Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Eric Wieser, Joachim Breitner
-/
import Mathlib.GroupTheory.FreeGroup.Basic
#align_import group_theory.is_free_group from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242... | Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean | 226 | 227 | theorem unique_lift (f : Generators G → H) : ∃! F : G →* H, ∀ a, F (of a) = f a := by |
simpa only [Function.funext_iff] using lift.symm.bijective.existsUnique f
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Monotone functions on an order topology
This file contains lemmas about limits and contin... | Mathlib/Topology/Order/Monotone.lean | 58 | 62 | theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
(bdd : BddBelow (range g) := by | bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) := by
rw [iInf, Monotone.map_sInf_of_continuousAt' Cf Mf (range_nonempty g) bdd, ← range_comp, iInf]
rfl
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful... | Mathlib/CategoryTheory/Closed/Functor.lean | 91 | 97 | theorem coev_expComparison (A B : C) :
F.map ((exp.coev A).app B) ≫ (expComparison F A).app (A ⨯ B) =
(exp.coev _).app (F.obj B) ≫ (exp (F.obj A)).map (inv (prodComparison F A B)) := by |
convert unit_transferNatTrans _ _ (prodComparisonNatIso F A).inv B using 3
apply IsIso.inv_eq_of_hom_inv_id -- Porting note: was `ext`
dsimp
simp
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Oliver Nash
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.D... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 140 | 141 | theorem univBall_apply_zero (c : P) (r : ℝ) : univBall c r 0 = c := by |
unfold univBall; split_ifs <;> simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 513 | 515 | theorem rpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x := by |
have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn
rw [← rpow_natCast, ← rpow_mul hx, inv_mul_cancel hn0, rpow_one]
|
/-
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.NumberTheory.Liouville.Basic
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.LocallyCompactRegular
import Mathlib.Topology.Instanc... | Mathlib/NumberTheory/Liouville/Residual.lean | 25 | 31 | theorem setOf_liouville_eq_iInter_iUnion :
{ x | Liouville x } =
⋂ n : ℕ, ⋃ (a : ℤ) (b : ℤ) (_ : 1 < b),
ball ((a : ℝ) / b) (1 / (b : ℝ) ^ n) \ {(a : ℝ) / b} := by |
ext x
simp only [mem_iInter, mem_iUnion, Liouville, mem_setOf_eq, exists_prop, mem_diff,
mem_singleton_iff, mem_ball, Real.dist_eq, and_comm]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb537... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 289 | 319 | theorem restrictProd_injective : Function.Injective (restrictProd p q) := by |
by_cases hpq : p * q = 0
· have : Unique (p * q).Gal := by rw [hpq]; infer_instance
exact fun f g _ => Eq.trans (Unique.eq_default f) (Unique.eq_default g).symm
intro f g hfg
classical
simp only [restrictProd, restrictDvd_def] at hfg
simp only [dif_neg hpq, MonoidHom.prod_apply, Prod.mk.inj_iff] at hfg... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 499 | 514 | theorem eq_one_div_of_mul_eq_one_right (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) :
J = 1 / I := by |
have hI : I ≠ 0 := ne_zero_of_mul_eq_one I J h
suffices h' : I * (1 / I) = 1 from
congr_arg Units.inv <| @Units.ext _ _ (Units.mkOfMulEqOne _ _ h) (Units.mkOfMulEqOne _ _ h') rfl
apply le_antisymm
· apply mul_le.mpr _
intro x hx y hy
rw [mul_comm]
exact (mem_div_iff_of_nonzero hI).mp hy x hx
... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 221 | 222 | theorem nonsing_inv_apply_not_isUnit (h : ¬IsUnit A.det) : A⁻¹ = 0 := by |
rw [inv_def, Ring.inverse_non_unit _ h, zero_smul]
|
/-
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.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179... | Mathlib/Data/Multiset/Powerset.lean | 204 | 208 | theorem powersetCardAux_cons (n : ℕ) (a : α) (l : List α) :
powersetCardAux (n + 1) (a :: l) =
powersetCardAux (n + 1) l ++ List.map (cons a) (powersetCardAux n l) := by |
simp only [powersetCardAux_eq_map_coe, sublistsLen_succ_cons, map_append, List.map_map,
append_cancel_left_eq]; rfl
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 287 | 288 | theorem edgeDensity_comm (s t : Finset α) : edgeDensity r s t = edgeDensity r t s := by |
rw [edgeDensity, mul_comm, card_interedges_comm hr, edgeDensity]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | Mathlib/CategoryTheory/Subobject/Basic.lean | 449 | 451 | theorem ofLE_refl {B : C} (X : Subobject B) : ofLE X X le_rfl = 𝟙 _ := by |
apply (cancel_mono X.arrow).mp
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib... | Mathlib/Data/Set/Pairwise/Basic.lean | 234 | 236 | theorem pairwise_subtype_iff_pairwise_set (s : Set α) (r : α → α → Prop) :
(Pairwise fun (x : s) (y : s) => r x y) ↔ s.Pairwise r := by |
simp only [Pairwise, Set.Pairwise, SetCoe.forall, Ne, Subtype.ext_iff, Subtype.coe_mk]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 371 | 380 | theorem listTransvecCol_mul_last_row_drop (i : Sum (Fin r) Unit) {k : ℕ} (hk : k ≤ r) :
(((listTransvecCol M).drop k).prod * M) (inr unit) i = M (inr unit) i := by |
-- Porting note: `apply` didn't work anymore, because of the implicit arguments
refine Nat.decreasingInduction' ?_ hk ?_
· intro n hn _ IH
have hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn
rw [List.drop_eq_get_cons hn']
simpa [listTransvecCol, Matrix.mul_assoc]
· simp... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 437 | 439 | theorem inner_smul_rotation_pi_div_two_left (x : V) (r : ℝ) :
⟪r • o.rotation (π / 2 : ℝ) x, x⟫ = 0 := by |
rw [inner_smul_left, inner_rotation_pi_div_two_left, mul_zero]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Mathlib.Init.Data.Int.Basic
import Mathlib.Init.ZeroOne
import Mathlib.Tactic.Lemma
import Mathlib.Tactic.TypeSta... | Mathlib/Algebra/Group/Defs.lean | 1,238 | 1,239 | theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by |
rw [← mul_left_inv a⁻¹, inv_eq_of_mul (mul_left_inv a)]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.Comma.Arrow
#align_import category_theory.comm_sq from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb... | Mathlib/CategoryTheory/CommSq.lean | 241 | 243 | theorem iff : HasLift sq ↔ Nonempty sq.LiftStruct := by |
constructor
exacts [fun h => h.exists_lift, fun h => mk 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.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
/-- The number of elements contained i... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 40 | 47 | theorem mem_range'_elems (r : Range) (h : x ∈ List.range' r.start r.numElems r.step) : x ∈ r := by |
obtain ⟨i, h', rfl⟩ := List.mem_range'.1 h
refine ⟨Nat.le_add_right .., ?_⟩
unfold numElems at h'; split at h'
· split at h' <;> [cases h'; simp_all]
· next step0 =>
refine Nat.not_le.1 fun h =>
Nat.not_le.2 h' <| (numElems_le_iff (Nat.pos_of_ne_zero step0)).2 h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.subtype from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185... | Mathlib/Data/Subtype.lean | 159 | 161 | theorem _root_.exists_subtype_mk_eq_iff {a : Subtype p} {b : α} :
(∃ h : p b, Subtype.mk b h = a) ↔ b = a := by |
simp only [@eq_comm _ b, exists_eq_subtype_mk_iff, @eq_comm _ _ a]
|
/-
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.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 129 | 131 | theorem get_map {β : Type*} (v : Vector α n) (f : α → β) (i : Fin n) :
(v.map f).get i = f (v.get i) := by |
cases v; simp [Vector.map, get_eq_get]; rfl
|
/-
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.Topology.Algebra.Module.Multilinear.Bounded
import Mathlib.Topology.Algebra.UniformConvergence
/-!
# Topology on continuous multilinear maps
In thi... | Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean | 119 | 123 | theorem continuous_eval_const (x : Π i, E i) :
Continuous fun p : ContinuousMultilinearMap 𝕜 E F ↦ p x := by |
letI := TopologicalAddGroup.toUniformSpace F
haveI := comm_topologicalAddGroup_is_uniform (G := F)
exact (uniformContinuous_eval_const x).continuous
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib... | Mathlib/Data/List/Sublists.lean | 305 | 312 | theorem sublistsLen_sublist_of_sublist (n) {l₁ l₂ : List α} (h : l₁ <+ l₂) :
sublistsLen n l₁ <+ sublistsLen n l₂ := by |
induction' n with n IHn generalizing l₁ l₂; · simp
induction' h with l₁ l₂ a _ IH l₁ l₂ a s IH; · rfl
· refine IH.trans ?_
rw [sublistsLen_succ_cons]
apply sublist_append_left
· simpa only [sublistsLen_succ_cons] using IH.append ((IHn s).map _)
|
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Field.Rat
im... | Mathlib/Algebra/Module/Basic.lean | 61 | 66 | theorem map_ratCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M]
[Module S M₂] (c : ℚ) (x : M) :
f ((c : R) • x) = (c : S) • f x := by |
rw [Rat.cast_def, Rat.cast_def, div_eq_mul_inv, div_eq_mul_inv, mul_smul, mul_smul,
map_intCast_smul f R S, map_inv_natCast_smul f R S]
|
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 96 | 99 | theorem prevD_eq (f : ∀ i j, C.X i ⟶ D.X j) {j j' : ι} (w : c.Rel j' j) :
prevD j f = f j j' ≫ D.d j' j := by |
obtain rfl := c.prev_eq' w
rfl
|
/-
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, Mario Carneiro, Anne Baanen
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tacti... | Mathlib/RingTheory/Ideal/Quotient.lean | 317 | 318 | theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by | ext; rfl
|
/-
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.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_characte... | Mathlib/NumberTheory/MulChar/Basic.lean | 357 | 361 | theorem inv_mul (χ : MulChar R R') : χ⁻¹ * χ = 1 := by |
ext x
rw [coeToFun_mul, Pi.mul_apply, inv_apply_eq_inv]
simp only [Ring.inverse_mul_cancel _ (IsUnit.map χ x.isUnit)]
rw [one_apply_coe]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import... | Mathlib/Algebra/Algebra/Operations.lean | 739 | 742 | theorem one_le_one_div {I : Submodule R A} : 1 ≤ 1 / I ↔ I ≤ 1 := by |
constructor; all_goals intro hI
· rwa [le_div_iff_mul_le, one_mul] at hI
· rwa [le_div_iff_mul_le, one_mul]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathl... | Mathlib/Analysis/BoxIntegral/Basic.lean | 300 | 301 | theorem HasIntegral.sub (h : HasIntegral I l f vol y) (h' : HasIntegral I l g vol y') :
HasIntegral I l (f - g) vol (y - y') := by | simpa only [sub_eq_add_neg] using h.add h'.neg
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 109 | 112 | theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
[CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive := by |
have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
rwa [U.adjoint_orthogonalProjection] at this
|
/-
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.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 406 | 410 | theorem sign_nonpos_iff : sign a ≤ 0 ↔ a ≤ 0 := by |
rcases lt_trichotomy 0 a with (h | h | h)
· simp [h, h.not_le]
· simp [← h]
· simp [h, h.le]
|
/-
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.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
/-... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 124 | 127 | theorem ae_empty_or_univ' (hf : QuasiErgodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s =ᵐ[μ] s) :
s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ := by |
obtain ⟨t, h₀, h₁, h₂⟩ := hf.toQuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae hs hs'
rcases hf.ae_empty_or_univ h₀ h₂ with (h₃ | h₃) <;> [left; right] <;> exact ae_eq_trans h₁.symm h₃
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 196 | 199 | theorem dist_self_center (z : ℍ) (r : ℝ) :
dist (z : ℂ) (z.center r) = z.im * (Real.cosh r - 1) := by |
rw [dist_of_re_eq (z.center_re r).symm, dist_comm, Real.dist_eq, mul_sub, mul_one]
exact abs_of_nonneg (sub_nonneg.2 <| le_mul_of_one_le_right z.im_pos.le (one_le_cosh _))
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,969 | 2,970 | theorem span_eq_take_drop (l : List α) : span p l = (takeWhile p l, dropWhile p l) := by |
simpa using span.loop_eq_take_drop p l []
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 322 | 324 | theorem hom_inv_whiskerRight' {X Y : C} (f : X ⟶ Y) [IsIso f] (Z : C) :
f ▷ Z ≫ inv f ▷ Z = 𝟙 (X ⊗ Z) := by |
rw [← comp_whiskerRight, IsIso.hom_inv_id, id_whiskerRight]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,659 | 1,665 | theorem getLast_map (f : α → β) {l : List α} (hl : l ≠ []) :
(l.map f).getLast (mt eq_nil_of_map_eq_nil hl) = f (l.getLast hl) := by |
induction' l with l_hd l_tl l_ih
· apply (hl rfl).elim
· cases l_tl
· simp
· simpa using l_ih _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 361 | 363 | theorem ofReal_mul' {p q : ℝ} (hq : 0 ≤ q) :
ENNReal.ofReal (p * q) = ENNReal.ofReal p * ENNReal.ofReal q := by |
rw [mul_comm, ofReal_mul hq, mul_comm]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... | Mathlib/Data/Nat/Fib/Basic.lean | 313 | 319 | theorem fib_succ_eq_succ_sum (n : ℕ) : fib (n + 1) = (∑ k ∈ Finset.range n, fib k) + 1 := by |
induction' n with n ih
· simp
· calc
fib (n + 2) = fib n + fib (n + 1) := fib_add_two
_ = (fib n + ∑ k ∈ Finset.range n, fib k) + 1 := by rw [ih, add_assoc]
_ = (∑ k ∈ Finset.range (n + 1), fib k) + 1 := by simp [Finset.range_add_one]
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 71 | 73 | theorem exists_reduced_word (w : W) : ∃ ω, ω.length = ℓ w ∧ w = π ω := by |
have := Nat.find_spec (cs.exists_word_with_prod w)
tauto
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 470 | 471 | theorem eval_natCast_mul {n : ℕ} : ((n : R[X]) * p).eval x = n * p.eval x := by |
rw [← C_eq_natCast, eval_C_mul]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 310 | 314 | theorem nhdsWithin_pi_eq' {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
(hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by |
simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←
iInf_principal_finite hI, ← iInf_inf_eq]
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 168 | 175 | theorem tendsto_approxOn_range_Lp_snorm [BorelSpace E] {f : β → E} (hp_ne_top : p ≠ ∞)
{μ : Measure β} (fmeas : Measurable f) [SeparableSpace (range f ∪ {0} : Set E)]
(hf : snorm f p μ < ∞) :
Tendsto (fun n => snorm (⇑(approxOn f fmeas (range f ∪ {0}) 0 (by simp) n) - f) p μ)
atTop (𝓝 0) := by |
refine tendsto_approxOn_Lp_snorm fmeas _ hp_ne_top ?_ ?_
· filter_upwards with x using subset_closure (by simp)
· simpa using hf
|
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
/-!
#... | Mathlib/Control/Bitraversable/Lemmas.lean | 79 | 83 | theorem tfst_tsnd {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
Comp.mk (tfst f <$> tsnd f' x)
= bitraverse (Comp.mk ∘ pure ∘ f) (Comp.mk ∘ map pure ∘ f') x := by |
rw [← comp_bitraverse]
simp only [Function.comp, map_pure]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
#align_import c... | Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 300 | 301 | theorem ι_fstSigmaMap (b) : Sigma.ι I.left b ≫ I.fstSigmaMap = I.fst b ≫ Sigma.ι I.right _ := by |
simp [fstSigmaMap]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.Adjunctions
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Algebra.Category.ModuleCat.Colimits
import Mathl... | Mathlib/RepresentationTheory/Rep.lean | 226 | 234 | theorem linearization_μ_inv_hom (X Y : Action (Type u) (MonCat.of G)) :
(inv ((linearization k G).μ X Y)).hom = (finsuppTensorFinsupp' k X.V Y.V).symm.toLinearMap := by |
-- Porting note (#11039): broken proof was
/- simp_rw [← Action.forget_map, Functor.map_inv, Action.forget_map, linearization_μ_hom]
apply IsIso.inv_eq_of_hom_inv_id _
exact LinearMap.ext fun x => LinearEquiv.symm_apply_apply _ _-/
rw [← Action.forget_map, Functor.map_inv]
apply IsIso.inv_eq_of_hom_inv_id
ex... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 995 | 1,000 | theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
(hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ x ∂μ, f x ≠ g x) :
∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := by |
contrapose! h
simp only [not_frequently, Ne, Classical.not_not]
exact ae_eq_of_ae_le_of_lintegral_le h_le hfi hg h
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
... | Mathlib/Data/PEquiv.lean | 369 | 373 | theorem single_trans_of_mem (a : α) {b : β} {c : γ} {f : β ≃. γ} (h : c ∈ f b) :
(single a b).trans f = single a c := by |
ext
dsimp [single, PEquiv.trans]
split_ifs <;> simp_all
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-co... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 292 | 293 | theorem X_inv (s : σ) : (X s : MvPowerSeries σ k)⁻¹ = 0 := by |
rw [inv_eq_zero, constantCoeff_X]
|
/-
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 | 363 | 368 | theorem map_exact {X Y Z : A} (f : X ⟶ Y) (g : Y ⟶ Z) (e1 : Exact f g) :
Exact (L.map f) (L.map g) := by |
let hcoker := isColimitOfHasCokernelOfPreservesColimit L f
let hker := isLimitOfHasKernelOfPreservesLimit L g
refine (exact_iff' _ _ hker hcoker).2 ⟨by simp [← L.map_comp, e1.1], ?_⟩
simp only [Fork.ι_ofι, Cofork.π_ofπ, ← L.map_comp, kernel_comp_cokernel _ _ e1, L.map_zero]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 177 | 181 | theorem measurable_measure_prod_mk_left [SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s) :
Measurable fun x => ν (Prod.mk x ⁻¹' s) := by |
rw [← sum_sFiniteSeq ν]
simp_rw [Measure.sum_apply_of_countable]
exact Measurable.ennreal_tsum (fun i ↦ measurable_measure_prod_mk_left_finite hs)
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7c... | Mathlib/Probability/Kernel/Composition.lean | 503 | 512 | theorem compProd_eq_sum_compProd_left (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ) :
κ ⊗ₖ η = kernel.sum fun n => seq κ n ⊗ₖ η := by |
by_cases h : IsSFiniteKernel η
swap
· simp_rw [compProd_of_not_isSFiniteKernel_right _ _ h]
simp
rw [compProd_eq_sum_compProd]
congr with n a s hs
simp_rw [kernel.sum_apply' _ _ hs, compProd_apply_eq_compProdFun _ _ _ hs,
compProdFun_tsum_right _ η a hs]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 324 | 326 | theorem univ_mem_measurableCylinders (α : ι → Type*) [∀ i, MeasurableSpace (α i)] :
Set.univ ∈ measurableCylinders α := by |
rw [← compl_empty]; exact compl_mem_measurableCylinders (empty_mem_measurableCylinders α)
|
/-
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 | 595 | 597 | theorem sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 := by |
rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq x, sq, sq, ← sub_eq_iff_eq_add, sub_self]
exact ⟨fun h => by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ Eq.symm⟩
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 929 | 944 | theorem measurableSet_of_differentiableAt_of_isComplete_with_param
(hf : Continuous f.uncurry) {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
MeasurableSet {p : α × E | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} := by |
have : {p : α × E | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K}
= {p : α × E | p.2 ∈ D (f p.1) K} := by simp [← differentiable_set_eq_D K hK]
rw [this]
simp only [D, mem_iInter, mem_iUnion]
simp only [setOf_forall, setOf_exists]
refine MeasurableSet.iInter (fun _ ↦ ?_)
refine Mea... |
/-
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.NumberTheory.LegendreSymbol.AddCharacter
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.Algebra.CharP.CharAndCard
#align_import numb... | Mathlib/NumberTheory/GaussSum.lean | 252 | 337 | theorem FiniteField.two_pow_card {F : Type*} [Fintype F] [Field F] (hF : ringChar F ≠ 2) :
(2 : F) ^ (Fintype.card F / 2) = χ₈ (Fintype.card F) := by |
have hp2 : ∀ n : ℕ, (2 ^ n : F) ≠ 0 := fun n => pow_ne_zero n (Ring.two_ne_zero hF)
obtain ⟨n, hp, hc⟩ := FiniteField.card F (ringChar F)
-- we work in `FF`, the eighth cyclotomic field extension of `F`
let FF := (Polynomial.cyclotomic 8 F).SplittingField
have hchar := Algebra.ringChar_eq F FF
have FFp :=... |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Complement
import Mathlib.GroupTheory.Sylow
import Mathlib.GroupTheory.Subgroup.Center
#align_import group_theory.transfer from "leanpro... | Mathlib/GroupTheory/Transfer.lean | 118 | 139 | theorem transfer_eq_prod_quotient_orbitRel_zpowers_quot [FiniteIndex H] (g : G)
[Fintype (Quotient (orbitRel (zpowers g) (G ⧸ H)))] :
transfer ϕ g =
∏ q : Quotient (orbitRel (zpowers g) (G ⧸ H)),
ϕ
⟨q.out'.out'⁻¹ * g ^ Function.minimalPeriod (g • ·) q.out' * q.out'.out',
Quot... |
classical
letI := H.fintypeQuotientOfFiniteIndex
calc
transfer ϕ g = ∏ q : G ⧸ H, _ := transfer_def ϕ (transferTransversal H g) g
_ = _ := ((quotientEquivSigmaZMod H g).symm.prod_comp _).symm
_ = _ := Finset.prod_sigma _ _ _
_ = _ := by
refine Fintype.prod_congr _ _ (fun q => ... |
/-
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.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousFunction.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#alig... | Mathlib/Topology/Homotopy/Basic.lean | 260 | 277 | theorem symm_trans {f₀ f₁ f₂ : C(X, Y)} (F : Homotopy f₀ f₁) (G : Homotopy f₁ f₂) :
(F.trans G).symm = G.symm.trans F.symm := by |
ext ⟨t, _⟩
rw [trans_apply, symm_apply, trans_apply]
simp only [coe_symm_eq, symm_apply]
split_ifs with h₁ h₂ h₂
· have ht : (t : ℝ) = 1 / 2 := by linarith
norm_num [ht]
· congr 2
apply Subtype.ext
simp only [coe_symm_eq]
linarith
· congr 2
apply Subtype.ext
simp only [coe_symm_eq... |
/-
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.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 404 | 414 | theorem eq_rnDeriv (t : SignedMeasure α) (f : α → ℝ) (hfi : Integrable f μ)
(htμ : t ⟂ᵥ μ.toENNRealVectorMeasure) (hadd : s = t + μ.withDensityᵥ f) :
f =ᵐ[μ] s.rnDeriv μ := by |
set f' := hfi.1.mk f
have hadd' : s = t + μ.withDensityᵥ f' := by
convert hadd using 2
exact WithDensityᵥEq.congr_ae hfi.1.ae_eq_mk.symm
have := haveLebesgueDecomposition_mk μ hfi.1.measurable_mk htμ hadd'
refine (Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) hfi ?_).symm
rw [← add_rig... |
/-
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 | 383 | 385 | theorem Base.ncard_diff_comm (hB₁ : M.Base B₁) (hB₂ : M.Base B₂) :
(B₁ \ B₂).ncard = (B₂ \ B₁).ncard := by |
rw [ncard_def, hB₁.encard_diff_comm hB₂, ← ncard_def]
|
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Basic
/-! We specialize the theory fo analytic functions to the case of functions that admit a
development given by a *finite* formal mu... | Mathlib/Analysis/Analytic/CPolynomial.lean | 521 | 524 | theorem isOpen_cPolynomialAt : IsOpen { x | CPolynomialAt 𝕜 f x } := by |
rw [isOpen_iff_mem_nhds]
rintro x ⟨p, n, r, hr⟩
exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.cPolynomialAt_of_mem hy
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set α` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 490 | 501 | theorem restrict_trim {μ : OuterMeasure α} {s : Set α} (hs : MeasurableSet s) :
(restrict s μ).trim = restrict s μ.trim := by |
refine le_antisymm (fun t => ?_) (le_trim_iff.2 fun t ht => ?_)
· rw [restrict_apply]
rcases μ.exists_measurable_superset_eq_trim (t ∩ s) with ⟨t', htt', ht', hμt'⟩
rw [← hμt']
rw [inter_subset] at htt'
refine (measure_mono htt').trans ?_
rw [trim_eq _ (hs.compl.union ht'), restrict_apply, unio... |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 209 | 210 | theorem inf_symmDiff_symmDiff : (a ⊓ b) ∆ (a ∆ b) = a ⊔ b := by |
rw [symmDiff_comm, symmDiff_symmDiff_inf]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 173 | 179 | theorem noncommProd_mem (S : Submonoid M) {ι : Type*} (t : Finset ι) (f : ι → M) (comm)
(h : ∀ c ∈ t, f c ∈ S) : t.noncommProd f comm ∈ S := by |
apply multiset_noncommProd_mem
intro y
rw [Multiset.mem_map]
rintro ⟨x, ⟨hx, rfl⟩⟩
exact h x hx
|
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Lemmas
/-!
# `compute_degree` and `monicity`: tactics for explicit polynomials
This file defines two related tactics: `comp... | Mathlib/Tactic/ComputeDegree.lean | 184 | 185 | theorem coeff_sub_of_eq {n : ℕ} {a b : R} {f g : R[X]} (hf : f.coeff n = a) (hg : g.coeff n = b) :
(f - g).coeff n = a - b := by | subst hf hg; apply coeff_sub
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 281 | 285 | theorem submodule_comap_mul_reverse (p q : Submodule R (CliffordAlgebra Q)) :
(p * q).comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) =
q.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) *
p.comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) := by |
simp_rw [← submodule_map_reverse_eq_comap, submodule_map_mul_reverse]
|
/-
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.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,867 | 1,873 | theorem liminf_preimage_iterate_ae_eq {f : α → α} (hf : QuasiMeasurePreserving f μ μ)
(hs : f ⁻¹' s =ᵐ[μ] s) : liminf (α := Set α) (fun n => (preimage f)^[n] s) atTop =ᵐ[μ] s := by |
rw [← ae_eq_set_compl_compl, @Filter.liminf_compl (Set α)]
rw [← ae_eq_set_compl_compl, ← preimage_compl] at hs
convert hf.limsup_preimage_iterate_ae_eq hs
ext1 n
simp only [← Set.preimage_iterate_eq, comp_apply, preimage_compl]
|
/-
Copyright (c) 2018 Mitchell Rowett. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Rowett, Scott Morrison
-/
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Group... | Mathlib/GroupTheory/Coset.lean | 266 | 277 | theorem rightCoset_eq_iff {x y : α} : op x • (s : Set α) = op y • s ↔ y * x⁻¹ ∈ s := by |
rw [Set.ext_iff]
simp_rw [mem_rightCoset_iff, SetLike.mem_coe]
constructor
· intro h
apply (h y).mpr
rw [mul_right_inv]
exact s.one_mem
· intro h z
rw [← inv_mul_cancel_left y x⁻¹]
rw [← mul_assoc]
exact s.mul_mem_cancel_right h
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 234 | 236 | theorem mem_filterAt_iff {x : α} {s : Set (Set α)} :
s ∈ v.filterAt x ↔ ∃ ε > (0 : ℝ), ∀ a ∈ v.setsAt x, a ⊆ closedBall x ε → a ∈ s := by |
simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf]
|
/-
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.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 76 | 78 | theorem separable_X : (X : R[X]).Separable := by |
rw [separable_def, derivative_X]
exact isCoprime_one_right
|
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Basic
/-!
# Lebesgue and Bochner integrals of conditional kernels
Integrals of `ProbabilityTheory.kernel.condKernel` an... | Mathlib/Probability/Kernel/Disintegration/Integral.lean | 284 | 290 | theorem Integrable.norm_integral_condKernel {f : α × Ω → E} (hf_int : Integrable f ρ) :
Integrable (fun x ↦ ‖∫ y, f (x, y) ∂ρ.condKernel x‖) ρ.fst := by |
refine hf_int.integral_norm_condKernel.mono hf_int.1.integral_condKernel.norm ?_
refine Filter.eventually_of_forall fun x ↦ ?_
rw [norm_norm]
refine (norm_integral_le_integral_norm _).trans_eq (Real.norm_of_nonneg ?_).symm
exact integral_nonneg_of_ae (Filter.eventually_of_forall fun y ↦ norm_nonneg _)
|
/-
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.Translations
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Algebra.ContinuedFrac... | Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean | 219 | 242 | theorem of_correctness_of_nth_stream_eq_none (nth_stream_eq_none : IntFractPair.stream v n = none) :
v = (of v).convergents (n - 1) := by |
induction n with
| zero => contradiction
-- IntFractPair.stream v 0 ≠ none
| succ n IH =>
let g := of v
change v = g.convergents n
have :
IntFractPair.stream v n = none ∨ ∃ ifp, IntFractPair.stream v n = some ifp ∧ ifp.fr = 0 :=
IntFractPair.succ_nth_stream_eq_none_iff.1 nth_stream_eq_n... |
/-
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.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Galois
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fie... | Mathlib/FieldTheory/IsSepClosed.lean | 168 | 179 | theorem algebraMap_surjective
[IsSepClosed k] [Algebra k K] [IsSeparable k K] :
Function.Surjective (algebraMap k K) := by |
refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (IsSeparable.isIntegral k x)
have hsep : (minpoly k x).Separable := IsSeparable.separable k x
have h : (minpoly k x).degree = 1 :=
degree_eq_one_of_irreducible k (minpoly.irreducible (IsSeparable.isIntegr... |
/-
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.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621... | Mathlib/Topology/Bornology/Constructions.lean | 88 | 91 | theorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t := by |
rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
rcases t.eq_empty_or_nonempty with (rfl | ht); · simp
simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or_iff]
|
/-
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 | 719 | 722 | theorem isBounded_iff_nndist {s : Set α} :
IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by |
simp only [isBounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
NNReal.coe_mk, exists_prop]
|
/-
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 | 169 | 172 | theorem recF_eq {α : Type _} (g : F α → α) (x : q.P.W) :
recF g x = g (abs (q.P.map (recF g) x.dest)) := by |
cases x
rfl
|
/-
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.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "lea... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 87 | 94 | theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞ := by |
rw [LpAddConst]
split_ifs with h
· apply ENNReal.rpow_lt_top_of_nonneg _ ENNReal.two_ne_top
simp only [one_div, sub_nonneg]
apply one_le_inv (ENNReal.toReal_pos h.1.ne' (h.2.trans ENNReal.one_lt_top).ne)
simpa using ENNReal.toReal_mono ENNReal.one_ne_top h.2.le
· exact ENNReal.one_lt_top
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Mo... | Mathlib/Data/Nat/Nth.lean | 183 | 190 | theorem exists_lt_card_nth_eq {x} (h : p x) :
∃ n, (∀ hf : (setOf p).Finite, n < hf.toFinset.card) ∧ nth p n = x := by |
refine (setOf p).finite_or_infinite.elim (fun hf => ?_) fun hf => ?_
· rcases exists_lt_card_finite_nth_eq hf h with ⟨n, hn, hx⟩
exact ⟨n, fun _ => hn, hx⟩
· rw [← @Set.mem_setOf_eq _ _ p, ← range_nth_of_infinite hf] at h
rcases h with ⟨n, hx⟩
exact ⟨n, fun hf' => absurd hf' hf, hx⟩
|
/-
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 | 216 | 217 | theorem restrict_apply_eq_zero' (hs : MeasurableSet s) : μ.restrict s t = 0 ↔ μ (t ∩ s) = 0 := by |
rw [restrict_apply' hs]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 165 | 169 | theorem eq_freeOn_iff : M = freeOn E ↔ M.E = E ∧ M.Indep E := by |
refine ⟨?_, fun h ↦ ?_⟩
· rintro rfl; simp [Subset.rfl]
simp only [eq_iff_indep_iff_indep_forall, freeOn_ground, freeOn_indep_iff, h.1, true_and]
exact fun I hIX ↦ iff_of_true (h.2.subset hIX) hIX
|
/-
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.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Ta... | Mathlib/Logic/Relation.lean | 345 | 350 | theorem cases_head (h : ReflTransGen r a b) : a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b := by |
induction h using Relation.ReflTransGen.head_induction_on
· left
rfl
· right
exact ⟨_, by assumption, by assumption⟩;
|
/-
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 | 335 | 340 | theorem volume_preimage_mul_right {a : ℝ} (h : a ≠ 0) (s : Set ℝ) :
volume ((· * a) ⁻¹' s) = ENNReal.ofReal (abs a⁻¹) * volume s :=
calc
volume ((· * a) ⁻¹' s) = Measure.map (· * a) volume s :=
((Homeomorph.mulRight₀ a h).toMeasurableEquiv.map_apply s).symm
_ = ENNReal.ofReal (abs a⁻¹) * volume s :=... | rw [map_volume_mul_right h]; rfl
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.monad fro... | Mathlib/Algebra/MvPolynomial/Monad.lean | 230 | 231 | theorem bind₂_comp_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T) :
(bind₂ g).comp (bind₂ f) = bind₂ ((bind₂ g).comp f) := by | ext : 2 <;> 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.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 379 | 417 | theorem openCover_TFAE {X Y : Scheme.{u}} [IsAffine Y] (f : X ⟶ Y) :
List.TFAE
[sourceAffineLocally (@P) f,
∃ 𝒰 : Scheme.OpenCover.{u} X, ∀ i : 𝒰.J, sourceAffineLocally (@P) (𝒰.map i ≫ f),
∀ (𝒰 : Scheme.OpenCover.{u} X) (i : 𝒰.J), sourceAffineLocally (@P) (𝒰.map i ≫ f),
∀ {U : Sc... |
tfae_have 1 → 4
· intro H U g hg V
-- Porting note: this has metavariable if I put it directly into rw
have := (hP.affine_openCover_TFAE f).out 0 3
rw [this] at H
haveI : IsAffine _ := V.2
rw [← Category.assoc]
-- Porting note: Lean could find this previously
have : IsOpenImmersion <| (... |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import d... | Mathlib/Data/PNat/Factors.lean | 141 | 146 | theorem coe_prod (v : PrimeMultiset) : (v.prod : ℕ) = (v : Multiset ℕ).prod := by |
let h : (v.prod : ℕ) = ((v.map Coe.coe).map Coe.coe).prod :=
PNat.coeMonoidHom.map_multiset_prod v.toPNatMultiset
rw [Multiset.map_map] at h
have : (Coe.coe : ℕ+ → ℕ) ∘ (Coe.coe : Nat.Primes → ℕ+) = Coe.coe := funext fun p => rfl
rw [this] at h; exact h
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 1,129 | 1,131 | theorem contractNth_apply_of_gt (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n)
(h : (j : ℕ) < k) : contractNth j op g k = g k.succ := by |
rw [contractNth, if_neg (not_lt_of_gt h), if_neg (Ne.symm <| ne_of_lt h)]
|
/-
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, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 106 | 110 | theorem unit_mul_mem_iff_mem {x y : α} (hy : IsUnit y) : y * x ∈ I ↔ x ∈ I := by |
refine ⟨fun h => ?_, fun h => I.mul_mem_left y h⟩
obtain ⟨y', hy'⟩ := hy.exists_left_inv
have := I.mul_mem_left y' h
rwa [← mul_assoc, hy', one_mul] at this
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 704 | 719 | theorem aleph0_le_cof {o} : ℵ₀ ≤ cof o ↔ IsLimit o := by |
rcases zero_or_succ_or_limit o with (rfl | ⟨o, rfl⟩ | l)
· simp [not_zero_isLimit, Cardinal.aleph0_ne_zero]
· simp [not_succ_isLimit, Cardinal.one_lt_aleph0]
· simp [l]
refine le_of_not_lt fun h => ?_
cases' Cardinal.lt_aleph0.1 h with n e
have := cof_cof o
rw [e, ord_nat] at this
cases n
... |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 977 | 984 | theorem closedBall_iSup {ι : Sort*} {p : ι → Seminorm 𝕜 E} (hp : BddAbove (range p)) (e : E)
{r : ℝ} (hr : 0 < r) : closedBall (⨆ i, p i) e r = ⋂ i, closedBall (p i) e r := by |
cases isEmpty_or_nonempty ι
· rw [iSup_of_empty', iInter_of_empty, Seminorm.sSup_empty]
exact closedBall_bot _ hr
· ext x
have := Seminorm.bddAbove_range_iff.mp hp (x - e)
simp only [mem_closedBall, mem_iInter, Seminorm.iSup_apply hp, ciSup_le_iff this]
|
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
impor... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 951 | 955 | theorem coeFn_abs {β} [TopologicalSpace β] [Lattice β] [TopologicalLattice β] [AddGroup β]
[TopologicalAddGroup β] (f : α →ₘ[μ] β) : ⇑|f| =ᵐ[μ] fun x => |f x| := by |
simp_rw [abs]
filter_upwards [AEEqFun.coeFn_sup f (-f), AEEqFun.coeFn_neg f] with x hx_sup hx_neg
rw [hx_sup, hx_neg, Pi.neg_apply]
|
/-
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.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.ord... | Mathlib/Algebra/Order/Archimedean.lean | 120 | 122 | theorem exists_nat_ge [OrderedSemiring α] [Archimedean α] (x : α) : ∃ n : ℕ, x ≤ n := by |
nontriviality α
exact (Archimedean.arch x one_pos).imp fun n h => by rwa [← nsmul_one]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,094 | 1,096 | theorem mk_surjective_eq_zero_iff :
#{f : α → β | Surjective f} = 0 ↔ #α < #β ∨ (#α ≠ 0 ∧ #β = 0) := by |
rw [mk_surjective_eq_zero_iff_lift, lift_lt]
|
/-
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.Nat.Gcd
import Batteries.Data.Int.DivMod
import Batteries.Lean.Float
/-! # Basics for the Rational Numbers -/
/--
Rational numbers, implemente... | .lake/packages/batteries/Batteries/Data/Rat/Basic.lean | 184 | 210 | theorem add.aux (a b : Rat) {g ad bd} (hg : g = a.den.gcd b.den)
(had : ad = a.den / g) (hbd : bd = b.den / g) :
let den := ad * b.den; let num := a.num * bd + b.num * ad
num.natAbs.gcd g = num.natAbs.gcd den := by |
intro den num
have ae : ad * g = a.den := had ▸ Nat.div_mul_cancel (hg ▸ Nat.gcd_dvd_left ..)
have be : bd * g = b.den := hbd ▸ Nat.div_mul_cancel (hg ▸ Nat.gcd_dvd_right ..)
have hden : den = ad * bd * g := by rw [Nat.mul_assoc, be]
rw [hden, Nat.Coprime.gcd_mul_left_cancel_right]
have cop : ad.Coprime bd... |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathl... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 225 | 243 | theorem add_eq_max_of_ne {q r : ℚ} (hne : padicNorm p q ≠ padicNorm p r) :
padicNorm p (q + r) = max (padicNorm p q) (padicNorm p r) := by |
wlog hlt : padicNorm p r < padicNorm p q
· rw [add_comm, max_comm]
exact this hne.symm (hne.lt_or_lt.resolve_right hlt)
have : padicNorm p q ≤ max (padicNorm p (q + r)) (padicNorm p r) :=
calc
padicNorm p q = padicNorm p (q + r + (-r)) := by ring_nf
_ ≤ max (padicNorm p (q + r)) (padicNorm p ... |
/-
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 | 1,369 | 1,369 | theorem frequently_false (f : Filter α) : ¬∃ᶠ _ in f, False := by | simp
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"0... | Mathlib/Algebra/Squarefree/Basic.lean | 201 | 205 | theorem IsRadical.squarefree (h0 : x ≠ 0) (h : IsRadical x) : Squarefree x := by |
rintro z ⟨w, rfl⟩
specialize h 2 (z * w) ⟨w, by simp_rw [pow_two, mul_left_comm, ← mul_assoc]⟩
rwa [← one_mul (z * w), mul_assoc, mul_dvd_mul_iff_right, ← isUnit_iff_dvd_one] at h
rw [mul_assoc, mul_ne_zero_iff] at h0; exact h0.2
|
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
#align_import algebra.grou... | Mathlib/Algebra/Order/Ring/Basic.lean | 35 | 35 | theorem map_sub_swap (x y : R) : f (x - y) = f (y - x) := by | rw [← map_neg, neg_sub]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,019 | 1,022 | theorem prodMonoidal_leftUnitor_inv_fst (X : C₁ × C₂) :
((λ_ X).inv : X ⟶ 𝟙_ _ ⊗ X).1 = (λ_ X.1).inv := by |
cases X
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 268 | 272 | theorem IsTopologicalBasis.inf_induced {γ} [s : TopologicalSpace β] {B₁ : Set (Set α)}
{B₂ : Set (Set β)} (h₁ : IsTopologicalBasis B₁) (h₂ : IsTopologicalBasis B₂) (f₁ : γ → α)
(f₂ : γ → β) :
IsTopologicalBasis (t := induced f₁ t ⊓ induced f₂ s) (image2 (f₁ ⁻¹' · ∩ f₂ ⁻¹' ·) B₁ B₂) := by |
simpa only [image2_image_left, image2_image_right] using (h₁.induced f₁).inf (h₂.induced f₂)
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/... | Mathlib/Algebra/Polynomial/Mirror.lean | 199 | 206 | theorem mirror_mul_of_domain : (p * q).mirror = p.mirror * q.mirror := by |
by_cases hp : p = 0
· rw [hp, zero_mul, mirror_zero, zero_mul]
by_cases hq : q = 0
· rw [hq, mul_zero, mirror_zero, mul_zero]
rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add]
rw [mul_assoc, ← mul_assoc q.reverse, ← X_pow_mul (p := reverse q)]
repeat' rw [mul_assoc]... |
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 971 | 974 | theorem Basis.equivFun_ofEquivFun [Finite ι] (e : M ≃ₗ[R] ι → R) :
(Basis.ofEquivFun e).equivFun = e := by |
ext j
simp_rw [Basis.equivFun_apply, Basis.ofEquivFun_repr_apply]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 23 | 24 | theorem then_eq_eq {o₁ o₂ : Ordering} : o₁.then o₂ = eq ↔ o₁ = eq ∧ o₂ = eq := by |
cases o₁ <;> simp [«then»]
|
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