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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | Mathlib/Order/Filter/Prod.lean | 373 | 375 | theorem pure_prod {a : α} {f : Filter β} : pure a ×ˢ f = map (Prod.mk a) f := by | rw [prod_eq, map_pure, pure_seq_eq_map] |
/-
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.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 331 | 333 | theorem lebesgue_number_lemma_of_emetric_nhds {c : α → Set α} (hs : IsCompact s)
(hc : ∀ x ∈ s, c x ∈ 𝓝 x) : ∃ δ > 0, ∀ x ∈ s, ∃ y, ball x δ ⊆ c y := by | simpa only [ball, UniformSpace.ball, preimage_setOf_eq, edist_comm] |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | Mathlib/Analysis/Convex/Between.lean | 776 | 780 | theorem Wbtw.collinear {x y z : P} (h : Wbtw R x y z) : Collinear R ({x, y, z} : Set P) := by | rw [collinear_iff_exists_forall_eq_smul_vadd]
refine ⟨x, z -ᵥ x, ?_⟩
intro p hp
simp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at hp |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
/-!
# One-dimensional de... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 274 | 282 | theorem differentiableOn_neg : DifferentiableOn 𝕜 (Neg.neg : 𝕜 → 𝕜) s :=
DifferentiableOn.neg differentiableOn_id
lemma differentiableAt_comp_neg {a : 𝕜} :
DifferentiableAt 𝕜 (fun x ↦ f (-x)) a ↔ DifferentiableAt 𝕜 f (-a) := by | refine ⟨fun H ↦ ?_, fun H ↦ H.comp a differentiable_neg.differentiableAt⟩
convert ((neg_neg a).symm ▸ H).comp (-a) differentiable_neg.differentiableAt
ext
simp only [Function.comp_apply, neg_neg] |
/-
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.Convex.Gauge
import Mathlib.Analysis.Normed.Module.Convex
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neum... | Mathlib/Analysis/Convex/GaugeRescale.lean | 41 | 44 | theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by | simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self] |
/-
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.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | Mathlib/Topology/Compactness/Compact.lean | 424 | 428 | theorem IsCompact.eventually_forall_of_forall_eventually {x₀ : X} {K : Set Y} (hK : IsCompact K)
{P : X → Y → Prop} (hP : ∀ y ∈ K, ∀ᶠ z : X × Y in 𝓝 (x₀, y), P z.1 z.2) :
∀ᶠ x in 𝓝 x₀, ∀ y ∈ K, P x y := by | simp only [nhds_prod_eq, ← eventually_iSup, ← hK.prod_nhdsSet_eq_biSup] at hP
exact hP.curry.mono fun _ h ↦ h.self_of_nhdsSet |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | Mathlib/LinearAlgebra/Prod.lean | 211 | 211 | theorem coprod_inl_inr : coprod (inl R M M₂) (inr R M M₂) = LinearMap.id := by | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 222 | 223 | theorem U_neg_sub_two (n : ℤ) : U R (-n - 2) = -U R n := by | simpa [sub_eq_add_neg, add_comm] using U_neg R (n + 2) |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
/-! # Subtypes of cond... | Mathlib/Order/CompleteLatticeIntervals.lean | 62 | 64 | theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by | simp [sSup, ht] |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | Mathlib/NumberTheory/Bernoulli.lean | 122 | 124 | theorem sum_bernoulli' (n : ℕ) : (∑ k ∈ range n, (n.choose k : ℚ) * bernoulli' k) = n := by | cases n with | zero => simp | succ n =>
suffices |
/-
Copyright (c) 2023 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bo... | Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 28 | 41 | theorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) :
∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc
_ = ∫ a in p.support, f a ∂(p.toMeasure) := by | rw [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by
apply integral_countable f p.support_countable
rwa [IntegrableOn, restrict_toMeasure_support p]
_ = ∑' (a : support p), (p a).toReal • f a := by
congr with x; congr 2
apply PMF.toMeasure_apply_singleto... |
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Tactic.Ring
/-!
# Hyperoperation sequence
This file defines the Hyperoperation sequence.
`hyperoperation 0 m k = k + ... | Mathlib/Data/Nat/Hyperoperation.lean | 53 | 55 | theorem hyperoperation_one : hyperoperation 1 = (· + ·) := by | ext m k
induction' k with bn bih |
/-
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.Field.Basic
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Module.NatInt
import Mathlib.GroupTheory.QuotientGroup.Defs
impor... | Mathlib/Algebra/CharZero/Quotient.lean | 54 | 64 | theorem zmultiples_zsmul_eq_zsmul_iff {ψ θ : R ⧸ AddSubgroup.zmultiples p} {z : ℤ} (hz : z ≠ 0) :
z • ψ = z • θ ↔ ∃ k : Fin z.natAbs, ψ = θ + ((k : ℕ) • (p / z) : R) := by | induction ψ using Quotient.inductionOn
induction θ using Quotient.inductionOn
-- Porting note: Introduced Zp notation to shorten lines
let Zp := AddSubgroup.zmultiples p
have : (Quotient.mk _ : R → R ⧸ Zp) = ((↑) : R → R ⧸ Zp) := rfl
simp only [Zp, this]
simp_rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGr... |
/-
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 | 240 | 243 | theorem angle_sub_lt_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
angle x (x - y) < π / 2 := by | rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg] |
/-
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.Data.Fintype.Pigeonhole
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Constructi... | Mathlib/FieldTheory/PrimitiveElement.lean | 380 | 384 | theorem primitive_element_iff_algHom_eq_of_eval' (α : E) :
F⟮α⟯ = ⊤ ↔ Function.Injective fun φ : E →ₐ[F] A ↦ φ α := by | classical
simp_rw [primitive_element_iff_minpoly_natDegree_eq, ← card_rootSet_eq_natDegree (K := A)
(Algebra.IsSeparable.isSeparable F α) (hA _), ← toFinset_card, |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | Mathlib/Data/Nat/PartENat.lean | 648 | 654 | theorem withTopEquiv_symm_coe (n : Nat) : withTopEquiv.symm n = n := by | simp
theorem withTopEquiv_symm_zero : withTopEquiv.symm 0 = 0 := by
simp
theorem withTopEquiv_symm_one : withTopEquiv.symm 1 = 1 := by |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | Mathlib/Analysis/RCLike/Basic.lean | 268 | 269 | theorem I_im' (z : K) : im (I : K) * im z = im z := by | rw [mul_comm, I_im] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 402 | 402 | theorem arcsin_eq_arccos {x : ℝ} (h : 0 ≤ x) : arcsin x = arccos (√(1 - x ^ 2)) := by | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | Mathlib/Data/Nat/Choose/Basic.lean | 54 | 54 | theorem choose_zero_right (n : ℕ) : choose n 0 = 1 := by | cases n <;> rfl |
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.IsPrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
import Mathlib.Data.Z... | Mathlib/RingTheory/ChainOfDivisors.lean | 66 | 81 | theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
∃ c : Fin (n + 1) → Associates M,
c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by | refine ⟨fun i => p ^ (i : ℕ), ?_, fun n m h => ?_, @fun y => ⟨fun h => ?_, ?_⟩⟩
· dsimp only
rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
· exact Associates.dvdNotUnit_iff_lt.mp
⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
not_isUnit_o... |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Heather Macbeth
-/
import Mathlib.Data.Matrix.Defs
import Mathlib.Topology.Algebra.IsUniformGroup.Basic
import Mathlib.Topology.UniformSpace.Pi
/-!
# Uniform space structure... | Mathlib/Topology/UniformSpace/Matrix.lean | 37 | 40 | theorem uniformContinuous {β : Type*} [UniformSpace β] {f : β → Matrix m n 𝕜} :
UniformContinuous f ↔ ∀ i j, UniformContinuous fun x => f x i j := by | simp only [UniformContinuous, Matrix.uniformity, Filter.tendsto_iInf, Filter.tendsto_comap_iff]
apply Iff.intro <;> intro a <;> apply a |
/-
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.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 868 | 872 | theorem isBasis_iff (hX : X ⊆ M.E := by | aesop_mat) :
M.IsBasis I X ↔ (M.Indep I ∧ I ⊆ X ∧ ∀ J, M.Indep J → I ⊆ J → J ⊆ X → I = J) := by
rw [isBasis_iff', and_iff_left hX]
theorem isBasis'_iff_isBasis_inter_ground : M.IsBasis' I X ↔ M.IsBasis I (X ∩ M.E) := 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.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invariants (f : α → α)... | Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 50 | 54 | theorem le_invariants_iterate (f : α → α) (n : ℕ) :
invariants f ≤ invariants (f^[n]) := by | induction n with
| zero => simp [invariants_le]
| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _) |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | Mathlib/Order/Heyting/Basic.lean | 326 | 328 | theorem Codisjoint.himp_eq_right (h : Codisjoint a b) : b ⇨ a = a := by | conv_rhs => rw [← @top_himp _ _ a]
rw [← h.eq_top, sup_himp_self_left] |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | Mathlib/Data/Matrix/RowCol.lean | 261 | 264 | theorem updateRow_apply [DecidableEq m] {i' : m} :
updateRow M i b i' j = if i' = i then b j else M i' j := by | by_cases h : i' = i
· rw [h, updateRow_self, if_pos rfl] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | Mathlib/NumberTheory/ArithmeticFunction.lean | 760 | 768 | theorem map_lcm [CommGroupWithZero R] {f : ArithmeticFunction R}
(hf : f.IsMultiplicative) {x y : ℕ} (hf_gcd : f (x.gcd y) ≠ 0) :
f (x.lcm y) = f x * f y / f (x.gcd y) := by | rw [← hf.lcm_apply_mul_gcd_apply, mul_div_cancel_right₀ _ hf_gcd]
theorem eq_zero_of_squarefree_of_dvd_eq_zero [MonoidWithZero R] {f : ArithmeticFunction R}
(hf : IsMultiplicative f) {m n : ℕ} (hn : Squarefree n) (hmn : m ∣ n)
(h_zero : f m = 0) :
f n = 0 := by |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 60 | 62 | theorem fixedBy_inv (g : G) : fixedBy α g⁻¹ = fixedBy α g := by | ext
rw [mem_fixedBy, mem_fixedBy, inv_smul_eq_iff, eq_comm] |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 171 | 176 | theorem infEdist_closure_pos_iff_not_mem_closure {x : α} {E : Set α} :
0 < infEdist x (closure E) ↔ x ∉ closure E := by | rw [infEdist_closure, infEdist_pos_iff_not_mem_closure]
theorem exists_real_pos_lt_infEdist_of_not_mem_closure {x : α} {E : Set α} (h : x ∉ closure E) :
∃ ε : ℝ, 0 < ε ∧ ENNReal.ofReal ε < infEdist x E := by |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... | Mathlib/Data/ENNReal/Basic.lean | 650 | 656 | theorem le_of_top_imp_top_of_toNNReal_le {a b : ℝ≥0∞} (h : a = ⊤ → b = ⊤)
(h_nnreal : a ≠ ⊤ → b ≠ ⊤ → a.toNNReal ≤ b.toNNReal) : a ≤ b := by | by_contra! hlt
lift b to ℝ≥0 using hlt.ne_top
lift a to ℝ≥0 using mt h coe_ne_top
refine hlt.not_le ?_
simpa using h_nnreal |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | Mathlib/Data/Real/GoldenRatio.lean | 70 | 72 | theorem one_sub_gold : 1 - ψ = φ := by | linarith [gold_add_goldConj] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | Mathlib/NumberTheory/Divisors.lean | 84 | 86 | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by | simp [properDivisors]
@[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 Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 358 | 371 | theorem Submodule.rank_le (s : Submodule R M) : Module.rank R s ≤ Module.rank R M := by | rw [← rank_top R M]
exact rank_mono le_top
theorem LinearMap.lift_rank_le_of_surjective (f : M →ₗ[R] M') (h : Surjective f) :
lift.{v} (Module.rank R M') ≤ lift.{v'} (Module.rank R M) := by
rw [← rank_range_of_surjective f h]
apply lift_rank_range_le
theorem LinearMap.rank_le_of_surjective (f : M →ₗ[R] M₁) ... |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 54 | 54 | theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by | rw [smeval_def] |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | Mathlib/Data/Finset/Prod.lean | 233 | 235 | theorem inter_product [DecidableEq α] [DecidableEq β] : (s ∩ s') ×ˢ t = s ×ˢ t ∩ s' ×ˢ t := by | ext ⟨x, y⟩
simp only [← and_and_right, mem_inter, mem_product] |
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | Mathlib/Algebra/Homology/Homotopy.lean | 386 | 390 | theorem nullHomotopicMap'_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ := by | simp only [nullHomotopicMap']
rw [nullHomotopicMap_f_of_not_rel_right r₁₀ hk₁] |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-!
# Integrals involving the Gamma function
In this file, we... | Mathlib/MeasureTheory/Integral/Gamma.lean | 21 | 37 | theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : -1 < q) :
∫ x in Ioi (0 : ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q + 1) / p) := by | calc
_ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by
rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),
abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]
refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
rw [← rpow_mul (le_... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 640 | 647 | theorem setToFun_congr_left (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ T' C') (h : T = T') (f : α → E) :
setToFun μ T hT f = setToFun μ T' hT' f := by | by_cases hf : Integrable f μ
· simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h]
· simp_rw [setToFun_undef _ hf]
theorem setToFun_congr_left' (hT : DominatedFinMeasAdditive μ T C) |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 581 | 584 | theorem fromEdgeSet_empty : fromEdgeSet (∅ : Set (Sym2 V)) = ⊥ := by | ext v w
simp only [fromEdgeSet_adj, Set.mem_empty_iff_false, false_and, bot_adj] |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Logic.Enco... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 224 | 231 | theorem iterate_succ_toZ (i : ι) (hi : i0 ≤ i) : succ^[(toZ i0 i).toNat] i0 = i := by | rw [toZ_of_ge hi, Int.toNat_natCast]
exact Nat.find_spec (exists_succ_iterate_of_le hi)
theorem iterate_pred_toZ (i : ι) (hi : i < i0) : pred^[(-toZ i0 i).toNat] i0 = i := by
rw [toZ_of_lt hi, neg_neg, Int.toNat_natCast]
exact Nat.find_spec (exists_pred_iterate_of_le hi.le) |
/-
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 `ℤ`-value... | Mathlib/Data/Int/Log.lean | 231 | 231 | theorem self_le_zpow_clog {b : ℕ} (hb : 1 < b) (r : R) : r ≤ (b : R) ^ clog b r := by | |
/-
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.Analysis.Normed.Group.Continuity
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Topology.MetricSpace.Thick... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean | 142 | 155 | theorem tendsto_measure_cthickening_of_isClosed {μ : Measure α} {s : Set α}
(hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) (h's : IsClosed s) :
Tendsto (fun r => μ (cthickening r s)) (𝓝 0) (𝓝 (μ s)) := by | convert tendsto_measure_cthickening hs
exact h's.closure_eq.symm
/-- If a set has a thickening with finite measure, then the measures of its `r`-thickenings
converge to the measure of its closure as `r > 0` tends to `0`. -/
theorem tendsto_measure_thickening {μ : Measure α} {s : Set α}
(hs : ∃ R > 0, μ (thickeni... |
/-
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.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | Mathlib/MeasureTheory/Function/EssSup.lean | 205 | 211 | theorem essInf_antitone_measure {f : α → β} (hμν : μ ≪ ν) : essInf f ν ≤ essInf f μ := by | refine liminf_le_liminf_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr hμν) ?_ ?_
all_goals isBoundedDefault
lemma essSup_eq_iSup (hμ : ∀ a, μ {a} ≠ 0) (f : α → β) : essSup f μ = ⨆ i, f i := by
rw [essSup, ae_eq_top.2 hμ, limsup_top_eq_iSup] |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | Mathlib/Algebra/Polynomial/Basic.lean | 681 | 681 | theorem toFinsupp_C_mul_X (a : R) : Polynomial.toFinsupp (C a * X) = Finsupp.single 1 a := by | |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Tactic.Abel
/-!
# Circulant matrices
This file contains the defini... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 167 | 170 | theorem circulant_isSymm_iff [SubtractionMonoid n] {v : n → α} :
(circulant v).IsSymm ↔ ∀ i, v (-i) = v i := by | rw [IsSymm, transpose_circulant, circulant_inj, funext_iff] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... | Mathlib/Data/Nat/Squarefree.lean | 94 | 103 | theorem squarefree_and_prime_pow_iff_prime {n : ℕ} : Squarefree n ∧ IsPrimePow n ↔ Prime n := by | refine ⟨?_, fun hn => ⟨hn.squarefree, hn.isPrimePow⟩⟩
rw [isPrimePow_nat_iff]
rintro ⟨h, p, k, hp, hk, rfl⟩
rw [squarefree_pow_iff hp.ne_one hk.ne'] at h
rwa [h.2, pow_one]
/-- Assuming that `n` has no factors less than `k`, returns the smallest prime `p` such that
`p^2 ∣ n`. -/
def minSqFacAux : ℕ → ℕ → Opt... |
/-
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.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 84 | 85 | theorem to_real_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by | simp [toComplex_def] |
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Dynamics.Minimal
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Group.Meas... | Mathlib/MeasureTheory/Group/Action.lean | 114 | 126 | theorem smul_mem_ae (c : G) {s : Set α} : c • s ∈ ae μ ↔ s ∈ ae μ := by | simp only [mem_ae_iff, ← smul_set_compl, measure_smul_eq_zero_iff]
@[to_additive (attr := simp)]
theorem smul_ae (c : G) : c • ae μ = ae μ := by
ext s
simp only [mem_smul_filter, preimage_smul, smul_mem_ae]
@[to_additive (attr := simp)]
theorem eventuallyConst_smul_set_ae (c : G) {s : Set α} :
EventuallyConst... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | Mathlib/Algebra/Divisibility/Units.lean | 214 | 215 | theorem IsRelPrime.mul_dvd_of_right_isPrimal (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z)
(hy : IsPrimal y) : x * y ∣ z := by | |
/-
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, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
/-!
# Extensionality lemmas for monoid and group structures
In this file we prove extensionality lemmas... | Mathlib/Algebra/Group/Ext.lean | 137 | 156 | theorem Group.ext {G : Type*} ⦃g₁ g₂ : Group G⦄
(h_mul : (letI := g₁; HMul.hMul : G → G → G) = (letI := g₂; HMul.hMul : G → G → G)) :
g₁ = g₂ := by | have h₁ : g₁.one = g₂.one := congr_arg (·.one) (Monoid.ext h_mul)
let f : @MonoidHom G G g₁.toMulOneClass g₂.toMulOneClass :=
@MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)
(fun x y => congr_fun (congr_fun h_mul x) y)
exact
Group.toDivInvMonoid_injective
(DivInvMonoid.ext h_mul
(fu... |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Shapes.Prod... | Mathlib/CategoryTheory/Limits/Lattice.lean | 85 | 93 | theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by | trans
· exact
(IsColimit.coconePointUniqueUpToIso (colimit.isColimit _)
(finiteColimitCocone (Discrete.functor f)).isColimit).to_eq
change Finset.univ.sup (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.sup f
simp only [← Finset.sup_map, Finset.univ_map_equiv_to_embedding]
rfl |
/-
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.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 191 | 195 | theorem fg_of_isUnit (I : FractionalIdeal S P) (h : IsUnit I) : FG (I : Submodule R P) :=
fg_unit h.unit
theorem _root_.Ideal.fg_of_isUnit (inj : Function.Injective (algebraMap R P)) (I : Ideal R)
(h : IsUnit (I : FractionalIdeal S P)) : I.FG := by | |
/-
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
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | Mathlib/Topology/ContinuousOn.lean | 1,128 | 1,134 | theorem continuousAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {x : α × β} :
ContinuousAt f x ↔ ContinuousAt (f ⟨x.1, ·⟩) x.2 := by | simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_left
theorem continuousAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {x : α × β} :
ContinuousAt f x ↔ ContinuousAt (f ⟨·, x.2⟩) x.1 := by
simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_... |
/-
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.JacobiSymbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 92 | 94 | theorem jacobiSymNat.mod_left (a b ab : ℕ) (r : ℤ) (hab : a % b = ab) (hr : jacobiSymNat ab b = r) :
jacobiSymNat a b = r := by | rw [← hr, jacobiSymNat, jacobiSymNat, _root_.jacobiSym.mod_left a b, ← hab]; rfl |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 619 | 623 | theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k := by | rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff] |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | Mathlib/Data/List/Rotate.lean | 132 | 133 | theorem rotate_append_length_eq (l l' : List α) : (l ++ l').rotate l.length = l' ++ l := by | rw [rotate_eq_rotate'] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | Mathlib/Analysis/SpecificLimits/Basic.lean | 365 | 376 | theorem NNReal.summable_geometric {r : ℝ≥0} (hr : r < 1) : Summable fun n : ℕ ↦ r ^ n :=
⟨_, NNReal.hasSum_geometric hr⟩
theorem tsum_geometric_nnreal {r : ℝ≥0} (hr : r < 1) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ :=
(NNReal.hasSum_geometric hr).tsum_eq
/-- The series `pow r` converges to `(1-r)⁻¹`. For `r < 1` the RHS is ... | rcases lt_or_le r 1 with hr | hr
· rcases ENNReal.lt_iff_exists_coe.1 hr with ⟨r, rfl, hr'⟩ |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 266 | 277 | theorem one_div_zpow {A : M} (n : ℤ) : (1 / A) ^ n = 1 / A ^ n := by | simp only [one_div, inv_zpow]
@[simp]
theorem transpose_zpow (A : M) : ∀ n : ℤ, (A ^ n)ᵀ = Aᵀ ^ n
| (n : ℕ) => by rw [zpow_natCast, zpow_natCast, transpose_pow]
| -[n+1] => by rw [zpow_negSucc, zpow_negSucc, transpose_nonsing_inv, transpose_pow]
@[simp]
theorem conjTranspose_zpow [StarRing R] (A : M) : ∀ n : ℤ, (... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 295 | 297 | theorem valuation_lt_one_iff_dvd (r : R) :
v.valuation K r < 1 ↔ v.asIdeal ∣ Ideal.span {r} := by | rw [valuation_of_algebraMap]; exact v.intValuation_lt_one_iff_dvd r |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 746 | 746 | theorem mul_eq_of_eq_div (h : a = c / b) : a * b = c := by | simp [h] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 93 | 96 | theorem toNNReal_strict_mono (hb : b ≠ ∞) (h : a < b) : a.toNNReal < b.toNNReal := by | simpa [← ENNReal.coe_lt_coe, hb, h.ne_top]
@[simp] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | Mathlib/GroupTheory/Perm/Support.lean | 575 | 580 | theorem card_support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
#(g.extendDomain f).support = #g.support := by | simp
end ExtendDomain
section Card |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 907 | 908 | theorem tan_periodic : Function.Periodic tan π := by | simpa only [Function.Periodic, tan_eq_sin_div_cos] using sin_antiperiodic.div cos_antiperiodic |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Tactic.Zify
/-!
# Factorial with big operators
This fi... | Mathlib/Data/Nat/Factorial/BigOperators.lean | 31 | 31 | theorem prod_factorial_dvd_factorial_sum : (∏ i ∈ s, (f i)!) ∣ (∑ i ∈ s, f i)! := by | |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
/-! # Measurability of the line derivative
We prove in `me... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 83 | 90 | theorem measurable_lineDeriv_uncurry [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) := by | borelize 𝕜
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
exact (measurable_deriv_with_param this).comp measurable_prodMk_right |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.A... | Mathlib/Algebra/Ring/Divisibility/Basic.lean | 141 | 143 | theorem dvd_sub_comm : a ∣ b - c ↔ a ∣ c - b := by | rw [← dvd_neg, neg_sub]
end NonUnitalRing |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | Mathlib/MeasureTheory/Integral/Average.lean | 319 | 320 | theorem average_eq_integral [IsProbabilityMeasure μ] (f : α → E) : ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by | rw [average, measure_univ, inv_one, one_smul] |
/-
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 | 179 | 182 | theorem pullbackDiagonalMapIso.inv_snd_fst :
(pullbackDiagonalMapIso f i i₁ i₂).inv ≫ pullback.snd _ _ ≫ pullback.fst _ _ =
pullback.fst _ _ := by | delta pullbackDiagonalMapIso |
/-
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 | 605 | 608 | theorem ncard_insert_le (a : α) (s : Set α) : (insert a s).ncard ≤ s.ncard + 1 := by | obtain hs | hs := s.finite_or_infinite
· to_encard_tac; rw [hs.cast_ncard_eq, (hs.insert _).cast_ncard_eq]; apply encard_insert_le
rw [(hs.mono (subset_insert a s)).ncard] |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | Mathlib/Analysis/RCLike/Basic.lean | 318 | 319 | theorem conj_smul (r : ℝ) (z : K) : conj (r • z) = r • conj z := by | rw [conj_eq_re_sub_im, conj_eq_re_sub_im, smul_re, smul_im, ofReal_mul, ofReal_mul, |
/-
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
/-!
# Met... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 45 | 47 | theorem cosh_half_dist (z w : ℍ) :
cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * √(z.im * w.im)) := by | rw [← sq_eq_sq₀, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt] |
/-
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.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 73 | 73 | theorem oangle_self (x : V) : o.oangle x x = 0 := by | |
/-
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.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.Cover.Open
/-!
# Gluing Schemes
Given a family of gluing data of schemes, we m... | Mathlib/AlgebraicGeometry/Gluing.lean | 296 | 298 | theorem glued_cover_cocycle_snd (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd _ _ =
pullback.snd _ _ := by | |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 76 | 78 | theorem mul_self : t * t = 1 := by | rcases ht with ⟨w, i, rfl⟩
simp |
/-
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.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 919 | 923 | theorem IsBasis.dep_of_ssubset (hI : M.IsBasis I X) (hIY : I ⊂ Y) (hYX : Y ⊆ X) : M.Dep Y := by | have : X ⊆ M.E := hI.subset_ground
rw [← not_indep_iff]
exact fun hY ↦ hIY.ne (hI.eq_of_subset_indep hY hIY.subset hYX) |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 643 | 647 | theorem hausdorffMeasure_le_one_of_subsingleton {s : Set X} (hs : s.Subsingleton) {d : ℝ}
(hd : 0 ≤ d) : μH[d] s ≤ 1 := by | rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩)
· simp only [measure_empty, zero_le]
· rw [(subsingleton_iff_singleton hx).1 hs] |
/-
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.Data.Finsupp.Defs
/-!
# Locus of unequal values of finitely supported functions
Let `α N` be two Types, assume that `N` has a `0` and let `f g : α →₀ N... | Mathlib/Data/Finsupp/NeLocus.lean | 58 | 61 | theorem nonempty_neLocus_iff {f g : α →₀ N} : (f.neLocus g).Nonempty ↔ f ≠ g :=
Finset.nonempty_iff_ne_empty.trans neLocus_eq_empty.not
theorem neLocus_comm : f.neLocus g = g.neLocus f := by | |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Nikolas Kuhn
-/
import Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves
/-!
# Description of the covering sieves of the coherent topology
This file characterises the cov... | Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean | 82 | 99 | theorem coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily (S : Sieve X) :
(S ∈ (coherentTopology C) X) ↔
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
EffectiveEpiFamily Y π ∧ (∀ a : α, (S.arrows) (π a)) ) := by | constructor
· intro h
induction h with
| of Y T hS =>
obtain ⟨a, h, Y', π, h', _⟩ := hS
refine ⟨a, h, Y', π, inferInstance, fun a' ↦ ?_⟩
obtain ⟨rfl, _⟩ := h'
exact ⟨Y' a', 𝟙 Y' a', π a', Presieve.ofArrows.mk a', by simp⟩
| top Y =>
exact ⟨Unit, inferInstance, fun _ => Y, fu... |
/-
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.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
/-!
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 110 | 112 | theorem sin_eq_neg_one_iff {x : ℂ} : sin x = -1 ↔ ∃ k : ℤ, -(π / 2) + k * (2 * π) = x := by | rw [← neg_eq_iff_eq_neg, ← cos_add_pi_div_two, cos_eq_one_iff]
simp only [← sub_eq_neg_add, sub_eq_iff_eq_add] |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 464 | 467 | theorem liftPropAt_symm_of_mem_maximalAtlas [HasGroupoid M G] {x : H}
(hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G)
(hx : x ∈ e.target) : LiftPropAt Q e.symm x := by | suffices h : Q (e ∘ e.symm) univ x by |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | Mathlib/Data/Nat/Choose/Basic.lean | 224 | 232 | theorem ascFactorial_eq_factorial_mul_choose (n k : ℕ) :
(n + 1).ascFactorial k = k ! * (n + k).choose k := by | rw [Nat.mul_comm]
apply Nat.mul_right_cancel (n + k - k).factorial_pos
rw [choose_mul_factorial_mul_factorial <| Nat.le_add_left k n, Nat.add_sub_cancel_right,
← factorial_mul_ascFactorial, Nat.mul_comm]
theorem ascFactorial_eq_factorial_mul_choose' (n k : ℕ) :
n.ascFactorial k = k ! * (n + k - 1).choose k... |
/-
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.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Prod
/-!
# The multiplicative and additive convolution of measures
In this file w... | Mathlib/MeasureTheory/Group/Convolution.lean | 75 | 77 | theorem mconv_zero (μ : Measure M) : μ ∗ (0 : Measure M) = (0 : Measure M) := by | unfold mconv
simp |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 500 | 502 | theorem one_div_le_one_div_of_neg_of_le (hb : b < 0) (h : a ≤ b) : 1 / b ≤ 1 / a := by | rwa [div_le_iff_of_neg' hb, ← div_eq_mul_one_div, div_le_one_of_neg (h.trans_lt hb)] |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | Mathlib/Algebra/Lie/Solvable.lean | 158 | 161 | theorem derivedSeries_eq_derivedSeriesOfIdeal_map (k : ℕ) :
(derivedSeries R I k).map I.incl = derivedSeriesOfIdeal R L k I := by | rw [derivedSeries_eq_derivedSeriesOfIdeal_comap, map_comap_incl, inf_eq_right]
apply derivedSeriesOfIdeal_le_self |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | Mathlib/Order/Filter/NAry.lean | 124 | 125 | theorem map₂_pure : map₂ m (pure a) (pure b) = pure (m a b) := by | rw [map₂_pure_right, map_pure] |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 84 | 85 | theorem eq_div_of_mul_eq_left {a b c : R} (hb : b ≠ 0) (h : a * b = c) : a = c / b := by | rw [← h, mul_div_cancel_right₀ _ hb] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, 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 | 115 | 133 | theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 := by | classical
refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)
split_ifs with h <;> simp only [h, zero_le', le_rfl]
theorem apply_ne_top (p : PMF α) (a : α) : p a ≠ ∞ :=
ne_of_lt (lt_of_le_of_lt (p.coe_le_one a) ENNReal.one_lt_top)
theorem apply_lt_top (p : PMF α) (a : α) : p a < ∞ :=
lt_... |
/-
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
/-!
# Met... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 313 | 320 | theorem isometry_real_vadd (a : ℝ) : Isometry (a +ᵥ · : ℍ → ℍ) :=
Isometry.of_dist_eq fun y₁ y₂ => by simp only [dist_eq, coe_vadd, vadd_im, dist_add_left]
theorem isometry_pos_mul (a : { x : ℝ // 0 < x }) : Isometry (a • · : ℍ → ℍ) := by | refine Isometry.of_dist_eq fun y₁ y₂ => ?_
simp only [dist_eq, coe_pos_real_smul, pos_real_im]; congr 2
rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (mul_self_nonneg _), Real.sqrt_mul_self_eq_abs,
Real.norm_eq_abs, mul_left_comm] |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël 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 | 375 | 378 | theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := by | simpa [← coe_subset] using Set.Ico_subset_Ici_self
theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := by |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | Mathlib/RingTheory/Coprime/Basic.lean | 351 | 353 | theorem neg_left {x y : R} (h : IsCoprime x y) : IsCoprime (-x) y := by | obtain ⟨a, b, h⟩ := h
use -a, b |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 374 | 378 | theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] (hp : p ≠ ∞) :
DenseRange (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) := by | refine (BoundedContinuousFunction.toLp_denseRange _ _ 𝕜 hp).mono ?_
refine range_subset_iff.2 fun f ↦ ?_
exact ⟨f.toContinuousMap, rfl⟩ |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | Mathlib/Data/Finset/Sups.lean | 452 | 458 | theorem disjSups_singleton : ({a} ○ {b} : Finset α) = if Disjoint a b then {a ⊔ b} else ∅ := by | split_ifs with h <;> simp [disjSups, filter_singleton, h]
theorem disjSups_union_left : (s₁ ∪ s₂) ○ t = s₁ ○ t ∪ s₂ ○ t := by
simp [disjSups, filter_union, image_union]
theorem disjSups_union_right : s ○ (t₁ ∪ t₂) = s ○ t₁ ∪ s ○ t₂ := by |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
-/
import Mathlib.Algebra.Algebra.ZMod
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheo... | Mathlib/FieldTheory/Finite/GaloisField.lean | 146 | 148 | theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by | have hp : 1 < p := h_prime.out.one_lt
have h1 : roots (X ^ p - X : (ZMod p)[X]) = Finset.univ.val := by |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | Mathlib/Data/Nat/Digits.lean | 210 | 215 | theorem digits_ofDigits (b : ℕ) (h : 1 < b) (L : List ℕ) (w₁ : ∀ l ∈ L, l < b)
(w₂ : ∀ h : L ≠ [], L.getLast h ≠ 0) : digits b (ofDigits b L) = L := by | induction' L with d L ih
· dsimp [ofDigits]
simp
· dsimp [ofDigits] |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | Mathlib/Data/Finset/Card.lean | 554 | 556 | theorem filter_card_add_filter_neg_card_eq_card
(p : α → Prop) [DecidablePred p] [∀ x, Decidable (¬p x)] :
#(s.filter p) + #(s.filter fun a ↦ ¬ p a) = #s := by | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 425 | 430 | theorem det_add_replicateCol_mul_replicateRow {ι : Type*} [Unique ι]
{A : Matrix m m α} (hA : IsUnit A.det) (u v : m → α) :
(A + replicateCol ι u * replicateRow ι v).det =
A.det * (1 + replicateRow ι v * A⁻¹ * replicateCol ι u).det := by | nth_rewrite 1 [← Matrix.mul_one A]
rwa [← Matrix.mul_nonsing_inv_cancel_left A (replicateCol ι u * replicateRow ι v), |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Subsemigroup.Membership
import Mathlib.Algebra.Group... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | 372 | 374 | theorem mem_map_equiv {f : R ≃+* S} {K : NonUnitalSubsemiring R} {x : S} :
x ∈ K.map (f : R →ₙ+* S) ↔ f.symm x ∈ K := by | convert @Set.mem_image_equiv _ _ (↑K) f.toEquiv x |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 293 | 297 | theorem iteratedDeriv_succ' : iteratedDeriv (n + 1) f = iteratedDeriv n (deriv f) := by | rw [iteratedDeriv_eq_iterate, iteratedDeriv_eq_iterate]; rfl
lemma AnalyticAt.hasFPowerSeriesAt {𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[CharZero 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜} (h : AnalyticAt 𝕜 f x) : |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Uthaiwat, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.I... | Mathlib/RingTheory/Polynomial/Nilpotent.lean | 174 | 182 | theorem not_isUnit_of_natDegree_pos_of_isReduced [IsReduced R] (p : R[X])
(hpl : 0 < p.natDegree) : ¬ IsUnit p := by | simp only [ne_eq, isNilpotent_iff_eq_zero, not_and, not_forall, exists_prop,
Polynomial.isUnit_iff_coeff_isUnit_isNilpotent]
intro _
refine ⟨p.natDegree, hpl.ne', ?_⟩
contrapose! hpl
simp only [coeff_natDegree, leadingCoeff_eq_zero] at hpl
simp [hpl] |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | Mathlib/RingTheory/Coprime/Lemmas.lean | 73 | 76 | theorem IsCoprime.prod_right_iff : IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i) := by | simpa only [isCoprime_comm] using IsCoprime.prod_left_iff (R := R)
theorem IsCoprime.of_prod_left (H1 : IsCoprime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) : |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 175 | 177 | theorem natDegree_natCast (n : ℕ) : natDegree (n : R[X]) = 0 := by | simp only [← C_eq_natCast, natDegree_C] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.