Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
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/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | Mathlib/Topology/PartialHomeomorph.lean | 492 | 494 | theorem iff_symm_preimage_eq' :
e.IsImage s t β e.target β© e.symm β»ΒΉ' (e.source β© s) = e.target β© t := by | rw [iff_symm_preimage_eq, β image_source_inter_eq, β image_source_inter_eq'] |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 543 | 546 | theorem pentagon_inv_hom_hom_hom_hom :
(Ξ±_ W X Y).inv β· Z β« (Ξ±_ (W β X) Y Z).hom β« (Ξ±_ W X (Y β Z)).hom =
(Ξ±_ W (X β Y) Z).hom β« W β (Ξ±_ X Y Z).hom := by | simp [β cancel_epi ((Ξ±_ W X Y).hom β· Z)] |
/-
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, YaΓ«l Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 218 | 219 | theorem top_bihimp : β€ β a = a := by | rw [bihimp_comm, bihimp_top] |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A cβ cβ cβ`: a basis... | Mathlib/Algebra/QuaternionBasis.lean | 114 | 114 | theorem lift_one : q.lift (1 : β[R,cβ,cβ,cβ]) = 1 := by | simp [lift] |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Eric Wieser
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
impor... | Mathlib/Data/Real/Pointwise.lean | 37 | 46 | theorem Real.sInf_smul_of_nonneg (ha : 0 β€ a) (s : Set β) : sInf (a β’ s) = a β’ sInf s := by | obtain rfl | hs := s.eq_empty_or_nonempty
Β· rw [smul_set_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
Β· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddBelow s
Β· exact ((OrderIso.smulRight ha').map_csInf' hs h).symm
Β· rw [Real.sInf_of_not_bddBelow (mt (bddB... |
/-
Copyright (c) 2024 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.MeasureTheory.Integral.PeakFunction
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
/-!
# Fourier inversion formula
In a fin... | Mathlib/Analysis/Fourier/Inversion.lean | 186 | 190 | theorem Continuous.fourier_inversion_inv (h : Continuous f)
(hf : Integrable f) (h'f : Integrable (π f)) :
π (πβ» f) = f := by | ext v
exact hf.fourier_inversion_inv h'f h.continuousAt |
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Probability.IdentDistrib
import Mathlib.Probability.Independence.Integrable
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mat... | Mathlib/Probability/StrongLaw.lean | 605 | 617 | theorem strong_law_ae_real {Ξ© : Type*} {m : MeasurableSpace Ξ©} {ΞΌ : Measure Ξ©}
(X : β β Ξ© β β) (hint : Integrable (X 0) ΞΌ)
(hindep : Pairwise ((IndepFun Β· Β· ΞΌ) on X))
(hident : β i, IdentDistrib (X i) (X 0) ΞΌ ΞΌ) :
βα΅ Ο βΞΌ, Tendsto (fun n : β => (β i β range n, X i Ο) / n) atTop (π ΞΌ[X 0]) := by | let mΞ© : MeasureSpace Ξ© := β¨ΞΌβ©
-- first get rid of the trivial case where the space is not a probability space
by_cases h : βα΅ Ο, X 0 Ο = 0
Β· have I : βα΅ Ο, β i, X i Ο = 0 := by
rw [ae_all_iff]
intro i
exact (hident i).symm.ae_snd (p := fun x β¦ x = 0) measurableSet_eq h
filter_upwards [I] wi... |
/-
Copyright (c) 2018 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | Mathlib/Topology/MetricSpace/Isometry.lean | 46 | 48 | theorem isometry_iff_dist_eq [PseudoMetricSpace Ξ±] [PseudoMetricSpace Ξ²] {f : Ξ± β Ξ²} :
Isometry f β β x y, dist (f x) (f y) = dist x y := by | simp only [isometry_iff_nndist_eq, β coe_nndist, NNReal.coe_inj] |
/-
Copyright (c) 2022 MarΓa InΓ©s de Frutos-FernΓ‘ndez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: MarΓa InΓ©s de Frutos-FernΓ‘ndez
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 412 | 420 | theorem count_pow_self (n : β) :
count K v ((v.asIdeal : FractionalIdeal Rβ° K) ^ n) = n := by | rw [count_pow, count_self, mul_one]
/-- `val_v(Iβ»βΏ) = -val_v(IβΏ)` for every `n β β€`. -/
theorem count_neg_zpow (n : β€) (I : FractionalIdeal Rβ° K) :
count K v (I ^ (-n)) = - count K v (I ^ n) := by
by_cases hI : I = 0
Β· by_cases hn : n = 0 |
/-
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.Group.Embedding
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.Algebra.Group.Pointwise.F... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 25 | 30 | theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
(a * b).support β a.support * b.support := by | rw [MonoidAlgebra.mul_def]
exact support_sum.trans <| biUnion_subset.2 fun _x hx β¦
support_sum.trans <| biUnion_subset.2 fun _y hy β¦
support_single_subset.trans <| singleton_subset_iff.2 <| mem_imageβ_of_mem hx hy |
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Set.Finite.Range
import Mathlib.Logic.Equiv.Embedding
/-!
# Numb... | Mathlib/Data/Fintype/CardEmbedding.lean | 36 | 50 | theorem card_embedding_eq {Ξ± Ξ² : Type*} [Fintype Ξ±] [Fintype Ξ²] [emb : Fintype (Ξ± βͺ Ξ²)] :
βΞ± βͺ Ξ²β = βΞ²β.descFactorial βΞ±β := by | rw [Subsingleton.elim emb Embedding.fintype]
refine Fintype.induction_empty_option (P := fun t β¦ βt βͺ Ξ²β = βΞ²β.descFactorial βtβ)
(fun Ξ±β Ξ±β hβ e ih β¦ ?_) (?_) (fun Ξ³ h ih β¦ ?_) Ξ± <;> dsimp only at * <;> clear! Ξ±
Β· letI := Fintype.ofEquiv _ e.symm
rw [β card_congr (Equiv.embeddingCongr e (Equiv.refl Ξ²))... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 690 | 693 | theorem affineCombination_affineCombinationSingleWeights [DecidableEq ΞΉ] (p : ΞΉ β P) {i : ΞΉ}
(hi : i β s) : s.affineCombination k p (affineCombinationSingleWeights k i) = p i := by | refine s.affineCombination_of_eq_one_of_eq_zero _ _ hi (by simp) ?_
rintro j - hj |
/-
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 Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 318 | 323 | theorem compl_compl_image [BooleanAlgebra Ξ±] (S : Set Ξ±) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by | rw [β image_comp, compl_comp_compl, image_id]
theorem image_insert_eq {f : Ξ± β Ξ²} {a : Ξ±} {s : Set Ξ±} :
f '' insert a s = insert (f a) (f '' s) := by |
/-
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.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
/-!
# Birkhoff average in a normed space
In this file we prove some lemmas about ... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 106 | 122 | theorem uniformEquicontinuous_birkhoffAverage (hf : LipschitzWith 1 f) (hg : UniformContinuous g) :
UniformEquicontinuous (birkhoffAverage π f g) := by | refine Metric.uniformity_basis_dist_le.uniformEquicontinuous_iff_right.2 fun Ξ΅ hΞ΅ β¦ ?_
rcases (uniformity_basis_edist_le.uniformContinuous_iff Metric.uniformity_basis_dist_le).1 hg Ξ΅ hΞ΅
with β¨Ξ΄, hΞ΄β, hδΡβ©
refine mem_uniformity_edist.2 β¨Ξ΄, hΞ΄β, fun {x y} h n β¦ ?_β©
calc
dist (birkhoffAverage π f g n x) (bi... |
/-
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.CategoryTheory.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 158 | 161 | theorem pullbackDiagonalMapIso.hom_fst :
(pullbackDiagonalMapIso f i iβ iβ).hom β« pullback.fst _ _ =
pullback.snd _ _ β« pullback.fst _ _ := by | delta pullbackDiagonalMapIso |
/-
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.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-!
# Action of `Mα΅α΅α΅` on `Lα΅` spaces
In this file we... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 78 | 79 | theorem smul_Lp_add (c : Mα΅α΅α΅) : β f g : Lp E p ΞΌ, c β’ (f + g) = c β’ f + c β’ g := by | rintro β¨β¨β©, _β© β¨β¨β©, _β©; rfl |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 110 | 113 | theorem ascPochhammer_ne_zero_eval_zero {n : β} (h : n β 0) : (ascPochhammer S n).eval 0 = 0 := by | simp [ascPochhammer_eval_zero, h]
theorem ascPochhammer_succ_right (n : β) : |
/-
Copyright (c) 2022 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 300 | 305 | theorem succ_nadd : succ a β― b = succ (a β― b) := by | rw [β one_nadd (a β― b), β nadd_assoc, one_nadd]
@[simp]
theorem nadd_nat (n : β) : a β― n = a + n := by
induction' n with n hn
Β· simp |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 170 | 172 | theorem withDensity_indicator {s : Set Ξ±} (hs : MeasurableSet s) (f : Ξ± β ββ₯0β) :
ΞΌ.withDensity (s.indicator f) = (ΞΌ.restrict s).withDensity f := by | ext1 t ht |
/-
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 SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 131 | 131 | theorem zero_rpow_le_one (x : β) : (0 : β) ^ x β€ 1 := by | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | Mathlib/Algebra/Polynomial/Monic.lean | 405 | 411 | theorem nextCoeff_map (hf : Injective f) (p : R[X]) : (p.map f).nextCoeff = f p.nextCoeff := by | unfold nextCoeff
rw [natDegree_map_eq_of_injective hf]
split_ifs <;> simp [*]
theorem leadingCoeff_of_injective (hf : Injective f) (p : R[X]) :
leadingCoeff (p.map f) = f (leadingCoeff p) := by |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# LindelΓΆf sets and LindelΓΆf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 364 | 369 | theorem IsLindelof.countable_of_discrete [DiscreteTopology X] (hs : IsLindelof s) :
s.Countable := by | have : β x : X, ({x} : Set X) β π x := by simp [nhds_discrete]
rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with β¨t, ht, _, hssubtβ©
rw [biUnion_of_singleton] at hssubt
exact ht.mono hssubt |
/-
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 | 166 | 168 | theorem frobeniusMorphism_iso_of_expComparison_iso (h : L β£ F) (A : C)
[i : IsIso (expComparison F A).natTrans] : IsIso (frobeniusMorphism F h A).natTrans := by | rw [β frobeniusMorphism_mate F h] at i |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | Mathlib/Data/Nat/Choose/Sum.lean | 94 | 115 | theorem sum_range_choose_halfway (m : β) : (β i β range (m + 1), (2 * m + 1).choose i) = 4 ^ m :=
have : (β i β range (m + 1), (2 * m + 1).choose (2 * m + 1 - i)) =
β i β range (m + 1), (2 * m + 1).choose i :=
sum_congr rfl fun i hi β¦ choose_symm <| by linarith [mem_range.1 hi]
mul_right_injectiveβ two_ne... | rw [two_mul, this]
_ = (β i β range (m + 1), (2 * m + 1).choose i) +
β i β Ico (m + 1) (2 * m + 2), (2 * m + 1).choose i := by
rw [range_eq_Ico, sum_Ico_reflect _ _ (by omega)]
congr
omega
_ = β i β range (2 * m + 2), (2 * m + 1).choose i := sum_range_add_sum_Ico _ (by om... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.... | Mathlib/GroupTheory/ClassEquation.lean | 47 | 70 | theorem Group.nat_card_center_add_sum_card_noncenter_eq_card [Finite G] :
Nat.card (Subgroup.center G) + βαΆ x β noncenter G, Nat.card x.carrier = Nat.card G := by | classical
cases nonempty_fintype G
rw [@Nat.card_eq_fintype_card G, β sum_conjClasses_card_eq_card, β
Finset.sum_sdiff (ConjClasses.noncenter G).toFinset.subset_univ]
simp only [Nat.card_eq_fintype_card, Set.toFinset_card]
congr 1
swap
Β· convert finsum_cond_eq_sum_of_cond_iff _ _
simp [Set.mem_toFin... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 479 | 480 | theorem iInf_neBot_of_directed {f : ΞΉ β Filter Ξ±} [hn : Nonempty Ξ±] (hd : Directed (Β· β₯ Β·) f)
(hb : β i, NeBot (f i)) : NeBot (iInf f) := by | |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Measure.Ty... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 44 | 54 | theorem borel_eq_generateFrom_Iio_rat : borel β = .generateFrom (β a : β, {Iio (a : β)}) := by | rw [borel_eq_generateFrom_Iio]
refine le_antisymm
(generateFrom_le ?_)
(generateFrom_mono <| iUnion_subset fun q β¦ singleton_subset_iff.mpr <| mem_range_self _)
rintro _ β¨a, rflβ©
have : IsLUB (range ((β) : β β β) β© Iio a) a := by
simp [isLUB_iff_le_iff, mem_upperBounds, β le_iff_forall_rat_lt_imp_le]
... |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 148 | 161 | theorem lfpApprox_eq_of_mem_fixedPoints {a b : Ordinal} (h_init : x β€ f x) (h_ab : a β€ b)
(h : lfpApprox f x a β fixedPoints f) : lfpApprox f x b = lfpApprox f x a := by | rw [mem_fixedPoints_iff] at h
induction b using Ordinal.induction with | h b IH =>
apply le_antisymm
Β· conv => left; rw [lfpApprox]
apply sSup_le
simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq,
forall_eq_or_imp, forall_exists_index, and_imp, forall_apply_eq_imp_iff... |
/-
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.Nodup
import Mathlib.Data.List.Lattice
import Batteries.Data.List.Pairwise
/-!
# Erasure of duplicates in a list
This file proves basic res... | Mathlib/Data/List/Dedup.lean | 135 | 140 | theorem Disjoint.union_eq {xs ys : List Ξ±} (h : Disjoint xs ys) :
xs βͺ ys = xs.dedup ++ ys := by | induction xs with
| nil => simp
| cons x xs ih =>
rw [cons_union] |
/-
Copyright (c) 2022 MarΓa InΓ©s de Frutos-FernΓ‘ndez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: MarΓa InΓ©s de Frutos-FernΓ‘ndez
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 290 | 291 | theorem valuation_le_one (r : R) : v.valuation K r β€ 1 := by | rw [valuation_of_algebraMap]; exact v.intValuation_le_one r |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | Mathlib/Algebra/Lie/Basic.lean | 412 | 414 | theorem coe_injective : @Function.Injective (Lβ βββ
Rβ Lβ) (Lβ β Lβ) (β) := by | rintro β¨β¨β¨f, _β©, _β©, _β© β¨β¨β¨g, _β©, _β©, _β© h
congr |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... | Mathlib/MeasureTheory/Integral/Marginal.lean | 171 | 175 | theorem lmarginal_erase (f : (β i, X i) β ββ₯0β) (hf : Measurable f) {i : Ξ΄}
(hi : i β s) (x : β i, X i) :
(β«β―β«β»_s, f βΞΌ) x = β«β» xα΅’, (β«β―β«β»_(erase s i), f βΞΌ) (Function.update x i xα΅’) βΞΌ i := by | simpa [insert_erase hi] using lmarginal_insert _ hf (not_mem_erase i s) x |
/-
Copyright (c) 2022 Mantas BakΕ‘ys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas BakΕ‘ys
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.Da... | Mathlib/Algebra/Order/Rearrangement.lean | 243 | 247 | theorem Monovary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : Monovary f g) :
β i, f (Ο i) β’ g i = β i, f i β’ g i β Monovary (f β Ο) g := by | simp [(hfg.monovaryOn _).sum_comp_perm_smul_eq_sum_smul_iff fun _ _ β¦ mem_univ _]
/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and |
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Tactic.Field... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 181 | 187 | theorem treesOfNumNodesEq_card_eq_catalan (n : β) : #(treesOfNumNodesEq n) = catalan n := by | induction n using Nat.case_strong_induction_on with
| hz => simp
| hi n ih =>
rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
Β· apply sum_congr rfl
rintro β¨i, jβ© H |
/-
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 SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 93 | 94 | theorem log_one : log 1 = 0 := by | simp [log] |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | Mathlib/Algebra/Lie/Basic.lean | 176 | 176 | theorem lie_skew : -β
y, xβ = β
x, yβ := by | |
/-
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.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | Mathlib/SetTheory/Lists.lean | 304 | 305 | theorem sizeof_pos {b} (l : Lists' Ξ± b) : 0 < SizeOf.sizeOf l := by | cases l <;> simp only [Lists'.atom.sizeOf_spec, Lists'.nil.sizeOf_spec, Lists'.cons'.sizeOf_spec, |
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 768 | 771 | theorem coe_abs_toReal_of_sign_nonneg {ΞΈ : Angle} (h : 0 β€ ΞΈ.sign) : β|ΞΈ.toReal| = ΞΈ := by | rw [abs_eq_self.2 (toReal_nonneg_iff_sign_nonneg.2 h), coe_toReal]
theorem neg_coe_abs_toReal_of_sign_nonpos {ΞΈ : Angle} (h : ΞΈ.sign β€ 0) : -β|ΞΈ.toReal| = ΞΈ := by |
/-
Copyright (c) 2023 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Boris Bolvig Kjær, Jon Eugster, Sina Hazratpour
-/
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.Topology.Category.CompHaus.EffectiveEpi... | Mathlib/Topology/Category/Profinite/EffectiveEpi.lean | 69 | 82 | theorem effectiveEpiFamily_tfae
{Ξ± : Type} [Finite Ξ±] {B : Profinite.{u}}
(X : Ξ± β Profinite.{u}) (Ο : (a : Ξ±) β (X a βΆ B)) :
TFAE
[ EffectiveEpiFamily X Ο
, Epi (Sigma.desc Ο)
, β b : B, β (a : Ξ±) (x : X a), Ο a x = b
] := by | tfae_have 2 β 1
| _ => by
simpa [β effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc Ο)).out 0 1]
tfae_have 1 β 2 := fun _ β¦ inferInstance
tfae_have 3 β 1 := by
erw [((CompHaus.effectiveEpiFamily_tfae |
/-
Copyright (c) 2022 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 309 | 309 | theorem nat_nadd (n : β) : βn β― a = a + n := by | rw [nadd_comm, nadd_nat] |
/-
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.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 407 | 409 | theorem det_nonsing_inv : Aβ»ΒΉ.det = Ring.inverse A.det := by | by_cases h : IsUnit A.det
Β· cases h.nonempty_invertible |
/-
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
/-!
# Monad operations on `MvPolynomial`
... | Mathlib/Algebra/MvPolynomial/Monad.lean | 273 | 275 | theorem bindβ_monomial (f : Ο β MvPolynomial Ο R) (d : Ο ββ β) (r : R) :
bindβ f (monomial d r) = C r * β i β d.support, f i ^ d i := by | simp only [monomial_eq, map_mul, bindβ_C_right, Finsupp.prod, map_prod, |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
impo... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 40 | 45 | theorem isIntegral (hpos : 0 < n) : IsIntegral β€ ΞΌ := by | use X ^ n - 1
constructor
Β· exact monic_X_pow_sub_C 1 (ne_of_lt hpos).symm
Β· simp only [((IsPrimitiveRoot.iff_def ΞΌ n).mp h).left, evalβ_one, evalβ_X_pow, evalβ_sub,
sub_self] |
/-
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.EuclideanDomain.Int
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Coprim... | Mathlib/Data/ZMod/Coprime.lean | 24 | 28 | theorem eq_zero_iff_gcd_ne_one {a : β€} {p : β} [pp : Fact p.Prime] :
(a : ZMod p) = 0 β a.gcd p β 1 := by | rw [Ne, Int.gcd_comm, β Int.isCoprime_iff_gcd_eq_one,
(Nat.prime_iff_prime_int.1 pp.1).coprime_iff_not_dvd, Classical.not_not,
intCast_zmod_eq_zero_iff_dvd] |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, RΓ©my Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | Mathlib/Probability/Process/Stopping.lean | 703 | 724 | theorem stoppedProcess_eq_stoppedValue {u : ΞΉ β Ξ© β Ξ²} {Ο : Ξ© β ΞΉ} :
stoppedProcess u Ο = fun i => stoppedValue u fun Ο => min i (Ο Ο) :=
rfl
theorem stoppedValue_stoppedProcess {u : ΞΉ β Ξ© β Ξ²} {Ο Ο : Ξ© β ΞΉ} :
stoppedValue (stoppedProcess u Ο) Ο = stoppedValue u fun Ο => min (Ο Ο) (Ο Ο) :=
rfl
theorem sto... | simp [stoppedProcess, min_eq_left h]
theorem stoppedProcess_eq_of_ge {u : ΞΉ β Ξ© β Ξ²} {Ο : Ξ© β ΞΉ} {i : ΞΉ} {Ο : Ξ©} (h : Ο Ο β€ i) :
stoppedProcess u Ο i Ο = u (Ο Ο) Ο := by simp [stoppedProcess, min_eq_right h]
section ProgMeasurable
variable [MeasurableSpace ΞΉ] [TopologicalSpace ΞΉ] [OrderTopology ΞΉ] [SecondCountab... |
/-
Copyright (c) 2023 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 170 | 171 | theorem lineDerivWithin_univ : lineDerivWithin π f univ x v = lineDeriv π f x v := by | simp [lineDerivWithin, lineDeriv] |
/-
Copyright (c) 2023 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | Mathlib/Probability/Independence/Kernel.lean | 304 | 327 | theorem indep_bot_left (m' : MeasurableSpace Ξ©) {_mΞ© : MeasurableSpace Ξ©}
{ΞΊ : Kernel Ξ± Ξ©} {ΞΌ : Measure Ξ±} [IsZeroOrMarkovKernel ΞΊ] :
Indep β₯ m' ΞΊ ΞΌ := (indep_bot_right m').symm
theorem indepSet_empty_right {_mΞ© : MeasurableSpace Ξ©}
{ΞΊ : Kernel Ξ± Ξ©} {ΞΌ : Measure Ξ±} [IsZeroOrMarkovKernel ΞΊ] (s : Set Ξ©) :
... | simp only [IndepSet, generateFrom_singleton_empty]
exact indep_bot_right _
theorem indepSet_empty_left {_mΞ© : MeasurableSpace Ξ©} {ΞΊ : Kernel Ξ± Ξ©}
{ΞΌ : Measure Ξ±} [IsZeroOrMarkovKernel ΞΊ] (s : Set Ξ©) :
IndepSet β
s ΞΊ ΞΌ :=
(indepSet_empty_right s).symm
theorem indepSets_of_indepSets_of_le_left {sβ sβ sβ : S... |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | Mathlib/NumberTheory/LucasLehmer.lean | 145 | 145 | theorem sZMod_eq_s (p' : β) (i : β) : sZMod (p' + 2) i = (s i : ZMod (2 ^ (p' + 2) - 1)) := by | |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | Mathlib/Data/Matrix/Kronecker.lean | 132 | 136 | theorem kroneckerMap_diagonal_left [Zero Ξ±] [Zero Ξ³] [DecidableEq l] (f : Ξ± β Ξ² β Ξ³)
(hf : β b, f 0 b = 0) (a : l β Ξ±) (B : Matrix m n Ξ²) :
kroneckerMap f (diagonal a) B =
Matrix.reindex (Equiv.prodComm _ _) (Equiv.prodComm _ _)
(blockDiagonal fun i => B.map fun b => f (a i) b) := by | |
/-
Copyright (c) 2023 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne
-/
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Probability.Kernel.CondDistrib
import Mathlib.Probability.ConditionalProbability
/-!
# Kernel as... | Mathlib/Probability/Kernel/Condexp.lean | 44 | 49 | theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id [TopologicalSpace F]
(hm : m β€ mΞ©) (hf : AEStronglyMeasurable f ΞΌ) :
AEStronglyMeasurable[m.prod mΞ©] (fun x : Ξ© Γ Ξ© => f x.2)
(@Measure.map Ξ© (Ξ© Γ Ξ©) mΞ© (m.prod mΞ©) (fun Ο => (id Ο, id Ο)) ΞΌ) := by | rw [β aestronglyMeasurable_comp_snd_map_prodMk_iff (measurable_id'' hm)] at hf
simp_rw [id] at hf β’ |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.LocalProperties.Basic
/-!
# The meta properties of surjective ring homomorphisms.
## Main results
Let `R` be a commutative ring, `M` be a subm... | Mathlib/RingTheory/RingHom/Surjective.lean | 36 | 45 | theorem surjective_stableUnderComposition : StableUnderComposition surjective := by | introv R hf hg; exact hg.comp hf
theorem surjective_respectsIso : RespectsIso surjective := by
apply surjective_stableUnderComposition.respectsIso
intros _ _ _ _ e
exact e.surjective
theorem surjective_isStableUnderBaseChange : IsStableUnderBaseChange surjective := by
refine IsStableUnderBaseChange.mk _ surje... |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 45 | 47 | theorem derivativeFun_coe (f : R[X]) : (f : Rβ¦Xβ§).derivativeFun = derivative f := by | ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative] |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
/-!
# Destuttering of Lists
This file proves theorems about `List.destutter` (in `Data.List.Defs`), which greedily remov... | Mathlib/Data/List/Destutter.lean | 60 | 61 | theorem destutter'_sublist (a) : l.destutter' R a <+ a :: l := by | induction' l with b l hl generalizing a |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 210 | 212 | theorem adjMatrix_mul_apply [NonAssocSemiring Ξ±] (M : Matrix V V Ξ±) (v w : V) :
(G.adjMatrix Ξ± * M) v w = β u β G.neighborFinset v, M u w := by | simp [mul_apply, neighborFinset_eq_filter, sum_filter] |
/-
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 SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 663 | 666 | theorem rpow_le_one_of_one_le_of_nonpos {x z : β} (hx : 1 β€ x) (hz : z β€ 0) : x ^ z β€ 1 := by | convert rpow_le_rpow_of_exponent_le hx hz
exact (rpow_zero x).symm |
/-
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
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | Mathlib/Data/Set/Card.lean | 580 | 584 | theorem ncard_singleton_inter (a : Ξ±) (s : Set Ξ±) : ({a} β© s).ncard β€ 1 := by | rw [β Nat.cast_le (Ξ± := ββ), (toFinite _).cast_ncard_eq, Nat.cast_one]
apply encard_singleton_inter
@[simp] |
/-
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 HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | Mathlib/SetTheory/Ordinal/Exponential.lean | 284 | 291 | theorem log_zero_left : β b, log 0 b = 0 :=
log_of_left_le_one zero_le_one
@[simp]
theorem log_zero_right (b : Ordinal) : log b 0 = 0 := by | obtain hb | hb := lt_or_le 1 b
Β· rw [log_def hb, β Ordinal.le_zero, pred_le, succ_zero]
apply csInf_le' |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Jens Wagemaker, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.... | Mathlib/Algebra/BigOperators/Associated.lean | 82 | 100 | theorem divisor_closure_eq_closure [CancelCommMonoidWithZero Ξ±]
(x y : Ξ±) (hxy : x * y β closure { r : Ξ± | IsUnit r β¨ Prime r}) :
x β closure { r : Ξ± | IsUnit r β¨ Prime r} := by | obtain β¨m, hm, hprodβ© := exists_multiset_of_mem_closure hxy
induction m using Multiset.induction generalizing x y with
| empty =>
apply subset_closure
simp only [Set.mem_setOf]
simp only [Multiset.prod_zero] at hprod
left; exact isUnit_of_mul_eq_one _ _ hprod.symm
| cons c s hind =>
simp only ... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | Mathlib/Algebra/Polynomial/Div.lean | 417 | 422 | theorem modByMonic_one (p : R[X]) : p %β 1 = 0 :=
(modByMonic_eq_zero_iff_dvd (by convert monic_one (R := R))).2 (one_dvd _)
@[simp]
theorem divByMonic_one (p : R[X]) : p /β 1 = p := by | conv_rhs => rw [β modByMonic_add_div p monic_one]; simp |
/-
Copyright (c) 2022 Violeta HernΓ‘ndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta HernΓ‘ndez Palacios
-/
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic.Basic
/-!
### Cardinality of algebraic numbers
In this file, ... | Mathlib/Algebra/AlgebraicCard.lean | 45 | 54 | theorem cardinalMk_lift_le_mul :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } β€ Cardinal.lift.{v} #R[X] * β΅β := by | rw [β mk_uLift, β mk_uLift]
choose g hgβ hgβ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => ?_
rw [lift_le_aleph0, le_aleph0_iff_set_countable]
suffices MapsTo (β) (g β»ΒΉ' {f}) (f.rootSet A) from
this.countable_of_injOn Subtype.coe_inje... |
/-
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.Order.Basic
/-!
# Set neighborhoods of intervals
In this file we prove basic theorems about `πΛ’ s`,
where `s` is one of the intervals
`Se... | Mathlib/Topology/Order/NhdsSet.lean | 44 | 45 | theorem nhdsSet_Ioc (h : a < b) : πΛ’ (Ioc a b) = π b β π (Ioo a b) := by | rw [β Ioo_insert_right h, nhdsSet_insert, nhdsSet_Ioo] |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Data.Nat.Log
/-!
# Integer logarithms in a field with respect to a natural base
This file defines two `β€`-value... | Mathlib/Data/Int/Log.lean | 228 | 228 | theorem clog_of_left_le_one {b : β} (hb : b β€ 1) (r : R) : clog b r = 0 := by | |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 166 | 170 | theorem toOuterMeasure_apply_eq_zero_iff : p.toOuterMeasure s = 0 β Disjoint p.support s := by | rw [toOuterMeasure_apply, ENNReal.tsum_eq_zero]
exact funext_iff.symm.trans Set.indicator_eq_zero'
theorem toOuterMeasure_apply_eq_one_iff : p.toOuterMeasure s = 1 β p.support β s := by |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | Mathlib/Topology/Order.lean | 644 | 646 | theorem continuous_le_dom {tβ tβ : TopologicalSpace Ξ±} {tβ : TopologicalSpace Ξ²} (hβ : tβ β€ tβ)
(hβ : Continuous[tβ, tβ] f) : Continuous[tβ, tβ] f := by | rw [continuous_iff_le_induced] at hβ β’ |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 240 | 242 | theorem fderiv_add_const (c : F) : fderiv π (fun y => f y + c) x = fderiv π f x := by | simp only [β fderivWithin_univ, fderivWithin_add_const] |
/-
Copyright (c) 2021 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
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 (i : Fin r β Unit) :
((listTransvecCol M).prod * M) (inr unit) i = M (inr unit) i := by | simpa using listTransvecCol_mul_last_row_drop M i (zero_le _)
/-- Multiplying by all the matrices in `listTransvecCol M` kills all the coefficients in the
last column but the last one. -/
theorem listTransvecCol_mul_last_col (hM : M (inr unit) (inr unit) β 0) (i : Fin r) :
((listTransvecCol M).prod * M) (inl i) (i... |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, YaΓ«l Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... | Mathlib/Data/Nat/Log.lean | 251 | 252 | theorem clog_of_two_le {b n : β} (hb : 1 < b) (hn : 2 β€ n) :
clog b n = clog b ((n + b - 1) / b) + 1 := by | rw [clog, dif_pos (β¨hb, hnβ© : 1 < b β§ 1 < n)] |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# LindelΓΆf sets and LindelΓΆf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 237 | 242 | theorem IsLindelof.inter_iInter_nonempty {ΞΉ : Type v} (hs : IsLindelof s) (t : ΞΉ β Set X)
(htc : β i, IsClosed (t i)) (hst : β u : Set ΞΉ, u.Countable β§ (s β© β i β u, t i).Nonempty) :
(s β© β i, t i).Nonempty := by | contrapose! hst
rcases hs.elim_countable_subfamily_closed t htc hst with β¨u, β¨_, husubβ©β©
exact β¨u, fun _ β¦ husubβ© |
/-
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 SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 587 | 590 | theorem rpow_neg (x : ββ₯0β) (y : β) : x ^ (-y) = (x ^ y)β»ΒΉ := by | cases x with
| top =>
rcases lt_trichotomy y 0 with (H | H | H) <;> |
/-
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.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | Mathlib/Data/Sign.lean | 405 | 409 | theorem sign_mul_self (x : Ξ±) : sign x * x = |x| := by | rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]
@[simp]
theorem self_mul_sign (x : Ξ±) : x * sign x = |x| := by |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 55 | 58 | theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by | ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero] |
/-
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.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 218 | 221 | theorem exists_not_isUnit_of_not_isField [Nontrivial R] (hf : Β¬IsField R) :
β (x : R) (_hx : x β (0 : R)), Β¬IsUnit x := by | have : Β¬_ := fun h => hf β¨exists_pair_ne R, mul_comm, hβ©
simp_rw [isUnit_iff_exists_inv] |
/-
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.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | Mathlib/Analysis/Convex/Side.lean | 248 | 252 | theorem sSameSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v β s.direction) :
s.SSameSide x (v +α΅₯ y) β s.SSameSide x y := by | rw [sSameSide_comm, sSameSide_vadd_left_iff hv, sSameSide_comm]
theorem wOppSide_vadd_left_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v β s.direction) : |
/-
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 HΓΆlzl, Mario Carneiro, SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | Mathlib/Topology/MetricSpace/Bounded.lean | 531 | 537 | theorem Metric.cobounded_eq_cocompact [ProperSpace Ξ±] : cobounded Ξ± = cocompact Ξ± := by | nontriviality Ξ±; inhabit Ξ±
exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
fun _ _ β¦ (isCompact_closedBall _ _).compl_mem_cocompact
theorem tendsto_dist_right_cocompact_atTop [ProperSpace Ξ±] (x : Ξ±) :
Tendsto (dist Β· x) (cocompact Ξ±) atTop := |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | Mathlib/FieldTheory/Finite/Basic.lean | 225 | 228 | theorem pow_card (a : K) : a ^ q = a := by | by_cases h : a = 0; Β· rw [h]; apply zero_pow Fintype.card_ne_zero
rw [β Nat.succ_pred_eq_of_pos Fintype.card_pos, pow_succ, Nat.pred_eq_sub_one,
pow_card_sub_one_eq_one a h, one_mul] |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Na... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 537 | 548 | theorem norm_iteratedFDeriv_clm_apply {f : E β F βL[π] G} {g : E β F} {N : WithTop ββ} {n : β}
(hf : ContDiff π N f) (hg : ContDiff π N g) (x : E) (hn : n β€ N) :
βiteratedFDeriv π n (fun y : E => (f y) (g y)) xβ β€ β i β Finset.range (n + 1),
β(n.choose i) * βiteratedFDeriv π i f xβ * βiteratedFDeriv ... | simp only [β iteratedFDerivWithin_univ]
exact norm_iteratedFDerivWithin_clm_apply hf.contDiffOn hg.contDiffOn uniqueDiffOn_univ
(Set.mem_univ x) hn
theorem norm_iteratedFDerivWithin_clm_apply_const {f : E β F βL[π] G} {c : F} {s : Set E} {x : E}
{N : WithTop ββ} {n : β} (hf : ContDiffWithinAt π N f s x) (h... |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes HΓΆlzl, YaΓ«l Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 714 | 716 | theorem enorm'_le_iff_norm_le {x : E} {y : F} : βxββ β€ βyββ β βxβ β€ βyβ := by | simp only [β ofReal_norm']
refine β¨fun h β¦ ?_, fun h β¦ by gcongrβ© |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Violeta HernΓ‘ndez Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | Mathlib/Order/Cover.lean | 162 | 165 | theorem WCovBy.Ioc_subset (h : a β©Ώ b) : Ioc a b β {b} := by | rw [β Icc_diff_left, h.Icc_eq, diff_singleton_subset_iff]
end PartialOrder |
/-
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.CategoryTheory.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 165 | 168 | theorem pullbackDiagonalMapIso.hom_snd :
(pullbackDiagonalMapIso f i iβ iβ).hom β« pullback.snd _ _ =
pullback.snd _ _ β« pullback.snd _ _ := by | delta pullbackDiagonalMapIso |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, YaΓ«l Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | Mathlib/MeasureTheory/Integral/Average.lean | 341 | 347 | theorem average_congr {f g : Ξ± β E} (h : f =α΅[ΞΌ] g) : β¨ x, f x βΞΌ = β¨ x, g x βΞΌ := by | simp only [average_eq, integral_congr_ae h]
theorem setAverage_congr (h : s =α΅[ΞΌ] t) : β¨ x in s, f x βΞΌ = β¨ x in t, f x βΞΌ := by
simp only [setAverage_eq, setIntegral_congr_set h, measureReal_congr h]
theorem setAverage_congr_fun (hs : MeasurableSet s) (h : βα΅ x βΞΌ, x β s β f x = g x) : |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 125 | 128 | theorem mem_support_seq_iff : b β (seq q p).support β β f β q.support, b β f '' p.support := by | simp
end Seq |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | Mathlib/Algebra/Polynomial/Roots.lean | 428 | 434 | theorem aroots_zero (S) [CommRing S] [IsDomain S] [Algebra T S] : (0 : T[X]).aroots S = 0 := by | rw [β C_0, aroots_C]
@[simp]
theorem aroots_one [CommRing S] [IsDomain S] [Algebra T S] :
(1 : T[X]).aroots S = 0 :=
aroots_C 1 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, YaΓ«l Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | Mathlib/Order/Interval/Finset/Basic.lean | 842 | 845 | theorem Ico_inter_Ico {a b c d : Ξ±} : Ico a b β© Ico c d = Ico (max a c) (min b d) := by | rw [β coe_inj, coe_inter, coe_Ico, coe_Ico, coe_Ico, Set.Ico_inter_Ico]
theorem Ioc_inter_Ioc {a b c d : Ξ±} : Ioc a b β© Ioc c d = Ioc (max a c) (min b d) := by |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 232 | 236 | theorem angle_sub_le_pi_div_two_of_inner_eq_zero {x y : V} (h : βͺx, yβ« = 0) :
angle x (x - y) β€ Ο / 2 := by | rw [β neg_eq_zero, β inner_neg_right] at h
rw [sub_eq_add_neg]
exact angle_add_le_pi_div_two_of_inner_eq_zero h |
/-
Copyright (c) 2024 Thomas Browning, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.FixedPoints
impo... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 132 | 138 | theorem closure_of_isSwap_of_isPretransitive [Finite Ξ±] {S : Set (Perm Ξ±)} (hS : β Ο β S, Ο.IsSwap)
[MulAction.IsPretransitive (Subgroup.closure S) Ξ±] : Subgroup.closure S = β€ := by | simp [eq_top_iff', mem_closure_isSwap hS, orbit_eq_univ, Set.toFinite]
/-- A transitive permutation group generated by transpositions must be the whole symmetric group -/
theorem surjective_of_isSwap_of_isPretransitive [Finite Ξ±] (S : Set G)
(hS1 : β Ο β S, Perm.IsSwap (MulAction.toPermHom G Ξ± Ο)) (hS2 : Subgroup.... |
/-
Copyright (c) 2024 Miyahara KΕ. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara KΕ
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Defs
import Mathlib.Data.Set.Function
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variab... | Mathlib/Data/List/Iterate.lean | 21 | 22 | theorem length_iterate (f : Ξ± β Ξ±) (a : Ξ±) (n : β) : length (iterate f a n) = n := by | induction n generalizing a <;> 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 SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 815 | 818 | theorem rpow_pos {p : β} {x : ββ₯0β} (hx_pos : 0 < x) (hx_ne_top : x β β€) : 0 < x ^ p := by | rcases lt_or_le 0 p with hp_pos | hp_nonpos
Β· exact rpow_pos_of_nonneg hx_pos (le_of_lt hp_pos)
Β· rw [β neg_neg p, rpow_neg, ENNReal.inv_pos] |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... | Mathlib/Order/Disjoint.lean | 548 | 550 | theorem eq_top_of_isCompl_bot (h : IsCompl x β₯) : x = β€ := by | rw [β sup_bot_eq x, h.sup_eq_top]
theorem eq_top_of_bot_isCompl (h : IsCompl β₯ x) : x = β€ := |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 332 | 333 | theorem leftUnitor_inv_braiding (X : C) : (Ξ»_ X).inv β« (Ξ²_ (π_ C) X).hom = (Ο_ X).inv := by | simp |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 770 | 779 | theorem biproduct.ΞΉ_toSubtype_subtype (j : Subtype p) :
biproduct.ΞΉ f j β« biproduct.toSubtype f p = biproduct.ΞΉ (Subtype.restrict p f) j := by | classical
ext
rw [biproduct.toSubtype, Category.assoc, biproduct.lift_Ο, biproduct.ΞΉ_Ο, biproduct.ΞΉ_Ο]
split_ifs with hβ hβ hβ
exacts [rfl, False.elim (hβ (Subtype.ext hβ)), False.elim (hβ (congr_arg Subtype.val hβ)), rfl]
@[reassoc (attr := simp)]
theorem biproduct.ΞΉ_fromSubtype (j : Subtype p) : |
/-
Copyright (c) 2024 MarΓa InΓ©s de Frutos-FernΓ‘ndez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: MarΓa InΓ©s de Frutos-FernΓ‘ndez
-/
import Mathlib.Data.NNReal.Defs
import Mathlib.RingTheory.Valuation.Basic
/-!
# Rank one valuations
We define rank one valuations.
... | Mathlib/RingTheory/Valuation/RankOne.lean | 51 | 55 | theorem zero_of_hom_zero {x : Ξβ} (hx : hom v x = 0) : x = 0 := by | refine (eq_of_le_of_not_lt (zero_le' (a := x)) fun h_lt β¦ ?_).symm
have hs := strictMono v h_lt
rw [map_zero, hx] at hs
exact hs.false |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, YaΓ«l Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... | Mathlib/Algebra/Ring/BooleanRing.lean | 84 | 87 | theorem mul_add_mul : a * b + b * a = 0 := by | have : a + b = a + b + (a * b + b * a) :=
calc
a + b = (a + b) * (a + b) := by rw [mul_self] |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 636 | 640 | theorem LinearMap.toMatrix_mul (f g : Mβ ββ[R] Mβ) :
LinearMap.toMatrix vβ vβ (f * g) = LinearMap.toMatrix vβ vβ f * LinearMap.toMatrix vβ vβ g := by | rw [Module.End.mul_eq_comp, LinearMap.toMatrix_comp vβ vβ vβ f g]
lemma LinearMap.toMatrix_pow (f : Mβ ββ[R] Mβ) (k : β) : |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 84 | 88 | theorem mem_sup_left {S T : Submonoid M} : β {x : M}, x β S β x β S β T := by | rw [β SetLike.le_def]
exact le_sup_left
@[to_additive] |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Relations holding pairwise
This file develops pairwise relations ... | Mathlib/Data/Set/Pairwise/Basic.lean | 196 | 197 | theorem InjOn.pairwise_image {s : Set ΞΉ} (h : s.InjOn f) :
(f '' s).Pairwise r β s.Pairwise (r on f) := by | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.Invarian... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 140 | 164 | theorem Basis.le_span {J : Set M} (v : Basis ΞΉ R M) (hJ : span R J = β€) : #(range v) β€ #J := by | haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite J
Β· rw [β Cardinal.lift_le, Cardinal.mk_range_eq_of_injective v.injective, Cardinal.mk_fintype J]
convert Cardinal.lift_le.{v}.2 (basis_le_span' v hJ)
simp
Β· let S : J β Set ΞΉ := fun j => β(v.repr j).support
let S' : J β Set M := fu... |
/-
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.Sets.Closeds
import Mathlib.Topology.Sets.OpenCover
/-!
# Sober spaces
A quasi-sober space is a topological space where every irreducible closed s... | Mathlib/Topology/Sober.lean | 92 | 93 | theorem isGenericPoint_iff_forall_closed (hS : IsClosed S) (hxS : x β S) :
IsGenericPoint x S β β Z : Set Ξ±, IsClosed Z β x β Z β S β Z := by | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 118 | 122 | theorem braiding_naturality {X X' Y Y' : C} (f : X βΆ Y) (g : X' βΆ Y') :
(f β g) β« (braiding Y Y').hom = (braiding X X').hom β« (g β f) := by | rw [tensorHom_def' f g, tensorHom_def g f]
simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc] |
/-
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.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | Mathlib/Data/List/OfFn.lean | 116 | 119 | theorem mem_ofFn' {n} (f : Fin n β Ξ±) (a : Ξ±) : a β ofFn f β a β Set.range f := by | simp only [mem_iff_get, Set.mem_range, get_ofFn]
exact β¨fun β¨i, hiβ© => β¨Fin.cast (by simp) i, hiβ©, fun β¨i, hiβ© => β¨Fin.cast (by simp) i, hiβ©β© |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | Mathlib/Topology/Algebra/Monoid.lean | 821 | 827 | theorem tendsto_multiset_prod {f : ΞΉ β Ξ± β M} {x : Filter Ξ±} {a : ΞΉ β M} (s : Multiset ΞΉ) :
(β i β s, Tendsto (f i) x (π (a i))) β
Tendsto (fun b => (s.map fun c => f c b).prod) x (π (s.map a).prod) := by | rcases s with β¨lβ©
simpa using tendsto_list_prod l
@[to_additive] |
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