copyright (c) 2023 Kyle Miller, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Rémi Bottinelli
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
/-!
# Connectivity of subgraphs and induced graphs
## Main definitions
* `SimpleG... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 73 | 78 | theorem subgraphOfAdj_connected {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).Connected := by |
refine ⟨⟨?_⟩⟩
rintro ⟨a, ha⟩ ⟨b, hb⟩
simp only [subgraphOfAdj_verts, Set.mem_insert_iff, Set.mem_singleton_iff] at ha hb
obtain rfl | rfl := ha <;> obtain rfl | rfl := hb <;>
first | rfl | (apply Adj.reachable; simp)
|
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.Associated
import Mathlib.NumberTheory.Divisors
#align_import algebra.is_prime_pow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0... | Mathlib/Algebra/IsPrimePow.lean | 38 | 42 | theorem not_isPrimePow_zero [NoZeroDivisors R] : ¬IsPrimePow (0 : R) := by |
simp only [isPrimePow_def, not_exists, not_and', and_imp]
intro x n _hn hx
rw [pow_eq_zero hx]
simp
|
/-
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.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Properties ... | Mathlib/Topology/LocalAtTarget.lean | 29 | 34 | theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
Inducing (s.restrictPreimage f) := by |
simp_rw [← inducing_subtype_val.of_comp_iff, inducing_iff_nhds, restrictPreimage,
MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h ⊢
intro a
rw [← h, ← inducing_subtype_val.nhds_eq_comap]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# The rank of a linear map
## Main Definit... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 91 | 98 | theorem rank_add_le (f g : V →ₗ[K] V') : rank (f + g) ≤ rank f + rank g :=
calc
rank (f + g) ≤ Module.rank K (LinearMap.range f ⊔ LinearMap.range g : Submodule K V') := by |
refine rank_le_of_submodule _ _ ?_
exact LinearMap.range_le_iff_comap.2 <| eq_top_iff'.2 fun x =>
show f x + g x ∈ (LinearMap.range f ⊔ LinearMap.range g : Submodule K V') from
mem_sup.2 ⟨_, ⟨x, rfl⟩, _, ⟨x, rfl⟩, rfl⟩
_ ≤ rank f + rank g := Submodule.rank_add_le_rank_add_rank _ _
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Si... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 123 | 129 | theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fintype.card α) :
IsCyclic α := by |
classical
use x
simp_rw [← SetLike.mem_coe, ← Set.eq_univ_iff_forall]
rw [← Fintype.card_congr (Equiv.Set.univ α), ← Fintype.card_zpowers] at hx
exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | Mathlib/Analysis/NormedSpace/Dual.lean | 114 | 124 | theorem norm_le_dual_bound (x : E) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ f : Dual 𝕜 E, ‖f x‖ ≤ M * ‖f‖) :
‖x‖ ≤ M := by |
classical
by_cases h : x = 0
· simp only [h, hMp, norm_zero]
· obtain ⟨f, hf₁, hfx⟩ : ∃ f : E →L[𝕜] 𝕜, ‖f‖ = 1 ∧ f x = ‖x‖ := exists_dual_vector 𝕜 x h
calc
‖x‖ = ‖(‖x‖ : 𝕜)‖ := RCLike.norm_coe_norm.symm
_ = ‖f x‖ := by rw [hfx]
_ ≤ M * ‖f‖ := hM f
_ = M := by rw ... |
/-
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.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 58 | 61 | theorem arcsin_projIcc (x : ℝ) :
arcsin (projIcc (-1) 1 (neg_le_self zero_le_one) x) = arcsin x := by |
rw [arcsin, Function.comp_apply, IccExtend_val, Function.comp_apply, IccExtend,
Function.comp_apply]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/FunctionField.lean | 67 | 75 | theorem germ_injective_of_isIntegral [IsIntegral X] {U : Opens X.carrier} (x : U) :
Function.Injective (X.presheaf.germ x) := by |
rw [injective_iff_map_eq_zero]
intro y hy
rw [← (X.presheaf.germ x).map_zero] at hy
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ x.prop x.prop _ _ hy
cases Subsingleton.elim iU iV
haveI : Nonempty W := ⟨⟨_, hW⟩⟩
exact map_injective_of_isIntegral X iU e
|
/-
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.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 124 | 130 | theorem Sized.induction {t} (hl : @Sized α t) {C : Ordnode α → Prop} (H0 : C nil)
(H1 : ∀ l x r, C l → C r → C (.node' l x r)) : C t := by |
induction t with
| nil => exact H0
| node _ _ _ _ t_ih_l t_ih_r =>
rw [hl.eq_node']
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2)
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.LightProfinite.Limits
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
/-!
# Effective epimorphisms in `LightProfinite`
... | Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean | 54 | 58 | theorem effectiveEpi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
EffectiveEpi f ↔ Function.Surjective f := by |
refine ⟨fun h ↦ ?_, fun h ↦ ⟨⟨EffectiveEpi.struct f h⟩⟩⟩
rw [← epi_iff_surjective]
infer_instance
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
/-!
# Lemmas about booleans
These are the lemmas about booleans w... | Mathlib/Init/Data/Bool/Lemmas.lean | 103 | 103 | theorem coe_sort_true : (true : Prop) = True := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 66 | 70 | theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSe... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 96 | 106 | theorem mk_one_pow_eq_mk_choose_add :
(mk 1 : S⟦X⟧) ^ (d + 1) = (mk fun n => Nat.choose (d + n) d : S⟦X⟧) := by |
induction d with
| zero => ext; simp
| succ d hd =>
ext n
rw [pow_add, hd, pow_one, mul_comm, coeff_mul]
simp_rw [coeff_mk, Pi.one_apply, one_mul]
norm_cast
rw [Finset.sum_antidiagonal_choose_add, ← Nat.choose_succ_succ, Nat.succ_eq_add_one,
add_right_comm]
|
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanp... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 143 | 149 | theorem uniformContinuous_extend : UniformContinuous (pkg.extend f) := by |
by_cases hf : UniformContinuous f
· rw [pkg.extend_def hf]
exact uniformContinuous_uniformly_extend pkg.uniformInducing pkg.dense hf
· change UniformContinuous (ite _ _ _)
rw [if_neg hf]
exact uniformContinuous_of_const fun a b => by congr 1
|
/-
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.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 75 | 81 | theorem HasProd.update (hf : HasProd f a₁) (b : β) [DecidableEq β] (a : α) :
HasProd (update f b a) (a / f b * a₁) := by |
convert (hasProd_ite_eq b (a / f b)).mul hf with b'
by_cases h : b' = b
· rw [h, update_same]
simp [eq_self_iff_true, if_true, sub_add_cancel]
· simp only [h, update_noteq, if_false, Ne, one_mul, not_false_iff]
|
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 144 | 145 | theorem transReflReparamAux_zero : transReflReparamAux 0 = 0 := by |
set_option tactic.skipAssignedInstances false in norm_num [transReflReparamAux]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 95 | 97 | theorem χ₄_nat_one_mod_four {n : ℕ} (hn : n % 4 = 1) : χ₄ n = 1 := by |
rw [χ₄_nat_mod_four, hn]
rfl
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 175 | 176 | theorem tail_think (s : Computation α) : tail (think s) = s := by |
cases' s with f al; apply Subtype.eq; dsimp [tail, think]
|
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
... | Mathlib/Data/Nat/Dist.lean | 60 | 60 | theorem dist_tri_left' (n m : ℕ) : n ≤ dist n m + m := by | rw [dist_comm]; apply dist_tri_left
|
/-
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.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/ZetaValues.lean | 74 | 77 | theorem antideriv_bernoulliFun (k : ℕ) (x : ℝ) :
HasDerivAt (fun x => bernoulliFun (k + 1) x / (k + 1)) (bernoulliFun k x) x := by |
convert (hasDerivAt_bernoulliFun (k + 1) x).div_const _ using 1
field_simp [Nat.cast_add_one_ne_zero k]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
-/
namespace List
variable {α β : Type*}
#align list.length_enum_from... | Mathlib/Data/List/Enum.lean | 63 | 70 | theorem mk_mem_enumFrom_iff_le_and_get?_sub {n i : ℕ} {x : α} {l : List α} :
(i, x) ∈ enumFrom n l ↔ n ≤ i ∧ l.get? (i - n) = x := by |
if h : n ≤ i then
rcases Nat.exists_eq_add_of_le h with ⟨i, rfl⟩
simp [mk_add_mem_enumFrom_iff_get?, Nat.add_sub_cancel_left]
else
have : ∀ k, n + k ≠ i := by rintro k rfl; simp at h
simp [h, mem_iff_get?, this]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.... | Mathlib/Analysis/Convex/Topology.lean | 109 | 112 | theorem closure_openSegment (x y : E) : closure (openSegment 𝕜 x y) = [x -[𝕜] y] := by |
rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)]
exact (image_closure_of_isCompact (isBounded_Ioo _ _).isCompact_closure <|
Continuous.continuousOn <| by continuity).symm
|
/-
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.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift... | Mathlib/Order/WithBot.lean | 135 | 136 | theorem unbot'_eq_iff {d y : α} {x : WithBot α} : unbot' d x = y ↔ x = y ∨ x = ⊥ ∧ y = d := by |
induction x <;> simp [@eq_comm _ d]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"... | Mathlib/RingTheory/Polynomial/Tower.lean | 54 | 56 | theorem aeval_algebraMap_apply (x : A) (p : R[X]) :
aeval (algebraMap A B x) p = algebraMap A B (aeval x p) := by |
rw [aeval_def, aeval_def, hom_eval₂, ← IsScalarTower.algebraMap_eq]
|
/-
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.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 92 | 93 | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := by |
rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub]
|
/-
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 | 93 | 94 | theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Algebra.Ring.NegOnePow
/-!
# Miscellaneous results about determinant
In this file, we collect var... | Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean | 21 | 47 | theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : ℕ}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : ∑ j, M j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.submatrix (Fin.succAbove j₁) id).det =
Int.negOnePow (j₁ - j₂) • (M.submatrix (Fin.succAbove j₂) id).det := by |
suffices ∀ j, (M.submatrix (Fin.succAbove j) id).det =
Int.negOnePow j • (M.submatrix (Fin.succAbove 0) id).det by
rw [this j₁, this j₂, smul_smul, ← Int.negOnePow_add, sub_add_cancel]
intro j
induction j using Fin.induction with
| zero => rw [Fin.val_zero, Nat.cast_zero, Int.negOnePow_zero, one_smul... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Algebra.Order.Ring.Cano... | Mathlib/Algebra/Order/Ring/WithTop.lean | 89 | 91 | theorem mul_lt_top' [LT α] {a b : WithTop α} (ha : a < ⊤) (hb : b < ⊤) : a * b < ⊤ := by |
rw [WithTop.lt_top_iff_ne_top] at *
simp only [Ne, mul_eq_top_iff, *, and_false, false_and, or_self, not_false_eq_true]
|
/-
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.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 85 | 85 | theorem gcd_self_add_left (m n : ℕ) : gcd (m + n) m = gcd n m := by | rw [add_comm, gcd_add_self_left]
|
/-
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.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 81 | 89 | theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R}
(hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]
apply le_of_eq_of_le _ (Set.IsWF.min_le_min_of_subset (support_add_subset (x := x) (y := y)))
· simp
· simp [hy]
· exact (Set.IsWF.min_union _ _ _ _).symm
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Logic.Equiv.Embedding
#align_import data.fintype.card_embedding from "leanprover-community/mathlib"@"98e83c3d... | Mathlib/Data/Fintype/CardEmbedding.lean | 36 | 50 | theorem card_embedding_eq {α β : Type*} [Fintype α] [Fintype β] [emb : Fintype (α ↪ β)] :
‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖ := by |
rw [Subsingleton.elim emb Embedding.fintype]
refine Fintype.induction_empty_option (P := fun t ↦ ‖t ↪ β‖ = ‖β‖.descFactorial ‖t‖)
(fun α₁ α₂ h₂ e ih ↦ ?_) (?_) (fun γ h ih ↦ ?_) α <;> dsimp only <;> clear! α
· letI := Fintype.ofEquiv _ e.symm
rw [← card_congr (Equiv.embeddingCongr e (Equiv.refl β)), ... |
/-
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.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 119 | 119 | theorem cylinder_zero (x : ∀ n, E n) : cylinder x 0 = univ := by | simp [cylinder_eq_pi]
|
/-
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.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce608... | Mathlib/MeasureTheory/Group/AddCircle.lean | 34 | 48 | theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =ᵐ[volume] ball x ε := by |
rcases le_or_lt ε 0 with hε | hε
· rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closedBall,
min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ENNReal.ofReal_eq_zero]
exact mul_nonpos_of_nonneg_of_nonpos zero_le_two hε
· suffices volume (closedBall x ε) ≤ volume (ball x ε) by
exact (ae_eq_of_subse... |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 181 | 184 | theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=
calc
coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _
_ = if m = n then a else 0 := by | simp only [Finsupp.unique_single_eq_iff]
|
/-
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.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
/-!
# Lemmas for Red-black trees
The main theorem in this file is `WF_def`, which shows that the ... | .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 51 | 52 | theorem reverse_eq_iff {t t' : RBNode α} : t.reverse = t' ↔ t = t'.reverse := by |
constructor <;> rintro rfl <;> simp
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 87 | 88 | theorem gold_pow_sub_gold_pow (n : ℕ) : φ ^ (n + 2) - φ ^ (n + 1) = φ ^ n := by |
rw [goldenRatio]; ring_nf; norm_num; ring
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 95 | 100 | theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
(Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom := by |
change (N₁Γ₀.app K).inv = _
simp only [N₁Γ₀_app]
rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 91 | 95 | theorem bernoulli'_spec' (n : ℕ) :
(∑ k ∈ antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by |
refine ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans ?_).trans (bernoulli'_spec n)
refine sum_congr rfl fun x hx => ?_
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
|
/-
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.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 74 | 78 | theorem memberSubfamily_inter (a : α) (𝒜 ℬ : Finset (Finset α)) :
(𝒜 ∩ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∩ ℬ.memberSubfamily a := by |
unfold memberSubfamily
rw [filter_inter_distrib, image_inter_of_injOn _ _ ((erase_injOn' _).mono _)]
simp
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 92 | 93 | theorem mul_star_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (x * star x) := by |
simpa only [star_star] using star_mul_self (star x)
|
/-
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.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.Unifo... | Mathlib/Topology/Instances/Real.lean | 77 | 78 | theorem Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop := by |
simp only [← comap_dist_right_atTop (0 : ℝ), Real.dist_eq, sub_zero, comap_abs_atTop]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8... | Mathlib/LinearAlgebra/SModEq.lean | 87 | 89 | theorem add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by |
rw [SModEq.def] at hxy₁ hxy₂ ⊢
simp_rw [Quotient.mk_add, hxy₁, hxy₂]
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 137 | 138 | theorem neg {x : R} (hx : IsSelfAdjoint x) : IsSelfAdjoint (-x) := by |
simp only [isSelfAdjoint_iff, star_neg, hx.star_eq]
|
/-
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.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 56 | 59 | theorem restrict_toOuterMeasure_eq_toOuterMeasure_restrict (h : MeasurableSet s) :
(μ.restrict s).toOuterMeasure = OuterMeasure.restrict s μ.toOuterMeasure := by |
simp_rw [restrict, restrictₗ, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,
toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, μ.trimmed]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
/-!
# Functions invariant under (quasi)ergodic map
In this file we prove tha... | Mathlib/Dynamics/Ergodic/Function.lean | 27 | 35 | theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
{s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
(h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
(hg_eq : g ∘ f =ᵐ[μ] g) :
∃ c, g =ᵐ[μ] const α c := by |
refine exists_eventuallyEq_const_of_eventually_mem_of_forall_separating MeasurableSet hs ?_
refine fun U hU ↦ h.ae_mem_or_ae_nmem₀ (s := g ⁻¹' U) (hgm hU) ?_b
refine (hg_eq.mono fun x hx ↦ ?_).set_eq
rw [← preimage_comp, mem_preimage, mem_preimage, hx]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Heather Macbeth
-/
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Math... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 88 | 91 | theorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A := by |
rw [polynomial_comp_attachBound]
apply SetLike.coe_mem
|
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
/-!
# Theorems about membershi... | Mathlib/Data/Vector/Mem.lean | 48 | 49 | theorem mem_cons_iff (v : Vector α n) : a' ∈ (a ::ᵥ v).toList ↔ a' = a ∨ a' ∈ v.toList := by |
rw [Vector.toList_cons, List.mem_cons]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
#align_import algebr... | Mathlib/Algebra/Ring/Semiconj.lean | 48 | 49 | theorem neg_right (h : SemiconjBy a x y) : SemiconjBy a (-x) (-y) := by |
simp only [SemiconjBy, h.eq, neg_mul, mul_neg]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e1... | Mathlib/FieldTheory/Minpoly/Basic.lean | 106 | 111 | theorem map_ne_one [Nontrivial B] {R : Type*} [Semiring R] [Nontrivial R] (f : A →+* R) :
(minpoly A x).map f ≠ 1 := by |
by_cases hx : IsIntegral A x
· exact mt ((monic hx).eq_one_of_map_eq_one f) (ne_one A x)
· rw [eq_zero hx, Polynomial.map_zero]
exact zero_ne_one
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathli... | Mathlib/RingTheory/Localization/NumDen.lean | 97 | 105 | theorem isInteger_of_isUnit_den {x : K} (h : IsUnit (den A x : A)) : IsInteger A x := by |
cases' h with d hd
have d_ne_zero : algebraMap A K (den A x) ≠ 0 :=
IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A x).2
use ↑d⁻¹ * num A x
refine _root_.trans ?_ (mk'_num_den A x)
rw [map_mul, map_units_inv, hd]
apply mul_left_cancel₀ d_ne_zero
rw [← mul_assoc, mul_inv_cancel d_ne_zero, ... |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 40 | 42 | theorem isometry_iff_nndist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, nndist (f x) (f y) = nndist x y := by |
simp only [Isometry, edist_nndist, ENNReal.coe_inj]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 88 | 92 | theorem realize_relabel {t : L.Term α} {g : α → β} {v : β → M} :
(t.relabel g).realize v = t.realize (v ∘ g) := by |
induction' t with _ n f ts ih
· rfl
· simp [ih]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 62 | 64 | theorem star_algebraMap (r : R) :
star (algebraMap R (CliffordAlgebra Q) r) = algebraMap R (CliffordAlgebra Q) r := by |
rw [star_def, involute.commutes, reverse.commutes]
|
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.Derangements.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Tactic.Ring
#align_imp... | Mathlib/Combinatorics/Derangements/Finite.lean | 95 | 104 | theorem card_derangements_fin_eq_numDerangements {n : ℕ} :
card (derangements (Fin n)) = numDerangements n := by |
induction' n using Nat.strong_induction_on with n hyp
rcases n with _ | _ | n
-- knock out cases 0 and 1
· rfl
· rfl
-- now we have n ≥ 2. rewrite everything in terms of card_derangements, so that we can use
-- `card_derangements_fin_add_two`
rw [numDerangements_add_two, card_derangements_fin_add_two, ... |
/-
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.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Interval.Finset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith
... | Mathlib/Algebra/BigOperators/Intervals.lean | 118 | 121 | theorem prod_eq_prod_Ico_succ_bot {a b : ℕ} (hab : a < b) (f : ℕ → M) :
∏ k ∈ Ico a b, f k = f a * ∏ k ∈ Ico (a + 1) b, f k := by |
have ha : a ∉ Ico (a + 1) b := by simp
rw [← prod_insert ha, Nat.Ico_insert_succ_left hab]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 128 | 141 | theorem det_mul_aux {M N : Matrix n n R} {p : n → n} (H : ¬Bijective p) :
(∑ σ : Perm n, ε σ * ∏ x, M (σ x) (p x) * N (p x) x) = 0 := by |
obtain ⟨i, j, hpij, hij⟩ : ∃ i j, p i = p j ∧ i ≠ j := by
rw [← Finite.injective_iff_bijective, Injective] at H
push_neg at H
exact H
exact
sum_involution (fun σ _ => σ * Equiv.swap i j)
(fun σ _ => by
have : (∏ x, M (σ x) (p x)) = ∏ x, M ((σ * Equiv.swap i j) x) (p x) :=
Fi... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 107 | 109 | theorem χ₄_int_one_mod_four {n : ℤ} (hn : n % 4 = 1) : χ₄ n = 1 := by |
rw [χ₄_int_mod_four, hn]
rfl
|
/-
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.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 45 | 65 | theorem isOrdered_iff' [@TransCmp α cmp] {t : RBNode α} :
isOrdered cmp t L R ↔
(∀ a ∈ L, t.All (cmpLT cmp a ·)) ∧
(∀ a ∈ R, t.All (cmpLT cmp · a)) ∧
(∀ a ∈ L, ∀ b ∈ R, cmpLT cmp a b) ∧
Ordered cmp t := by |
induction t generalizing L R with
| nil =>
simp [isOrdered]; split <;> simp [cmpLT_iff]
next h => intro _ ha _ hb; cases h _ _ ha hb
| node _ l v r =>
simp [isOrdered, *]
exact ⟨
fun ⟨⟨Ll, lv, Lv, ol⟩, ⟨vr, rR, vR, or⟩⟩ => ⟨
fun _ h => ⟨Lv _ h, Ll _ h, (Lv _ h).trans_l vr⟩,
... |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprove... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 76 | 89 | theorem prime_of_aleph0_le (ha : ℵ₀ ≤ a) : Prime a := by |
refine ⟨(aleph0_pos.trans_le ha).ne', ?_, fun b c hbc => ?_⟩
· rw [isUnit_iff]
exact (one_lt_aleph0.trans_le ha).ne'
rcases eq_or_ne (b * c) 0 with hz | hz
· rcases mul_eq_zero.mp hz with (rfl | rfl) <;> simp
wlog h : c ≤ b
· cases le_total c b <;> [solve_by_elim; rw [or_comm]]
apply_assumption
... |
/-
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.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 117 | 119 | theorem DFinsupp.toFinsupp_support (f : Π₀ _ : ι, M) : f.toFinsupp.support = f.support := by |
ext
simp
|
/-
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.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
#align_import topology.is_locally_homeomorph from "leanprover-community/mathlib"@"e9... | Mathlib/Topology/IsLocalHomeomorph.lean | 45 | 59 | theorem isLocalHomeomorphOn_iff_openEmbedding_restrict {f : X → Y} :
IsLocalHomeomorphOn f s ↔ ∀ x ∈ s, ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f) := by |
refine ⟨fun h x hx ↦ ?_, fun h x hx ↦ ?_⟩
· obtain ⟨e, hxe, rfl⟩ := h x hx
exact ⟨e.source, e.open_source.mem_nhds hxe, e.openEmbedding_restrict⟩
· obtain ⟨U, hU, emb⟩ := h x hx
have : OpenEmbedding ((interior U).restrict f) := by
refine emb.comp ⟨embedding_inclusion interior_subset, ?_⟩
rw [... |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 106 | 109 | theorem induction_on₂ {β : LocalizedModule S M → LocalizedModule S M → Prop}
(h : ∀ (m m' : M) (s s' : S), β (mk m s) (mk m' s')) : ∀ x y, β x y := by |
rintro ⟨⟨m, s⟩⟩ ⟨⟨m', s'⟩⟩
exact h m m' s s'
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.or... | Mathlib/Algebra/Order/Group/MinMax.lean | 22 | 23 | theorem max_one_div_max_inv_one_eq_self (a : α) : max a 1 / max a⁻¹ 1 = a := by |
rcases le_total a 1 with (h | h) <;> simp [h]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 37 | 39 | theorem coeff_scaleRoots (p : R[X]) (s : R) (i : ℕ) :
(scaleRoots p s).coeff i = coeff p i * s ^ (p.natDegree - i) := by |
simp (config := { contextual := true }) [scaleRoots, coeff_monomial]
|
/-
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.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cb... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 58 | 61 | theorem normalizer_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
H.toLieSubmodule.normalizer = H.toLieSubmodule := by |
rw [← LieSubmodule.coe_toSubmodule_eq_iff, coe_normalizer_eq_normalizer,
IsCartanSubalgebra.self_normalizing, coe_toLieSubmodule]
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Floris van Doorn
-/
import Mathlib.Data.PNat.Basic
#align_import data.pnat.find from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-!
#... | Mathlib/Data/PNat/Find.lean | 71 | 76 | theorem find_eq_iff : PNat.find h = m ↔ p m ∧ ∀ n < m, ¬p n := by |
constructor
· rintro rfl
exact ⟨PNat.find_spec h, fun _ => PNat.find_min h⟩
· rintro ⟨hm, hlt⟩
exact le_antisymm (PNat.find_min' h hm) (not_lt.1 <| imp_not_comm.1 (hlt _) <| PNat.find_spec h)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
#align_import ... | Mathlib/RingTheory/Localization/Away/Basic.lean | 129 | 141 | theorem away_of_isUnit_of_bijective {R : Type*} (S : Type*) [CommRing R] [CommRing S]
[Algebra R S] {r : R} (hr : IsUnit r) (H : Function.Bijective (algebraMap R S)) :
IsLocalization.Away r S :=
{ map_units' := by |
rintro ⟨_, n, rfl⟩
exact (algebraMap R S).isUnit_map (hr.pow _)
surj' := fun z => by
obtain ⟨z', rfl⟩ := H.2 z
exact ⟨⟨z', 1⟩, by simp⟩
exists_of_eq := fun {x y} => by
erw [H.1.eq_iff]
rintro rfl
exact ⟨1, rfl⟩ }
|
/-
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.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-!
# Integrals involving the Gamma function
In this file, we... | Mathlib/MeasureTheory/Integral/Gamma.lean | 39 | 57 | theorem integral_rpow_mul_exp_neg_mul_rpow {p q b : ℝ} (hp : 0 < p) (hq : - 1 < q) (hb : 0 < b) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- b * x ^ p) =
b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p) := by |
calc
_ = ∫ x in Ioi (0:ℝ), b ^ (-p⁻¹ * q) * ((b ^ p⁻¹ * x) ^ q * rexp (-(b ^ p⁻¹ * x) ^ p)) := by
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [mul_rpow _ (le_of_lt hx), mul_rpow _ (le_of_lt hx), ← rpow_mul, ← rpow_mul,
inv_mul_cancel, rpow_one, mul_assoc, ← mul_assoc, ← rpo... |
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 84 | 87 | theorem toΓSpec_preim_basicOpen_eq (r : Γ.obj (op X)) :
X.toΓSpecFun ⁻¹' (basicOpen r).1 = (X.toRingedSpace.basicOpen r).1 := by |
ext
erw [X.toRingedSpace.mem_top_basicOpen]; apply not_mem_prime_iff_unit_in_stalk
|
/-
Copyright (c) 2023 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mat... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 132 | 136 | theorem listToPath_pathToList {x : SingleObj α} (p : Path (star α) x) :
listToPath (pathToList p) = p.cast rfl ext := by |
induction' p with y z p a ih
· rfl
· dsimp at *; rw [ih]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.WithOne.Defs
#align_import algebra.group.with_one.basic from "leanprover-commun... | Mathlib/Algebra/Group/WithOne/Basic.lean | 128 | 129 | theorem map_map (f : α →ₙ* β) (g : β →ₙ* γ) (x) : map g (map f x) = map (g.comp f) x := by |
induction x <;> rfl
|
/-
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.Matrix.Kronecker
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
#align_import linear_algebra.tensor_product... | Mathlib/LinearAlgebra/TensorProduct/Matrix.lean | 68 | 77 | theorem TensorProduct.toMatrix_assoc :
toMatrix ((bM.tensorProduct bN).tensorProduct bP) (bM.tensorProduct (bN.tensorProduct bP))
(TensorProduct.assoc R M N P) =
(1 : Matrix (ι × κ × τ) (ι × κ × τ) R).submatrix _root_.id (Equiv.prodAssoc _ _ _) := by |
ext ⟨i, j, k⟩ ⟨⟨i', j'⟩, k'⟩
simp_rw [toMatrix_apply, Basis.tensorProduct_apply, LinearEquiv.coe_coe,
TensorProduct.assoc_tmul, Basis.tensorProduct_repr_tmul_apply, Matrix.submatrix_apply,
Equiv.prodAssoc_apply, _root_.id, Basis.repr_self_apply, Matrix.one_apply, Prod.ext_iff,
ite_and, @eq_comm _ i', @... |
/-
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.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 55 | 58 | theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 115 | 118 | theorem basisSets_zero (U) (hU : U ∈ p.basisSets) : (0 : E) ∈ U := by |
rcases p.basisSets_iff.mp hU with ⟨ι', r, hr, hU⟩
rw [hU, mem_ball_zero, map_zero]
exact hr
|
/-
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.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fi... | Mathlib/Data/Fin/Tuple/Sort.lean | 99 | 102 | theorem monotone_proj (f : Fin n → α) : Monotone (graph.proj : graph f → α) := by |
rintro ⟨⟨x, i⟩, hx⟩ ⟨⟨y, j⟩, hy⟩ (_ | h)
· exact le_of_lt ‹_›
· simp [graph.proj]
|
/-
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.RingTheory.LocalProperties
#align_import ring_theory.ring_hom.surjective from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
/-!... | Mathlib/RingTheory/RingHom/Surjective.lean | 30 | 33 | theorem surjective_respectsIso : RespectsIso surjective := by |
apply surjective_stableUnderComposition.respectsIso
intros _ _ _ _ e
exact e.surjective
|
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import ... | Mathlib/Data/Nat/Choose/Multinomial.lean | 107 | 109 | theorem binomial_eq_choose [DecidableEq α] (h : a ≠ b) :
multinomial {a, b} f = (f a + f b).choose (f a) := by |
simp [binomial_eq h, choose_eq_factorial_div_factorial (Nat.le_add_right _ _)]
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Anal... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 44 | 45 | theorem hasSum_at_zero (a : ℕ → E) : HasSum (fun n => (0 : 𝕜) ^ n • a n) (a 0) := by |
convert hasSum_single (α := E) 0 fun b h ↦ _ <;> simp [*]
|
/-
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 | 126 | 128 | theorem linearIndependent_iff :
LinearIndependent R v ↔ ∀ l, Finsupp.total ι M R v l = 0 → l = 0 := by |
simp [LinearIndependent, LinearMap.ker_eq_bot']
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.FieldTheory.Separable
#align_import field_theory.separable_degree from "lea... | Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 111 | 116 | theorem _root_.Irreducible.hasSeparableContraction (q : ℕ) [hF : ExpChar F q] {f : F[X]}
(irred : Irreducible f) : HasSeparableContraction q f := by |
cases hF
· exact ⟨f, irred.separable, ⟨0, by rw [pow_zero, expand_one]⟩⟩
· rcases exists_separable_of_irreducible q irred ‹q.Prime›.ne_zero with ⟨n, g, hgs, hge⟩
exact ⟨g, hgs, n, hge⟩
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Data.List.Chain
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Category.ULift
... | Mathlib/CategoryTheory/IsConnected.lean | 115 | 117 | theorem any_functor_const_on_obj [IsPreconnected J] {α : Type u₂} (F : J ⥤ Discrete α) (j j' : J) :
F.obj j = F.obj j' := by |
ext; exact ((isoConstant F j').hom.app j).down.1
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
#align_import measure_theory.function.cond... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 76 | 93 | theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero' (hm : m ≤ m0) (f : Lp E' p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf_meas : AEStronglyMeasurabl... |
let f_meas : lpMeas E' 𝕜 m p μ := ⟨f, hf_meas⟩
-- Porting note: `simp only` does not call `rfl` to try to close the goal. See https://github.com/leanprover-community/mathlib4/issues/5025
have hf_f_meas : f =ᵐ[μ] f_meas := by simp only [Subtype.coe_mk]; rfl
refine hf_f_meas.trans ?_
refine lpMeas.ae_eq_zero_... |
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