Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 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, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 684 | 688 | theorem fst_inf_snd : Submodule.fst R M M₂ ⊓ Submodule.snd R M M₂ = ⊥ := by |
-- Porting note (#10936): was `tidy`
rw [eq_bot_iff]; rintro ⟨x, y⟩
simp only [fst, comap_bot, snd, ge_iff_le, mem_inf, mem_ker, snd_apply, fst_apply, mem_bot,
Prod.mk_eq_zero, and_comm, imp_self]
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 447 | 448 | theorem cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 := by |
simpa only [cos_zero] using (cos_periodic.int_mul n).add_antiperiod_eq cos_antiperiodic
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 954 | 957 | theorem sup'_map {s : Finset γ} {f : γ ↪ β} (g : β → α) (hs : (s.map f).Nonempty) :
(s.map f).sup' hs g = s.sup' (map_nonempty.1 hs) (g ∘ f) := by |
rw [← WithBot.coe_eq_coe, coe_sup', sup_map, coe_sup']
rfl
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-communi... | Mathlib/MeasureTheory/Group/Prod.lean | 296 | 303 | theorem ae_measure_preimage_mul_right_lt_top_of_ne_zero [IsMulLeftInvariant ν]
(sm : MeasurableSet s) (h2s : ν s ≠ 0) (h3s : ν s ≠ ∞) :
∀ᵐ x ∂μ, ν ((fun y => y * x) ⁻¹' s) < ∞ := by |
refine (ae_measure_preimage_mul_right_lt_top ν ν sm h3s).filter_mono ?_
refine (absolutelyContinuous_of_isMulLeftInvariant μ ν ?_).ae_le
refine mt ?_ h2s
intro hν
rw [hν, Measure.coe_zero, Pi.zero_apply]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
/-!
# Lemmas a... | Mathlib/Data/Int/Order/Units.lean | 37 | 37 | theorem units_inv_eq_self (u : ℤˣ) : u⁻¹ = u := by | rw [inv_eq_iff_mul_eq_one, units_mul_self]
|
/-
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.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 150 | 152 | theorem formPerm_apply_getLast (x : α) (xs : List α) :
formPerm (x :: xs) ((x :: xs).getLast (cons_ne_nil x xs)) = x := by |
induction' xs using List.reverseRecOn with xs y _ generalizing x <;> simp
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 158 | 161 | theorem StarConvex.mem (hs : StarConvex 𝕜 x s) (h : s.Nonempty) : x ∈ s := by |
obtain ⟨y, hy⟩ := h
convert hs hy zero_le_one le_rfl (add_zero 1)
rw [one_smul, zero_smul, add_zero]
|
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib... | Mathlib/NumberTheory/Multiplicity.lean | 82 | 146 | theorem odd_sq_dvd_geom_sum₂_sub (hp : Odd p) :
(p : R) ^ 2 ∣ (∑ i ∈ range p, (a + p * b) ^ i * a ^ (p - 1 - i)) - p * a ^ (p - 1) := by |
have h1 : ∀ (i : ℕ),
(p : R) ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * i + a ^ i) := by
intro i
calc
↑p ^ 2 ∣ (↑p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]
_ ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) := by
simp only [sq_dvd_add_pow_sub_sub (↑p * b) ... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Type... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 371 | 372 | theorem lift_map_single (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) (r : R) :
(lift R F).map (single f r) = r • F.map f := by | simp
|
/-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 209 | 221 | theorem injective_noncommPiCoprod_of_independent
(hind : CompleteLattice.Independent fun i => (ϕ i).range)
(hinj : ∀ i, Function.Injective (ϕ i)) : Function.Injective (noncommPiCoprod ϕ hcomm) := by |
classical
apply (MonoidHom.ker_eq_bot_iff _).mp
rw [eq_bot_iff]
intro f heq1
have : ∀ i, i ∈ Finset.univ → ϕ i (f i) = 1 :=
Subgroup.eq_one_of_noncommProd_eq_one_of_independent _ _ (fun _ _ _ _ h => hcomm h _ _)
_ hind (by simp) heq1
ext i
apply hinj
simp [this i (Finset.mem... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Convex.Slope
/-!
# Convexity of functions and derivatives
... | Mathlib/Analysis/Convex/Deriv.lean | 849 | 852 | theorem strictAntiOn_deriv (hfc : StrictConcaveOn ℝ S f) (hfd : ∀ x ∈ S, DifferentiableAt ℝ f x) :
StrictAntiOn (deriv f) S := by |
simpa only [Pi.neg_def, deriv.neg, neg_neg] using
(hfc.neg.strictMonoOn_deriv (fun x hx ↦ (hfd x hx).neg)).neg
|
/-
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 | 640 | 644 | theorem eq_C_of_degree_le_zero (h : degree p ≤ 0) : p = C (coeff p 0) := by |
ext (_ | n)
· simp
rw [coeff_C, if_neg (Nat.succ_ne_zero _), coeff_eq_zero_of_degree_lt]
exact h.trans_lt (WithBot.coe_lt_coe.2 n.succ_pos)
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 380 | 390 | theorem finprod_eq_prod_of_mulSupport_subset (f : α → M) {s : Finset α} (h : mulSupport f ⊆ s) :
∏ᶠ i, f i = ∏ i ∈ s, f i := by |
have A : mulSupport (f ∘ PLift.down) = Equiv.plift.symm '' mulSupport f := by
rw [mulSupport_comp_eq_preimage]
exact (Equiv.plift.symm.image_eq_preimage _).symm
have : mulSupport (f ∘ PLift.down) ⊆ s.map Equiv.plift.symm.toEmbedding := by
rw [A, Finset.coe_map]
exact image_subset _ h
rw [finprod_... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
#align_import computability.halting from "leanprover-community/mathlib"@"a50170a88a4757... | Mathlib/Computability/Halting.lean | 303 | 312 | theorem to_part {n f} (pf : @Partrec' n f) : _root_.Partrec f := by |
induction pf with
| prim hf => exact hf.to_prim.to_comp
| comp _ _ _ hf hg => exact (Partrec.vector_mOfFn hg).bind (hf.comp snd)
| rfind _ hf =>
have := hf.comp (vector_cons.comp snd fst)
have :=
((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp
this).to₂.partrec₂... |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-c... | Mathlib/RingTheory/Discriminant.lean | 88 | 89 | theorem discr_reindex (b : Basis ι A B) (f : ι ≃ ι') : discr A (b ∘ ⇑f.symm) = discr A b := by |
classical rw [← Basis.coe_reindex, discr_def, traceMatrix_reindex, det_reindex_self, ← discr_def]
|
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 439 | 441 | theorem rootMultiplicity_le_iff {p : R[X]} (p0 : p ≠ 0) (a : R) (n : ℕ) :
rootMultiplicity a p ≤ n ↔ ¬(X - C a) ^ (n + 1) ∣ p := by |
rw [← (le_rootMultiplicity_iff p0).not, not_le, Nat.lt_add_one_iff]
|
/-
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.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 1,233 | 1,236 | theorem QuotientMap.image_connectedComponent [TopologicalSpace β] {f : α → β} (hf : QuotientMap f)
(h_fibers : ∀ y : β, IsConnected (f ⁻¹' {y})) (a : α) :
f '' connectedComponent a = connectedComponent (f a) := by |
rw [← hf.preimage_connectedComponent h_fibers, image_preimage_eq _ hf.surjective]
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 340 | 341 | theorem sin_int_mul_pi_sub (x : ℝ) (n : ℤ) : sin (n * π - x) = -((-1) ^ n * sin x) := by |
simpa only [sin_neg, mul_neg, Int.coe_negOnePow] using sin_antiperiodic.int_mul_sub_eq n
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 799 | 810 | theorem Orthonormal.exists_orthonormalBasis_extension (hv : Orthonormal 𝕜 ((↑) : v → E)) :
∃ (u : Finset E) (b : OrthonormalBasis u 𝕜 E), v ⊆ u ∧ ⇑b = ((↑) : u → E) := by |
obtain ⟨u₀, hu₀s, hu₀, hu₀_max⟩ := exists_maximal_orthonormal hv
rw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hu₀] at hu₀_max
have hu₀_finite : u₀.Finite := hu₀.linearIndependent.setFinite
let u : Finset E := hu₀_finite.toFinset
let fu : ↥u ≃ ↥u₀ := hu₀_finite.subtypeEquivToFinset.symm
have hu :... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 692 | 697 | theorem Ico_eq_locus_Ioc_eq_iUnion_Ioo :
{ b | toIcoMod hp a b = toIocMod hp a b } = ⋃ z : ℤ, Set.Ioo (a + z • p) (a + p + z • p) := by |
ext1;
simp_rw [Set.mem_setOf, Set.mem_iUnion, ← Set.sub_mem_Ioo_iff_left, ←
not_modEq_iff_toIcoMod_eq_toIocMod, modEq_iff_not_forall_mem_Ioo_mod hp, not_forall,
Classical.not_not]
|
/-
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.Algebra.Classes
import Mathlib.Init.Data.Ordering.Basic
#align_import init.data.ordering.lemmas from "leanprover-community/lean"@"4bd31... | Mathlib/Init/Data/Ordering/Lemmas.lean | 20 | 22 | theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by |
by_cases c <;> simp [*]
|
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#a... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 58 | 62 | theorem tendstoInMeasure_iff_norm [SeminormedAddCommGroup E] {l : Filter ι} {f : ι → α → E}
{g : α → E} :
TendstoInMeasure μ f l g ↔
∀ ε, 0 < ε → Tendsto (fun i => μ { x | ε ≤ ‖f i x - g x‖ }) l (𝓝 0) := by |
simp_rw [TendstoInMeasure, dist_eq_norm]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.C... | Mathlib/CategoryTheory/Abelian/Exact.lean | 198 | 202 | theorem kernel.lift.inv [Mono f] (ex : Exact f g) :
have := isIso_kernel_lift_of_exact_of_mono _ _ ex
inv (kernel.lift _ _ ex.w) ≫ f = kernel.ι g := by |
have := isIso_kernel_lift_of_exact_of_mono _ _ ex
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 495 | 499 | theorem prod_range_div' (n : ℕ) (f : ℕ → G) :
((range n).map fun k ↦ f k / f (k + 1)).prod = f 0 / f n := by |
induction' n with n h
· exact (div_self' (f 0)).symm
· rw [range_succ, map_append, map_singleton, prod_append, prod_singleton, h, div_mul_div_cancel']
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 750 | 760 | theorem Monotone.tendsto_le_alternating_series
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfm : Monotone f) (k : ℕ) : l ≤ ∑ i ∈ range (2 * k), (-1) ^ i * f i := by |
have ha : Antitone (fun n ↦ ∑ i ∈ range (2 * n), (-1) ^ i * f i) := by
refine antitone_nat_of_succ_le (fun n ↦ ?_)
rw [show 2 * (n + 1) = 2 * n + 1 + 1 by ring, sum_range_succ, sum_range_succ]
simp_rw [_root_.pow_succ', show (-1 : E) ^ (2 * n) = 1 by simp, neg_one_mul, one_mul,
← sub_eq_add_neg, su... |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 726 | 728 | theorem abs_im_le_norm (z : K) : |im z| ≤ ‖z‖ := by |
rw [mul_self_le_mul_self_iff (abs_nonneg _) (norm_nonneg _), abs_mul_abs_self, mul_self_norm]
apply im_sq_le_normSq
|
/-
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 | 428 | 430 | theorem nontrivial_toCycle (f : Perm α) (hf : IsCycle f) : (toCycle f hf).Nontrivial := by |
obtain ⟨x, hx, -⟩ := id hf
simp [toCycle_eq_toList f hf x hx, hx, Cycle.nontrivial_coe_nodup_iff (nodup_toList _ _)]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Support
#align_import algebra.monoid_algebra.degree from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 244 | 246 | theorem supDegree_sub_le {f g : R'[A]} :
(f - g).supDegree D ≤ f.supDegree D ⊔ g.supDegree D := by |
rw [sub_eq_add_neg, ← supDegree_neg (f := g)]; apply supDegree_add_le
|
/-
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.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import a... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 172 | 180 | theorem integrable_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
Integrable fun x : ℝ => ↑x * cexp (-b * (x : ℂ) ^ 2) := by |
refine ⟨(continuous_ofReal.mul (Complex.continuous_exp.comp ?_)).aestronglyMeasurable, ?_⟩
· exact continuous_const.mul (continuous_ofReal.pow 2)
have := (integrable_mul_exp_neg_mul_sq hb).hasFiniteIntegral
rw [← hasFiniteIntegral_norm_iff] at this ⊢
convert this
rw [norm_mul, norm_mul, norm_cexp_neg_mul_s... |
/-
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.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 446 | 450 | theorem image_image₂_antidistrib {g : γ → δ} {f' : β' → α' → δ} {g₁ : β → β'} {g₂ : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g₁ b) (g₂ a)) :
(image₂ f s t).image g = image₂ f' (t.image g₁) (s.image g₂) := by |
rw [image₂_swap f]
exact image_image₂_distrib fun _ _ => h_antidistrib _ _
|
/-
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.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 227 | 230 | theorem linear_combination_coord_eq_self [Fintype ι] (b : AffineBasis ι k V) (v : V) :
∑ i, b.coord i v • b i = v := by |
have hb := b.affineCombination_coord_eq_self v
rwa [Finset.univ.affineCombination_eq_linear_combination _ _ (b.sum_coord_apply_eq_one v)] at hb
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
im... | Mathlib/CategoryTheory/Category/Basic.lean | 213 | 213 | theorem eq_whisker {f g : X ⟶ Y} (w : f = g) (h : Y ⟶ Z) : f ≫ h = g ≫ h := by | rw [w]
|
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_the... | Mathlib/GroupTheory/Perm/Fin.lean | 49 | 57 | theorem Equiv.Perm.decomposeFin_symm_apply_succ {n : ℕ} (e : Perm (Fin n)) (p : Fin (n + 1))
(x : Fin n) : Equiv.Perm.decomposeFin.symm (p, e) x.succ = swap 0 p (e x).succ := by |
refine Fin.cases ?_ ?_ p
· simp [Equiv.Perm.decomposeFin, EquivFunctor.map]
· intro i
by_cases h : i = e x
· simp [h, Equiv.Perm.decomposeFin, EquivFunctor.map]
· simp [h, Fin.succ_ne_zero, Equiv.Perm.decomposeFin, EquivFunctor.map,
swap_apply_def, Ne.symm h]
|
/-
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 | 238 | 242 | theorem natDegree_lt_natDegree {p q : R[X]} (hp : p ≠ 0) (hpq : p.degree < q.degree) :
p.natDegree < q.natDegree := by |
by_cases hq : q = 0
· exact (not_lt_bot <| hq ▸ hpq).elim
rwa [degree_eq_natDegree hp, degree_eq_natDegree hq, Nat.cast_lt] at hpq
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 849 | 854 | theorem continuous_symm_of_equiv_compact_to_t2 [CompactSpace X] [T2Space Y] {f : X ≃ Y}
(hf : Continuous f) : Continuous f.symm := by |
rw [continuous_iff_isClosed]
intro C hC
have hC' : IsClosed (f '' C) := (hC.isCompact.image hf).isClosed
rwa [Equiv.image_eq_preimage] at hC'
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 290 | 304 | theorem tendsto_logGammaSeq_of_le_one (hf_conv : ConvexOn ℝ (Ioi 0) f)
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (hx' : x ≤ 1) :
Tendsto (logGammaSeq x) atTop (𝓝 <| f x - f 1) := by |
refine tendsto_of_tendsto_of_tendsto_of_le_of_le' (f := logGammaSeq x)
(g := fun n ↦ f x - f 1 - x * (log (n + 1) - log n)) ?_ tendsto_const_nhds ?_ ?_
· have : f x - f 1 = f x - f 1 - x * 0 := by ring
nth_rw 2 [this]
exact Tendsto.sub tendsto_const_nhds (tendsto_log_nat_add_one_sub_log.const_mul _)
... |
/-
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.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 481 | 493 | theorem pullbackConeOfLeftLift_snd :
pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).snd = s.snd := by |
-- Porting note: `ext` did not pick up `NatTrans.ext`
refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext _ _ <| funext fun x => ?_
· change pullback.lift _ _ _ ≫ pullback.snd = _
simp
· change (_ ≫ _ ≫ _) ≫ _ = _
simp_rw [Category.assoc]
erw [s.snd.c.naturality_assoc]
erw [← s.pt.presheaf.ma... |
/-
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.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compac... | Mathlib/Topology/PartitionOfUnity.lean | 161 | 164 | theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x := by |
have H := f.sum_eq_one hx
contrapose! H
simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
|
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 733 | 745 | theorem comp_eq_zero_iff : p.comp q = 0 ↔ p = 0 ∨ p.eval (q.coeff 0) = 0 ∧ q = C (q.coeff 0) := by |
constructor
· intro h
have key : p.natDegree = 0 ∨ q.natDegree = 0 := by
rw [← mul_eq_zero, ← natDegree_comp, h, natDegree_zero]
replace key := Or.imp eq_C_of_natDegree_eq_zero eq_C_of_natDegree_eq_zero key
cases' key with key key
· rw [key, C_comp] at h
exact Or.inl (key.trans h)
·... |
/-
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, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,228 | 1,234 | theorem _root_.MeasureTheory.Integrable.hasSum_intervalIntegral (hfi : Integrable f μ) (y : ℝ) :
HasSum (fun n : ℤ => ∫ x in y + n..y + n + 1, f x ∂μ) (∫ x, f x ∂μ) := by |
simp_rw [integral_of_le (le_add_of_nonneg_right zero_le_one)]
rw [← integral_univ, ← iUnion_Ioc_add_intCast y]
exact
hasSum_integral_iUnion (fun i => measurableSet_Ioc) (pairwise_disjoint_Ioc_add_intCast y)
hfi.integrableOn
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 778 | 779 | theorem norm_sq_re_add_conj (x : K) : ‖x + conj x‖ ^ 2 = re (x + conj x) ^ 2 := by |
rw [add_conj, ← ofReal_ofNat, ← ofReal_mul, norm_ofReal, sq_abs, ofReal_re]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 379 | 380 | theorem ENNReal.tsum_geometric_add_one (r : ℝ≥0∞) : ∑' n : ℕ, r ^ (n + 1) = r * (1 - r)⁻¹ := by |
simp only [_root_.pow_succ', ENNReal.tsum_mul_left, ENNReal.tsum_geometric]
|
/-
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.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import a... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 110 | 127 | theorem integrable_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs : -1 < s) :
Integrable fun x : ℝ => x ^ s * exp (-b * x ^ 2) := by |
rw [← integrableOn_univ, ← @Iio_union_Ici _ _ (0 : ℝ), integrableOn_union,
integrableOn_Ici_iff_integrableOn_Ioi]
refine ⟨?_, integrableOn_rpow_mul_exp_neg_mul_sq hb hs⟩
rw [← (Measure.measurePreserving_neg (volume : Measure ℝ)).integrableOn_comp_preimage
(Homeomorph.neg ℝ).measurableEmbedding]
simp ... |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 623 | 647 | theorem AEMeasurable.isLUB {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α}
(hf : ∀ i, AEMeasurable (f i) μ) (hg : ∀ᵐ b ∂μ, IsLUB { a | ∃ i, f i b = a } (g b)) :
AEMeasurable g μ := by |
nontriviality α
haveI hα : Nonempty α := inferInstance
cases' isEmpty_or_nonempty ι with hι hι
· simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg
exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (ha _).antisymm (hb _))
let p : δ → (ι → α) → Prop := fun x f' => IsLUB { ... |
/-
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.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 445 | 452 | theorem map_eq_top_or_isMaximal_of_surjective {I : Ideal R} (H : IsMaximal I) :
map f I = ⊤ ∨ IsMaximal (map f I) := by |
refine or_iff_not_imp_left.2 fun ne_top => ⟨⟨fun h => ne_top h, fun J hJ => ?_⟩⟩
· refine
(relIsoOfSurjective f hf).injective
(Subtype.ext_iff.2 (Eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne ?_ ?_)) comap_top.symm))
· exact map_le_iff_le_comap.1 (le_of_lt hJ)
· exact fun h => hJ.right (le_map_... |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | Mathlib/Data/List/Sym.lean | 146 | 151 | theorem length_sym2 {xs : List α} : xs.sym2.length = Nat.choose (xs.length + 1) 2 := by |
induction xs with
| nil => rfl
| cons x xs ih =>
rw [List.sym2, length_append, length_map, length_cons,
Nat.choose_succ_succ, ← ih, Nat.choose_one_right]
|
/-
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.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#al... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 349 | 358 | theorem exists_root {f : Polynomial (AlgebraicClosureAux k)}
(hfm : f.Monic) (hfi : Irreducible f) : ∃ x : AlgebraicClosureAux k, f.eval x = 0 := by |
have : ∃ n p, Polynomial.map (ofStep k n) p = f := by
convert Ring.DirectLimit.Polynomial.exists_of f
obtain ⟨n, p, rfl⟩ := this
rw [monic_map_iff] at hfm
have := hfm.irreducible_of_irreducible_map (ofStep k n) p hfi
obtain ⟨x, hx⟩ := toStepSucc.exists_root k hfm this
refine ⟨ofStep k (n + 1) x, ?_⟩
... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 348 | 352 | theorem wOppSide_smul_vsub_vadd_left {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x := by |
refine ⟨p₂, hp₂, p₁, hp₁, ?_⟩
rw [vadd_vsub, ← neg_neg t, neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev]
exact SameRay.sameRay_nonneg_smul_left _ (neg_nonneg.2 ht)
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.comp... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 250 | 258 | theorem get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero {ifp_n : IntFractPair K}
(stream_nth_eq : IntFractPair.stream v n = some ifp_n) (nth_fr_ne_zero : ifp_n.fr ≠ 0) :
(of v).s.get? n = some ⟨1, (IntFractPair.of ifp_n.fr⁻¹).b⟩ :=
have : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr⁻¹... |
cases ifp_n
simp only [IntFractPair.stream, Nat.add_eq, add_zero, stream_nth_eq, Option.some_bind,
ite_eq_right_iff]
intro; contradiction
get?_of_eq_some_of_succ_get?_intFractPair_stream this
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
/--
A `Heap` is the nodes of the pairing heap.
Each node have two pointers: `child` going to the first c... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 154 | 156 | theorem Heap.size_deleteMin_lt {s : Heap α} (eq : s.deleteMin le = some (a, s')) :
s'.size < s.size := by |
cases s with cases eq | node a c => simp_arith [size_combine, size]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 248 | 250 | theorem floor_eq_iff (ha : 0 ≤ a) : ⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1 := by |
rw [← le_floor_iff ha, ← Nat.cast_one, ← Nat.cast_add, ← floor_lt ha, Nat.lt_add_one_iff,
le_antisymm_iff, and_comm]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
#align_import category_theory.limits.shapes.wide_pullbacks from "lean... | Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 364 | 365 | theorem lift_base : lift f fs w ≫ base arrows = f := by |
simp only [limit.lift_π, WidePullbackShape.mkCone_pt, WidePullbackShape.mkCone_π_app]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 192 | 198 | theorem recF_eq_of_Wequiv {α : Type u} (u : F α → α) (x y : q.P.W) :
Wequiv x y → recF u x = recF u y := by |
intro h
induction h with
| ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp, ih]
| abs a f a' f' h => simp only [recF_eq', abs_map, h]
| trans x y z _ _ ih₁ ih₂ => exact Eq.trans ih₁ ih₂
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Bhavik Mehta
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/... | Mathlib/Probability/CondCount.lean | 62 | 62 | theorem condCount_empty {s : Set Ω} : condCount s ∅ = 0 := by | simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4... | Mathlib/Data/Finset/Powerset.lean | 259 | 270 | theorem powersetCard_succ_insert [DecidableEq α] {x : α} {s : Finset α} (h : x ∉ s) (n : ℕ) :
powersetCard n.succ (insert x s) =
powersetCard n.succ s ∪ (powersetCard n s).image (insert x) := by |
rw [powersetCard_eq_filter, powerset_insert, filter_union, ← powersetCard_eq_filter]
congr
rw [powersetCard_eq_filter, filter_image]
congr 1
ext t
simp only [mem_powerset, mem_filter, Function.comp_apply, and_congr_right_iff]
intro ht
have : x ∉ t := fun H => h (ht H)
simp [card_insert_of_not_mem thi... |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 794 | 800 | theorem IntermediateField.isSeparable_adjoin_pair_of_separable {x y : E}
(hx : (minpoly F x).Separable) (hy : (minpoly F y).Separable) :
IsSeparable F F⟮x, y⟯ := by |
rw [← adjoin_simple_adjoin_simple]
replace hy := hy.map_minpoly F⟮x⟯
rw [← isSeparable_adjoin_simple_iff_separable] at hx hy
exact IsSeparable.trans F F⟮x⟯ F⟮x⟯⟮y⟯
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed_space.indicator_function from "l... | Mathlib/Analysis/NormedSpace/IndicatorFunction.lean | 44 | 46 | theorem norm_indicator_le_norm_self : ‖indicator s f a‖ ≤ ‖f a‖ := by |
rw [norm_indicator_eq_indicator_norm]
apply indicator_norm_le_norm_self
|
/-
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 | 211 | 216 | theorem degree_eq_one_iff : (minpoly A x).degree = 1 ↔ x ∈ (algebraMap A B).range := by |
refine ⟨minpoly.mem_range_of_degree_eq_one _ _, ?_⟩
rintro ⟨x, rfl⟩
haveI := Module.nontrivial A B
exact (degree_X_sub_C x ▸ minpoly.min A (algebraMap A B x) (monic_X_sub_C x) (by simp)).antisymm
(Nat.WithBot.add_one_le_of_lt <| minpoly.degree_pos isIntegral_algebraMap)
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import ... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 216 | 223 | theorem contMDiffAt_finsum {x₀ : M} {g : ι → M → F}
(hφ : ∀ i, x₀ ∈ tsupport (f i) → ContMDiffAt I 𝓘(ℝ, F) n (g i) x₀) :
ContMDiffAt I 𝓘(ℝ, F) n (fun x ↦ ∑ᶠ i, f i x • g i x) x₀ := by |
refine _root_.contMDiffAt_finsum (f.locallyFinite.smul_left _) fun i ↦ ?_
by_cases hx : x₀ ∈ tsupport (f i)
· exact ContMDiffAt.smul ((f i).smooth.of_le le_top).contMDiffAt (hφ i hx)
· exact contMDiffAt_of_not_mem (compl_subset_compl.mpr
(tsupport_smul_subset_left (f i) (g i)) hx) n
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 123 | 132 | theorem nodup_iff_get?_ne_get? {l : List α} :
l.Nodup ↔ ∀ i j : ℕ, i < j → j < l.length → l.get? i ≠ l.get? j := by |
rw [Nodup, pairwise_iff_get]
constructor
· intro h i j hij hj
rw [get?_eq_get (lt_trans hij hj), get?_eq_get hj, Ne, Option.some_inj]
exact h _ _ hij
· intro h i j hij
rw [Ne, ← Option.some_inj, ← get?_eq_get, ← get?_eq_get]
exact h i j hij j.2
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 571 | 572 | theorem closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by |
rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]
|
/-
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.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 633 | 638 | theorem X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic {K : Type*} [Field K] (f : K[X]) {a : K}
(hf : (X - C a) ∣ f /ₘ (X - C a)) : X - C a ∣ derivative f := by |
have key := divByMonic_add_X_sub_C_mul_derivate_divByMonic_eq_derivative f a
have ⟨u,hu⟩ := hf
rw [← key, hu, ← mul_add (X - C a) u _]
use (u + derivative ((X - C a) * u))
|
/-
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 | 223 | 224 | theorem forall_mem_image {f : α → β} {s : Set α} {p : β → Prop} :
(∀ y ∈ f '' s, p y) ↔ ∀ ⦃x⦄, x ∈ s → p (f x) := by | simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 1,314 | 1,322 | theorem exists_mem_eq_sup' (f : ι → α) : ∃ i, i ∈ s ∧ s.sup' H f = f i := by |
induction H using Finset.Nonempty.cons_induction with
| singleton c => exact ⟨c, mem_singleton_self c, rfl⟩
| cons c s hcs hs ih =>
rcases ih with ⟨b, hb, h'⟩
rw [sup'_cons hs, h']
cases le_total (f b) (f c) with
| inl h => exact ⟨c, mem_cons.2 (Or.inl rfl), sup_eq_left.2 h⟩
| inr h => exact... |
/-
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 | 537 | 537 | theorem not_disjoint_self_iff : ¬Disjoint f f ↔ f.NeBot := by | rw [disjoint_self, neBot_iff]
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 632 | 640 | theorem foldrAux_of_valid (f : Char → α → α) (l m r a) :
foldrAux f a ⟨l ++ m ++ r⟩ ⟨utf8Len l + utf8Len m⟩ ⟨utf8Len l⟩ = m.foldr f a := by |
rw [← m.reverse_reverse]
induction m.reverse generalizing r a with (unfold foldrAux; simp)
| cons c m IH =>
rw [if_pos (by exact Nat.lt_add_of_pos_right add_csize_pos)]
simp [← Nat.add_assoc, by simpa using prev_of_valid (l++m.reverse) c r]
simp [by simpa using get_of_valid (l++m.reverse) (c::r)]
... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 794 | 797 | theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ := by |
simp only [iInter_and, @iInter_comm _ ι]
|
/-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_imp... | Mathlib/Analysis/Convex/Slope.lean | 307 | 312 | theorem StrictConvexOn.secant_strict_mono_aux2 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
(hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) < (f z - f x) / (z - x) := by |
have hxy' : 0 < y - x := by linarith
have hxz' : 0 < z - x := by linarith
rw [div_lt_div_iff hxy' hxz']
linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]
|
/-
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 | 2,013 | 2,021 | theorem isBigO_iff_exists_eq_mul :
u =O[l] v ↔ ∃ φ : α → 𝕜, l.IsBoundedUnder (· ≤ ·) (norm ∘ φ) ∧ u =ᶠ[l] φ * v := by |
constructor
· rintro h
rcases h.exists_nonneg with ⟨c, hnnc, hc⟩
rcases hc.exists_eq_mul hnnc with ⟨φ, hφ, huvφ⟩
exact ⟨φ, ⟨c, hφ⟩, huvφ⟩
· rintro ⟨φ, ⟨c, hφ⟩, huvφ⟩
exact isBigO_iff_isBigOWith.2 ⟨c, isBigOWith_of_eq_mul φ hφ huvφ⟩
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,502 | 1,504 | theorem merge_loop_nil_right (s : α → α → Bool) (l t) :
merge.loop s l [] t = reverseAux t l := by |
cases l <;> rw [merge.loop]; intro; contradiction
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 60 | 62 | theorem hasMFDerivAt_iff_hasFDerivAt {f'} :
HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x f' ↔ HasFDerivAt f f' x := by |
rw [← hasMFDerivWithinAt_univ, hasMFDerivWithinAt_iff_hasFDerivWithinAt, hasFDerivWithinAt_univ]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,701 | 1,711 | theorem integral_map {β} [MeasurableSpace β] {φ : α → β} (hφ : AEMeasurable φ μ) {f : β → G}
(hfm : AEStronglyMeasurable f (Measure.map φ μ)) :
∫ y, f y ∂Measure.map φ μ = ∫ x, f (φ x) ∂μ :=
let g := hfm.mk f
calc
∫ y, f y ∂Measure.map φ μ = ∫ y, g y ∂Measure.map φ μ := integral_congr_ae hfm.ae_eq_mk
... | congr 1; exact Measure.map_congr hφ.ae_eq_mk
_ = ∫ x, g (hφ.mk φ x) ∂μ :=
(integral_map_of_stronglyMeasurable hφ.measurable_mk hfm.stronglyMeasurable_mk)
_ = ∫ x, g (φ x) ∂μ := integral_congr_ae (hφ.ae_eq_mk.symm.fun_comp _)
_ = ∫ x, f (φ x) ∂μ := integral_congr_ae <| ae_eq_comp hφ hfm.ae_eq_mk.symm
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.... | Mathlib/Order/Cover.lean | 439 | 440 | theorem CovBy.Iio_eq (h : a ⋖ b) : Iio b = Iic a := by |
rw [← Iic_union_Ioo_eq_Iio h.lt, h.Ioo_eq, union_empty]
|
/-
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, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 576 | 577 | theorem prod_map_fst : (prod p q).map (fst R M M₂) = p := by |
ext x; simp [(⟨0, zero_mem _⟩ : ∃ x, x ∈ q)]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 455 | 458 | theorem ListBlank.append_mk {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) :
ListBlank.append l₁ (ListBlank.mk l₂) = ListBlank.mk (l₁ ++ l₂) := by |
induction l₁ <;>
simp only [*, ListBlank.append, List.nil_append, List.cons_append, ListBlank.cons_mk]
|
/-
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.Sites.DenseSubsite
#align_import category_theory.sites.induced_topology from "leanprover-community/mathlib"@"ba43124c37cfe0009bbfc57505f9503a... | Mathlib/CategoryTheory/Sites/InducedTopology.lean | 112 | 121 | theorem Functor.locallyCoverDense_of_isCoverDense [Full G] [G.IsCoverDense K] :
LocallyCoverDense K G := by |
intro X T
refine K.superset_covering ?_ (K.bind_covering T.property
fun Y f _ => G.is_cover_of_isCoverDense _ Y)
rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', rfl : _ = _⟩, rfl⟩
use W; use G.preimage (f' ≫ f); use g
constructor
· simpa using T.val.downward_closed hf f'
· 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 | 281 | 288 | theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) :
map ((↑) : s → α) atTop = 𝓝[<] b := by |
rcases eq_empty_or_nonempty (Iio b) with (hb' | ⟨a, ha⟩)
· have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩
rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty]
· rw [← comap_coe_nhdsWithin_Iio_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem]
rw [Subtype.range_val]
exact (... |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathl... | Mathlib/Order/JordanHolder.lean | 287 | 292 | theorem isMaximal_eraseLast_last {s : CompositionSeries X} (h : 0 < s.length) :
IsMaximal s.eraseLast.last s.last := by |
have : s.length - 1 + 1 = s.length := by
conv_rhs => rw [← Nat.add_one_sub_one s.length]; rw [Nat.succ_sub h]
rw [last_eraseLast, last]
convert s.step ⟨s.length - 1, by omega⟩; ext; simp [this]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 571 | 571 | theorem zero_nmul (a) : 0 ⨳ a = 0 := by | rw [nmul_comm, nmul_zero]
|
/-
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.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 444 | 446 | theorem integral_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [SMulCommClass 𝕜 ℂ E] (a : 𝕜)
(f : ℂ → E) (c : ℂ) (R : ℝ) : (∮ z in C(c, R), a • f z) = a • ∮ z in C(c, R), f z := by |
simp only [circleIntegral, ← smul_comm a (_ : ℂ) (_ : E), intervalIntegral.integral_smul]
|
/-
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.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 282 | 286 | theorem map_map₂_antidistrib {n : γ → δ} {m' : β' → α' → δ} {n₁ : β → β'} {n₂ : α → α'}
(h_antidistrib : ∀ a b, n (m a b) = m' (n₁ b) (n₂ a)) :
(map₂ m f g).map n = map₂ m' (g.map n₁) (f.map n₂) := by |
rw [map₂_swap m]
exact map_map₂_distrib fun _ _ => h_antidistrib _ _
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 414 | 416 | theorem encard_le_encard_of_injOn (hf : MapsTo f s t) (f_inj : InjOn f s) :
s.encard ≤ t.encard := by |
rw [← f_inj.encard_image]; apply encard_le_card; rintro _ ⟨x, hx, rfl⟩; exact hf hx
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 332 | 332 | theorem zeroIsoTerminal_inv [HasTerminal C] : zeroIsoTerminal.inv = (0 : ⊤_ C ⟶ 0) := by | ext
|
/-
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 | 691 | 695 | theorem cos_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.cos (∡ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂ := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
cos_angle_mul_dist_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
|
/-
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.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,731 | 1,734 | theorem aevalTower_ofId :
aevalTower (Algebra.ofId S A) = (aeval : (σ → A) → MvPolynomial σ S →ₐ[S] A) := by |
ext
simp only [aeval_X, aevalTower_X]
|
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Per... | Mathlib/GroupTheory/Solvable.lean | 118 | 122 | theorem isSolvable_of_comm {G : Type*} [hG : Group G] (h : ∀ a b : G, a * b = b * a) :
IsSolvable G := by |
letI hG' : CommGroup G := { hG with mul_comm := h }
cases hG
exact CommGroup.isSolvable
|
/-
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 | 125 | 133 | theorem Orientation.map_of_isEmpty [IsEmpty ι] (x : Orientation R M ι) (f : M ≃ₗ[R] M) :
Orientation.map ι f x = x := by |
induction' x using Module.Ray.ind with g hg
rw [Orientation.map_apply]
congr
ext i
rw [AlternatingMap.compLinearMap_apply]
congr
simp only [LinearEquiv.coe_coe, eq_iff_true_of_subsingleton]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_e... | Mathlib/FieldTheory/PrimitiveElement.lean | 381 | 385 | theorem primitive_element_iff_minpoly_natDegree_eq (α : E) :
F⟮α⟯ = ⊤ ↔ (minpoly F α).natDegree = finrank F E := by |
rw [← adjoin.finrank (IsIntegral.of_finite F α), ← finrank_top F E]
refine ⟨fun h => ?_, fun h => eq_of_le_of_finrank_eq le_top h⟩
exact congr_arg (fun K : IntermediateField F E => finrank F K) h
|
/-
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 | 45 | 65 | theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
[BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
μ (Metric.ball 0 1) =
.ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) /... |
obtain hE | hE := subsingleton_or_nontrivial E
· rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← integral_univ, Set.univ_nonempty.eq_zero,
integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero, zero_div, zero_add,
Real.Gamma_one, div_one, norm_zero, Real.zero_rpow (ne_of_gt hp), neg_zer... |
/-
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.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheo... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 147 | 149 | theorem hasEigenvalue_iff_mem_spectrum [FiniteDimensional K V] {f : End K V} {μ : K} :
f.HasEigenvalue μ ↔ μ ∈ spectrum K f := by |
rw [spectrum.mem_iff, IsUnit.sub_iff, LinearMap.isUnit_iff_ker_eq_bot, HasEigenvalue, eigenspace]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 959 | 960 | theorem div_self {a : Ordinal} (h : a ≠ 0) : a / a = 1 := by |
simpa only [mul_one] using mul_div_cancel 1 h
|
/-
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.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 294 | 297 | theorem mkRat_mul_mkRat (n₁ n₂ : Int) (d₁ d₂) :
mkRat n₁ d₁ * mkRat n₂ d₂ = mkRat (n₁ * n₂) (d₁ * d₂) := by |
if z₁ : d₁ = 0 then simp [z₁] else if z₂ : d₂ = 0 then simp [z₂] else
rw [← normalize_eq_mkRat z₁, ← normalize_eq_mkRat z₂, normalize_mul_normalize, normalize_eq_mkRat]
|
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.CategoryTheory.Galois.Decomposition
import Mathlib.C... | Mathlib/CategoryTheory/Galois/Prorepresentability.lean | 385 | 391 | theorem FiberFunctor.isPretransitive_of_isGalois (X : C) [IsGalois X] :
MulAction.IsPretransitive (Aut F) (F.obj X) := by |
refine ⟨fun x y ↦ ?_⟩
obtain ⟨(φ : Aut X), h⟩ := MulAction.IsPretransitive.exists_smul_eq (M := Aut X) x y
obtain ⟨a, ha⟩ := AutGalois.π_surjective F ⟨X, x, inferInstance⟩ φ
use (autMulEquivAutGalois F).symm ⟨a⟩
simpa [mulAction_def, ha]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | Mathlib/Analysis/MeanInequalities.lean | 672 | 685 | theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by |
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
lift A to ℝ≥0 using hA
lift B to ℝ≥0 using hB
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at hf_sum hg_sum
obtain ⟨C, hC, H⟩ := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum
refine ⟨C... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 846 | 850 | theorem Basis.exists_basis_inter_eq_of_superset (hI : M.Basis I X) (hXY : X ⊆ Y)
(hY : Y ⊆ M.E := by | aesop_mat) : ∃ J, M.Basis J Y ∧ J ∩ X = I := by
obtain ⟨J, hJ, hIJ⟩ := hI.indep.subset_basis_of_subset (hI.subset.trans hXY)
refine ⟨J, hJ, subset_antisymm ?_ (subset_inter hIJ hI.subset)⟩
exact fun e he ↦ hI.mem_of_insert_indep he.2 (hJ.indep.subset (insert_subset he.1 hIJ))
|
/-
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 | 1,383 | 1,384 | theorem frequently_or_distrib_left {f : Filter α} [NeBot f] {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p ∨ q x) ↔ p ∨ ∃ᶠ x in f, q x := by | simp
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure fr... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 131 | 138 | theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity {f : α → ℝ≥0} {g : α → E}
(hf : AEMeasurable f μ) (hfg : Integrable (f • g) μ) :
μ.withDensityᵥ (f • g) = (μ.withDensity (fun x ↦ f x)).withDensityᵥ g := by |
ext s hs
rw [withDensityᵥ_apply hfg hs,
withDensityᵥ_apply ((integrable_withDensity_iff_integrable_smul₀ hf).mpr hfg) hs,
setIntegral_withDensity_eq_setIntegral_smul₀ hf.restrict _ hs]
rfl
|
/-
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 | 192 | 203 | theorem mapAccumr₂_bisim {ys : Vector β n} {f₁ : α → β → σ₁ → σ₁ × γ}
{f₂ : α → β → σ₂ → σ₂ × γ} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {s q} a b, R s q → R (f₁ a b s).1 (f₂ a b q).1 ∧ (f₁ a b s).2 = (f₂ a b q).2) :
R (mapAccumr₂ f₁ xs ys s₁).1 (mapAccumr₂ f₂ xs ys s₂).1
∧ ... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂
next => exact ⟨h₀, rfl⟩
next xs ys x y ih =>
rcases (hR x y h₀) with ⟨hR, _⟩
simp only [mapAccumr₂_snoc, ih hR, true_and]
congr 1
|
/-
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 | 402 | 405 | theorem IsLowerSet.not_bddBelow (hs : IsLowerSet s) : s.Nonempty → ¬BddBelow s := by |
rintro ⟨a, ha⟩ ⟨b, hb⟩
obtain ⟨c, hc⟩ := exists_lt b
exact hc.not_le (hb <| hs (hc.le.trans <| hb ha) ha)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 827 | 832 | theorem len_lt_of_mono {Δ' Δ : SimplexCategory} (i : Δ' ⟶ Δ) [hi : Mono i] (hi' : Δ ≠ Δ') :
Δ'.len < Δ.len := by |
rcases lt_or_eq_of_le (len_le_of_mono hi) with (h | h)
· exact h
· exfalso
exact hi' (by ext; exact h.symm)
|
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