Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Prepa... | Mathlib/Data/Multiset/FinsetOps.lean | 250 | 251 | theorem le_ndinter {s t u : Multiset α} : s ≤ ndinter t u ↔ s ≤ t ∧ s ⊆ u := by |
simp [ndinter, le_filter, subset_iff]
|
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#a... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 244 | 249 | theorem TendstoInMeasure.exists_seq_tendsto_ae' {u : Filter ι} [NeBot u] [IsCountablyGenerated u]
{f : ι → α → E} {g : α → E} (hfg : TendstoInMeasure μ f u g) :
∃ ns : ℕ → ι, ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) := by |
obtain ⟨ms, hms⟩ := hfg.exists_seq_tendstoInMeasure_atTop
obtain ⟨ns, -, hns⟩ := hms.exists_seq_tendsto_ae
exact ⟨ms ∘ ns, hns⟩
|
/-
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.MeasureTheory.Measure.AEDisjoint
import Mathlib.MeasureTheory.Constructions.EventuallyMeasurable
#align_import measur... | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 252 | 263 | theorem exists_subordinate_pairwise_disjoint [Countable ι] {s : ι → Set α}
(h : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s)) :
∃ t : ι → Set α,
(∀ i, t i ⊆ s i) ∧
(∀ i, s i =ᵐ[μ] t i) ∧ (∀ i, MeasurableSet (t i)) ∧ Pairwise (Disjoint on t) := by |
choose t ht_sub htm ht_eq using fun i => exists_measurable_subset_ae_eq (h i)
rcases exists_null_pairwise_disjoint_diff hd with ⟨u, hum, hu₀, hud⟩
exact
⟨fun i => t i \ u i, fun i => diff_subset.trans (ht_sub _), fun i =>
(ht_eq _).symm.trans (diff_null_ae_eq_self (hu₀ i)).symm, fun i => (htm i).diff (... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Int
import Mathlib.GroupTheory.GroupAction.Opposite
import Mathlib.Logic.Function.Iterate
#align_import algebra.hom.iterate from "lean... | Mathlib/Algebra/GroupPower/IterateHom.lean | 111 | 111 | theorem mul_right_iterate_apply_one : (· * a)^[n] 1 = a ^ n := by | simp [mul_right_iterate]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 102 | 103 | theorem gcd_sub_self_right {m n : ℕ} (h : m ≤ n) : gcd m (n - m) = gcd m n := by |
rw [gcd_comm, gcd_sub_self_left h, gcd_comm]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,140 | 1,142 | theorem mker_one : mker (1 : M →* N) = ⊤ := by |
ext
simp [mem_mker]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 667 | 670 | theorem not_modEq_iff_toIcoDiv_eq_toIocDiv :
¬a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b := by |
rw [not_modEq_iff_toIcoMod_eq_toIocMod hp, toIcoMod, toIocMod, sub_right_inj,
(zsmul_strictMono_left hp).injective.eq_iff]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathli... | Mathlib/Probability/Martingale/Basic.lean | 362 | 368 | theorem smul_nonneg {f : ι → Ω → F} {c : ℝ} (hc : 0 ≤ c) (hf : Supermartingale f ℱ μ) :
Supermartingale (c • f) ℱ μ := by |
refine ⟨hf.1.smul c, fun i j hij => ?_, fun i => (hf.2.2 i).smul c⟩
refine (condexp_smul c (f j)).le.trans ?_
filter_upwards [hf.2.1 i j hij] with _ hle
simp_rw [Pi.smul_apply]
exact smul_le_smul_of_nonneg_left hle hc
|
/-
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
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 987 | 991 | theorem cardFactors_multiset_prod {s : Multiset ℕ} (h0 : s.prod ≠ 0) :
Ω s.prod = (Multiset.map Ω s).sum := by |
induction s using Multiset.induction_on with
| empty => simp
| cons ih => simp_all [cardFactors_mul, not_or]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 99 | 101 | theorem inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c := by |
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
exact div_le_iff' h
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
#align_import ... | Mathlib/RingTheory/Localization/Away/Basic.lean | 253 | 263 | theorem selfZPow_mul_neg (d : ℤ) : selfZPow x B d * selfZPow x B (-d) = 1 := by |
by_cases hd : d ≤ 0
· erw [selfZPow_of_nonpos x B hd, selfZPow_of_nonneg, ← map_pow, Int.natAbs_neg,
IsLocalization.mk'_spec, map_one]
apply nonneg_of_neg_nonpos
rwa [neg_neg]
· erw [selfZPow_of_nonneg x B (le_of_not_le hd), selfZPow_of_nonpos, ← map_pow, Int.natAbs_neg,
@IsLocalization.mk'_s... |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system fro... | Mathlib/MeasureTheory/PiSystem.lean | 568 | 568 | theorem has_univ : d.Has univ := by | simpa using d.has_compl d.has_empty
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 425 | 442 | theorem testAgainstNN_lipschitz (μ : FiniteMeasure Ω) :
LipschitzWith μ.mass fun f : Ω →ᵇ ℝ≥0 => μ.testAgainstNN f := by |
rw [lipschitzWith_iff_dist_le_mul]
intro f₁ f₂
suffices abs (μ.testAgainstNN f₁ - μ.testAgainstNN f₂ : ℝ) ≤ μ.mass * dist f₁ f₂ by
rwa [NNReal.dist_eq]
apply abs_le.mpr
constructor
· have key' := μ.testAgainstNN_lipschitz_estimate f₂ f₁
rw [mul_comm] at key'
suffices ↑(μ.testAgainstNN f₂) ≤ ↑(μ... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 50 | 57 | theorem continuousAt_angle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∠ y.1 y.2.1 y.2.2) x := by |
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp
((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd... |
/-
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.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 774 | 776 | theorem val_mul_le {n : ℕ} (a b : ZMod n) : (a * b).val ≤ a.val * b.val := by |
rw [val_mul]
apply Nat.mod_le
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.Se... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 203 | 205 | theorem norm_one [Nontrivial E] : ‖(1 : E)‖ = 1 := by |
have : 0 < ‖(1 : E)‖ := norm_pos_iff.mpr one_ne_zero
rw [← mul_left_inj' this.ne', ← norm_star_mul_self, mul_one, star_one, one_mul]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 78 | 80 | theorem Algebra.isAlgebraic_iff : Algebra.IsAlgebraic R A ↔ (⊤ : Subalgebra R A).IsAlgebraic := by |
delta Subalgebra.IsAlgebraic
simp only [Algebra.isAlgebraic_def, Algebra.mem_top, forall_prop_of_true, iff_self_iff]
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd0... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 492 | 496 | theorem rowLens_length_ofRowLens {w : List ℕ} {hw : w.Sorted (· ≥ ·)} (hpos : ∀ x ∈ w, 0 < x) :
(ofRowLens w hw).rowLens.length = w.length := by |
simp only [length_rowLens, colLen, Nat.find_eq_iff, mem_cells, mem_ofRowLens,
lt_self_iff_false, IsEmpty.exists_iff, Classical.not_not]
exact ⟨not_false, fun n hn => ⟨hn, hpos _ (List.get_mem _ _ hn)⟩⟩
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 143 | 143 | theorem succ_eval : succ.eval = fun v => pure [v.headI.succ] := by | simp [eval]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 67 | 69 | theorem eq_emptyOn_or_nonempty (M : Matroid α) : M = emptyOn α ∨ Matroid.Nonempty M := by |
rw [← ground_eq_empty_iff]
exact M.E.eq_empty_or_nonempty.elim Or.inl (fun h ↦ Or.inr ⟨h⟩)
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontin... | Mathlib/Topology/Semicontinuous.lean | 1,015 | 1,019 | theorem UpperSemicontinuousAt.comp_continuousAt_of_eq {f : α → β} {g : ι → α} {y : α} {x : ι}
(hf : UpperSemicontinuousAt f y) (hg : ContinuousAt g x) (hy : g x = y) :
UpperSemicontinuousAt (fun x ↦ f (g x)) x := by |
rw [← hy] at hf
exact comp_continuousAt hf hg
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 755 | 757 | theorem lift_comp_reindex_symm (e : ι ≃ ι₂) (φ : MultilinearMap R s E) :
lift φ ∘ₗ (reindex R s e).symm = lift (domDomCongrLinearEquiv' R R s _ e φ) := by |
ext; simp [reindex]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Basic
#align_import data.bool.all_any from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Boolean quantifi... | Mathlib/Data/Bool/AllAny.lean | 48 | 48 | theorem any_iff_exists_prop : (any l fun a => p a) ↔ ∃ a ∈ l, p a := by | simp [any_iff_exists]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Si... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 525 | 531 | theorem isSimpleGroup_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
(h : Fintype.card α = p) : IsSimpleGroup α := by |
subst h
have : Nontrivial α := by
have h' := Nat.Prime.one_lt hp.out
exact Fintype.one_lt_card_iff_nontrivial.1 h'
exact ⟨fun H _ => H.eq_bot_or_eq_top_of_prime_card⟩
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Group.Lattice
import Mathlib.Algebra.Order.Monoid.Unbu... | Mathlib/Algebra/Order/Group/Abs.lean | 421 | 421 | theorem abs_add' (a b : α) : |a| ≤ |b| + |b + a| := by | simpa using abs_add (-b) (b + a)
|
/-
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.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,570 | 1,572 | theorem norm_iteratedFDeriv_zero : ‖iteratedFDeriv 𝕜 0 f x‖ = ‖f x‖ := by |
-- Porting note: added `comp_apply`.
rw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 257 | 260 | theorem inv_le_one_iff : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a := by |
rcases em (a = 1) with (rfl | ha)
· simp [le_rfl]
· simp only [Ne.le_iff_lt (Ne.symm ha), Ne.le_iff_lt (mt inv_eq_one.1 ha), inv_lt_one_iff]
|
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanpro... | Mathlib/Algebra/Polynomial/Splits.lean | 363 | 367 | theorem eq_X_sub_C_of_splits_of_single_root {x : K} {h : K[X]} (h_splits : Splits i h)
(h_roots : (h.map i).roots = {i x}) : h = C h.leadingCoeff * (X - C x) := by |
apply Polynomial.map_injective _ i.injective
rw [eq_prod_roots_of_splits h_splits, h_roots]
simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Ring.Pi
import Ma... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 584 | 587 | theorem smul_equiv_smul {G : Type*} [SMul G β] [IsScalarTower G β β] {f1 f2 : CauSeq β abv} (c : G)
(hf : f1 ≈ f2) : c • f1 ≈ c • f2 := by |
simpa [const_smul, smul_one_mul _ _] using
mul_equiv_mul (const_equiv.mpr <| Eq.refl <| c • (1 : β)) hf
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,450 | 1,477 | theorem prod_bij_ne_one {s : Finset α} {t : Finset γ} {f : α → β} {g : γ → β}
(i : ∀ a ∈ s, f a ≠ 1 → γ) (hi : ∀ a h₁ h₂, i a h₁ h₂ ∈ t)
(i_inj : ∀ a₁ h₁₁ h₁₂ a₂ h₂₁ h₂₂, i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂)
(i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ a h₁ h₂, i a h₁ h₂ = b) (h : ∀ a h₁ h₂, f a = g (i a h₁ h₂)) :
... |
classical
calc
∏ x ∈ s, f x = ∏ x ∈ s.filter fun x => f x ≠ 1, f x := by rw [prod_filter_ne_one]
_ = ∏ x ∈ t.filter fun x => g x ≠ 1, g x :=
prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
?_ ?_ ?_ ?_
_ = ∏ x ∈ t, g x := prod_filter_ne_one _
· intr... |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 1,071 | 1,073 | theorem pderivCLM_eq_lineDeriv (m : E) (f : 𝓢(E, F)) (x : E) :
pderivCLM 𝕜 m f x = lineDeriv ℝ f x m := by |
simp only [pderivCLM_apply, f.differentiableAt.lineDeriv_eq_fderiv]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Init.Align
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.Algebra.Category.ModuleCat.EpiMono
#align... | Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | 436 | 454 | theorem sub_of_eq_image {P Q : C} (f : P ⟶ Q) (x y : P) :
f x = f y → ∃ z, f z = 0 ∧ ∀ (R : C) (g : P ⟶ R), (g : P ⟶ R) y = 0 → g z = g x :=
Quotient.inductionOn₂ x y fun a a' h =>
match Quotient.exact h with
| ⟨R, p, q, ep, _, comm⟩ =>
let a'' : R ⟶ P := ↑(p ≫ a.hom) - ↑(q ≫ a'.hom)
⟨a'',
... |
apply (epi_iff_cancel_zero _).1 ep' _ (a'.hom ≫ g)
simpa using comm'
apply Quotient.sound
-- Can we prevent quotient.sound from giving us this weird `coe_b` thingy?
change app g (a'' : Over P) ≈ app g a
exact ⟨R, 𝟙 R, p, inferInstance, ep, by simp [a'', ... |
/-
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.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 474 | 474 | theorem mul_div (a b c : G) : a * (b / c) = a * b / c := by | simp only [mul_assoc, div_eq_mul_inv]
|
/-
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.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 93 | 95 | theorem triangle_succ (n : ℕ) : (n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n := by |
rw [← add_mul_div_left, Nat.mul_comm 2 n, ← Nat.mul_add, Nat.add_sub_cancel, Nat.mul_comm]
cases n <;> rfl; apply zero_lt_succ
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 89 | 93 | theorem rank_unit [StrongRankCondition R] [DecidableEq n] (A : (Matrix n n R)ˣ) :
(A : Matrix n n R).rank = Fintype.card n := by |
apply le_antisymm (rank_le_card_width (A : Matrix n n R)) _
have := rank_mul_le_left (A : Matrix n n R) (↑A⁻¹ : Matrix n n R)
rwa [← Units.val_mul, mul_inv_self, Units.val_one, rank_one] at this
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Covering.DensityTheorem
#align_import measure_theory.covering.liminf_limsup from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ce... | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | 232 | 243 | theorem blimsup_cthickening_ae_eq_blimsup_thickening {p : ℕ → Prop} {s : ℕ → Set α} {r : ℕ → ℝ}
(hr : Tendsto r atTop (𝓝 0)) (hr' : ∀ᶠ i in atTop, p i → 0 < r i) :
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => thickening (r i) (s i)) atTop p : Set α) := by |
refine eventuallyLE_antisymm_iff.mpr ⟨?_, HasSubset.Subset.eventuallyLE (?_ : _ ≤ _)⟩
· rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq μ p s (@one_half_pos ℝ _) r hr).symm
EventuallyEq.rfl]
apply HasSubset.Subset.eventuallyLE
change _ ≤ _
refine mono_blimsup' (hr'.mono fun i hi pi => cthic... |
/-
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 | 412 | 417 | theorem trim_mkMetric [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞) :
(mkMetric m : OuterMeasure X).trim = mkMetric m := by |
simp only [mkMetric, mkMetric'.eq_iSup_nat, trim_iSup]
congr 1 with n : 1
refine mkMetric'.trim_pre _ (fun s => ?_) _
simp
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62... | Mathlib/Algebra/Group/Support.lean | 262 | 266 | theorem mulSupport_pow [Monoid M] (f : α → M) (n : ℕ) :
(mulSupport fun x => f x ^ n) ⊆ mulSupport f := by |
induction' n with n hfn
· simp [pow_zero, mulSupport_one]
· simpa only [pow_succ'] using (mulSupport_mul f _).trans (union_subset Subset.rfl hfn)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 800 | 804 | theorem map_roots_le_of_injective [IsDomain A] [IsDomain B] (p : A[X]) {f : A →+* B}
(hf : Function.Injective f) : p.roots.map f ≤ (p.map f).roots := by |
by_cases hp0 : p = 0
· simp only [hp0, roots_zero, Multiset.map_zero, Polynomial.map_zero, le_rfl]
exact map_roots_le ((Polynomial.map_ne_zero_iff hf).mpr hp0)
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,782 | 1,784 | theorem comap_span (f : P₁ ≃ᵃ[k] P₂) (s : Set P₂) :
(affineSpan k s).comap (f : P₁ →ᵃ[k] P₂) = affineSpan k (f ⁻¹' s) := by |
rw [← map_symm, map_span, AffineEquiv.coe_coe, f.image_symm]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 338 | 341 | theorem eval₂_at_apply {S : Type*} [Semiring S] (f : R →+* S) (r : R) :
p.eval₂ f (f r) = f (p.eval r) := by |
rw [eval₂_eq_sum, eval_eq_sum, sum, sum, map_sum f]
simp only [f.map_mul, f.map_pow]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 224 | 224 | theorem sup_comm (a b : α) : a ⊔ b = b ⊔ a := by | apply le_antisymm <;> simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 369 | 373 | theorem floor_lt_ceil_of_lt_of_pos {a b : α} (h : a < b) (h' : 0 < b) : ⌊a⌋₊ < ⌈b⌉₊ := by |
rcases le_or_lt 0 a with (ha | ha)
· rw [floor_lt ha]
exact h.trans_le (le_ceil _)
· rwa [floor_of_nonpos ha.le, lt_ceil, Nat.cast_zero]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 329 | 329 | theorem succ_nadd : succ a ♯ b = succ (a ♯ b) := by | rw [← one_nadd (a ♯ b), ← nadd_assoc, one_nadd]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 334 | 336 | theorem thickening_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (x : E) :
thickening ε (closedBall x δ) = ball x (ε + δ) := by |
rw [← cthickening_singleton _ hδ, thickening_cthickening hε hδ, thickening_singleton]
|
/-
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.Measure.AEMeasurable
#align_import measure_theory.lattice from "leanprover-community/mathlib"@"a95b442734d137aef46c1871e147089877fd0f62"
/-... | Mathlib/MeasureTheory/Order/Lattice.lean | 248 | 252 | theorem Finset.measurable_range_sup' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :
Measurable ((range (n + 1)).sup' nonempty_range_succ f) := by |
simp_rw [← Nat.lt_succ_iff] at hf
refine Finset.measurable_sup' _ ?_
simpa [Finset.mem_range]
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 91 | 98 | theorem isOrthoᵢ_flip (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) {v : n → M₁} :
B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v := by |
simp_rw [isOrthoᵢ_def]
constructor <;> intro h i j hij
· rw [flip_apply]
exact h j i (Ne.symm hij)
simp_rw [flip_apply] at h
exact h j i (Ne.symm hij)
|
/-
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 | 899 | 900 | theorem hausdorffMeasure_preimage (e : X ≃ᵢ Y) (d : ℝ) (s : Set Y) : μH[d] (e ⁻¹' s) = μH[d] s := by |
rw [← e.image_symm, e.symm.hausdorffMeasure_image]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.G... | Mathlib/Algebra/Ring/Basic.lean | 112 | 113 | theorem inv_neg' (a : α) : (-a)⁻¹ = -a⁻¹ := by |
rw [eq_comm, eq_inv_iff_mul_eq_one, neg_mul, mul_neg, neg_neg, mul_left_inv]
|
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... | Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 282 | 291 | theorem sumInv_comp_sum : sumInv I M ∘ₗ sum I M = LinearMap.id := by |
ext j x
apply DirectSum.ext (AdicCompletion I R) (fun i ↦ ?_)
ext n
simp only [LinearMap.coe_comp, Function.comp_apply, sum_lof, map_mk, component_sumInv,
mk_apply_coe, AdicCauchySequence.map_apply_coe, Submodule.mkQ_apply, LinearMap.id_comp]
rw [DirectSum.component.of, DirectSum.component.of]
split
... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 352 | 392 | theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by |
rw [@BorelSpace.measurable_eq G _ _]
refine MeasurableSpace.generateFrom_le ?_
intro U hU
rw [μ.outerMeasure_caratheodory]
intro U'
rw [μ.outerMeasure_of_isOpen ((U' : Set G) ∩ U) (U'.isOpen.inter hU)]
simp only [innerContent, iSup_subtype']
rw [Opens.coe_mk]
haveI : Nonempty { L : Compacts G // (L :... |
/-
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.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,014 | 1,016 | theorem map_coeSubmodule (h : ↑(map f I) = f '' I) :
LieSubmodule.toSubmodule (map f I) = (LieSubmodule.toSubmodule I).map (f : L →ₗ[R] L') := by |
rw [SetLike.ext'_iff, LieSubmodule.coe_toSubmodule, h, Submodule.map_coe]; rfl
|
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be51... | Mathlib/Analysis/Convex/Intrinsic.lean | 302 | 314 | theorem intrinsicClosure_eq_closure : intrinsicClosure 𝕜 s = closure s := by |
ext x
simp only [mem_closure_iff, mem_intrinsicClosure]
refine ⟨?_, fun h => ⟨⟨x, _⟩, ?_, Subtype.coe_mk _ ?_⟩⟩
· rintro ⟨x, h, rfl⟩ t ht hx
obtain ⟨z, hz₁, hz₂⟩ := h _ (continuous_induced_dom.isOpen_preimage t ht) hx
exact ⟨z, hz₁, hz₂⟩
· rintro _ ⟨t, ht, rfl⟩ hx
obtain ⟨y, hyt, hys⟩ := h _ ht h... |
/-
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.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 551 | 557 | theorem dist_orthocenter_reflection_circumcenter_finset (t : Triangle ℝ P) {i₁ i₂ : Fin 3}
(h : i₁ ≠ i₂) :
dist t.orthocenter
(reflection (affineSpan ℝ (t.points '' ↑({i₁, i₂} : Finset (Fin 3)))) t.circumcenter) =
t.circumradius := by |
simp only [mem_singleton, coe_insert, coe_singleton, Set.mem_singleton_iff]
exact dist_orthocenter_reflection_circumcenter _ h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,627 | 1,632 | theorem TopologicalGroup.t2Space_of_one_sep (H : ∀ x : G, x ≠ 1 → ∃ U ∈ 𝓝 (1 : G), x ∉ U) :
T2Space G := by |
suffices T1Space G from inferInstance
refine t1Space_iff_specializes_imp_eq.2 fun x y hspec ↦ by_contra fun hne ↦ ?_
rcases H (x * y⁻¹) (by rwa [Ne, mul_inv_eq_one]) with ⟨U, hU₁, hU⟩
exact hU <| mem_of_mem_nhds <| hspec.map (continuous_mul_right y⁻¹) (by rwa [mul_inv_self])
|
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.R... | Mathlib/Dynamics/Newton.lean | 106 | 126 | theorem exists_unique_nilpotent_sub_and_aeval_eq_zero
(h : IsNilpotent (aeval x P)) (h' : IsUnit (aeval x <| derivative P)) :
∃! r, IsNilpotent (x - r) ∧ aeval r P = 0 := by |
simp_rw [(neg_sub _ x).symm, isNilpotent_neg_iff]
refine exists_unique_of_exists_of_unique ?_ fun r₁ r₂ ⟨hr₁, hr₁'⟩ ⟨hr₂, hr₂'⟩ ↦ ?_
· -- Existence
obtain ⟨n, hn⟩ := id h
refine ⟨P.newtonMap^[n] x, isNilpotent_iterate_newtonMap_sub_of_isNilpotent h n, ?_⟩
rw [← zero_dvd_iff, ← pow_eq_zero_of_le n.lt_... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-co... | Mathlib/RingTheory/Localization/Integer.lean | 85 | 87 | theorem exists_integer_multiple (a : S) : ∃ b : M, IsInteger R ((b : R) • a) := by |
simp_rw [Algebra.smul_def, mul_comm _ a]
apply exists_integer_multiple'
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-comm... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 108 | 112 | theorem lt_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_le_of_lt h.1 h.2
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 214 | 223 | theorem toPlus_eq_mk {X : C} {P : Cᵒᵖ ⥤ D} (x : P.obj (op X)) :
(J.toPlus P).app _ x = mk (Meq.mk ⊤ x) := by |
dsimp [mk, toPlus]
delta Cover.toMultiequalizer
simp only [comp_apply]
apply congr_arg
apply (Meq.equiv P ⊤).injective
ext i
rw [Meq.equiv_apply, Equiv.apply_symm_apply, ← comp_apply, Multiequalizer.lift_ι]
rfl
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 399 | 405 | theorem blockDiagonal_transpose (M : o → Matrix m n α) :
(blockDiagonal M)ᵀ = blockDiagonal fun k => (M k)ᵀ := by |
ext
simp only [transpose_apply, blockDiagonal_apply, eq_comm]
split_ifs with h
· rw [h]
· rfl
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex Kontorovich
-/
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be4... | Mathlib/Order/Filter/Pi.lean | 193 | 199 | theorem map_eval_pi (f : ∀ i, Filter (α i)) [∀ i, NeBot (f i)] (i : ι) :
map (eval i) (pi f) = f i := by |
refine le_antisymm (tendsto_eval_pi f i) fun s hs => ?_
rcases mem_pi.1 (mem_map.1 hs) with ⟨I, hIf, t, htf, hI⟩
rw [← image_subset_iff] at hI
refine mem_of_superset (htf i) ((subset_eval_image_pi ?_ _).trans hI)
exact nonempty_of_mem (pi_mem_pi hIf fun i _ => htf i)
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.angle.sphere from "leanprover... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 136 | 140 | theorem two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi {s : Sphere P} {p₁ p₂ p₃ : P}
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₃ : p₂ ≠ p₃)
(hp₁p₃ : p₁ ≠ p₃) : (2 : ℤ) • ∡ p₃ p₁ s.center + (2 : ℤ) • ∡ p₁ p₂ p₃ = π := by |
rw [← oangle_center_eq_two_zsmul_oangle hp₁ hp₂ hp₃ hp₂p₁ hp₂p₃,
oangle_eq_pi_sub_two_zsmul_oangle_center_right hp₁ hp₃ hp₁p₃, add_sub_cancel]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 97 | 102 | theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs,
← lintegral_add_left hf]
simp only [Pi.add_apply]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connect... | Mathlib/Order/Interval/Set/OrdConnected.lean | 340 | 345 | theorem ordConnected_of_uIcc_subset_left (h : ∀ y ∈ s, [[x, y]] ⊆ s) : OrdConnected s :=
ordConnected_iff_uIcc_subset.2 fun y hy z hz =>
calc
[[y, z]] ⊆ [[y, x]] ∪ [[x, z]] := uIcc_subset_uIcc_union_uIcc
_ = [[x, y]] ∪ [[x, z]] := by | rw [uIcc_comm]
_ ⊆ s := union_subset (h y hy) (h z hz)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 355 | 356 | theorem toDual_eq_equivFun [Finite ι] (m : M) (i : ι) : b.toDual m (b i) = b.equivFun m i := by |
rw [b.equivFun_apply, toDual_eq_repr]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 116 | 126 | theorem adic_module_basis :
I.ringFilterBasis.SubmodulesBasis fun n : ℕ => I ^ n • (⊤ : Submodule R M) :=
{ inter := fun i j =>
⟨max i j,
le_inf_iff.mpr
⟨smul_mono_left <| pow_le_pow_right (le_max_left i j),
smul_mono_left <| pow_le_pow_right (le_max_right i j)⟩⟩
smul := fu... | simpa [(I ^ i).mul_top] using a_in
exact smul_mem_smul a_in mem_top⟩ }
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.... | Mathlib/RingTheory/TensorProduct/Basic.lean | 535 | 536 | theorem intCast_def' (z : ℤ) : (z : A ⊗[R] B) = (1 : A) ⊗ₜ (z : B) := by |
rw [intCast_def, ← zsmul_one, smul_tmul, zsmul_one]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 1,240 | 1,242 | theorem Integrable.re (hf : Integrable f μ) : Integrable (fun x => RCLike.re (f x)) μ := by |
rw [← memℒp_one_iff_integrable] at hf ⊢
exact hf.re
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27dd... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 68 | 71 | theorem snorm_trim_ae (hm : m ≤ m0) {f : α → E} (hf : AEStronglyMeasurable f (μ.trim hm)) :
snorm f p (μ.trim hm) = snorm f p μ := by |
rw [snorm_congr_ae hf.ae_eq_mk, snorm_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk)]
exact snorm_trim hm hf.stronglyMeasurable_mk
|
/-
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
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_im... | Mathlib/RingTheory/Coprime/Lemmas.lean | 250 | 251 | theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by |
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 176 | 178 | theorem fromBlocks_submatrix_sum_swap_sum_swap {l m n o α : Type*} (A : Matrix n l α)
(B : Matrix n m α) (C : Matrix o l α) (D : Matrix o m α) :
(fromBlocks A B C D).submatrix Sum.swap Sum.swap = fromBlocks D C B A := by | simp
|
/-
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.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 708 | 715 | theorem comap_eq_sup_ker_of_disjoint {p : Submodule K V} [FiniteDimensional K p] {f : V →ₗ[K] V}
(h : ∀ x ∈ p, f x ∈ p) (h' : Disjoint p (ker f)) :
p.comap f = p ⊔ ker f := by |
refine le_antisymm (fun x hx ↦ ?_) (sup_le_iff.mpr ⟨h, ker_le_comap _⟩)
obtain ⟨⟨y, hy⟩, hxy⟩ :=
surjective_of_injective ((injective_restrict_iff_disjoint h).mpr h') ⟨f x, hx⟩
replace hxy : f y = f x := by simpa [Subtype.ext_iff] using hxy
exact Submodule.mem_sup.mpr ⟨y, hy, x - y, by simp [hxy], add_sub_c... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 784 | 785 | theorem mod_mem_mod [Mod α] (a b : Part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma % mb ∈ a % b := by | simp [mod_def]; aesop
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.DFinsupp.Basic
#align_import algebra.direct_sum.basic from "leanprover-community/mathlib"@"f7fc89d5d5ff1d... | Mathlib/Algebra/DirectSum/Basic.lean | 243 | 245 | theorem fromAddMonoid_of (i : ι) (f : γ →+ β i) : fromAddMonoid (of _ i f) = (of _ i).comp f := by |
rw [fromAddMonoid, toAddMonoid_of]
rfl
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.m... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 260 | 263 | theorem zpow_bit1 (A : M) (n : ℤ) : A ^ bit1 n = A ^ n * A ^ n * A := by |
rw [bit1, zpow_add_one_of_ne_neg_one, zpow_bit0]
intro h
simpa using congr_arg bodd h
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 1,375 | 1,376 | theorem le_to_int {m n : ZNum} : (m : ℤ) ≤ n ↔ m ≤ n := by |
rw [← not_lt]; exact not_congr lt_to_int
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | Mathlib/RingTheory/Kaehler.lean | 689 | 704 | theorem KaehlerDifferential.ker_map :
LinearMap.ker (KaehlerDifferential.map R S A B) =
(((kerTotal S B).restrictScalars A).comap finsupp_map).map
(Finsupp.total A (Ω[A⁄R]) A (D R A)) := by |
rw [← Submodule.map_comap_eq_of_surjective (total_surjective R A) (LinearMap.ker _)]
congr 1
ext x
simp only [Submodule.mem_comap, LinearMap.mem_ker, Finsupp.apply_total, ← kerTotal_eq,
Submodule.restrictScalars_mem]
simp only [Finsupp.total_apply, Function.comp_apply, LinearMap.coe_comp, Finsupp.lmapDom... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,291 | 2,291 | theorem ite_left (s t : Set α) : s.ite s t = s ∪ t := by | simp [Set.ite]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,287 | 1,293 | theorem isDomain_of_le_nonZeroDivisors [Algebra A S] {M : Submonoid A} [IsLocalization M S]
(hM : M ≤ nonZeroDivisors A) : IsDomain S := by |
apply @NoZeroDivisors.to_isDomain _ _ (id _) (id _)
· exact
⟨⟨(algebraMap A S) 0, (algebraMap A S) 1, fun h =>
zero_ne_one (IsLocalization.injective S hM h)⟩⟩
· exact noZeroDivisors_of_le_nonZeroDivisors _ hM
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 137 | 139 | theorem tendsto_dist_right_atTop_iff (c : α) {f : β → α} {l : Filter β} :
Tendsto (fun x ↦ dist (f x) c) l atTop ↔ Tendsto f l (cobounded α) := by |
rw [← comap_dist_right_atTop c, tendsto_comap_iff, Function.comp_def]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 649 | 655 | theorem LinearMap.toMatrix_mulVec_repr (f : M₁ →ₗ[R] M₂) (x : M₁) :
LinearMap.toMatrix v₁ v₂ f *ᵥ v₁.repr x = v₂.repr (f x) := by |
ext i
rw [← Matrix.toLin'_apply, LinearMap.toMatrix, LinearEquiv.trans_apply, Matrix.toLin'_toMatrix',
LinearEquiv.arrowCongr_apply, v₂.equivFun_apply]
congr
exact v₁.equivFun.symm_apply_apply x
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 768 | 769 | theorem toIocMod_zero_sub_comm (a b : α) : toIocMod hp 0 (a - b) = p - toIcoMod hp 0 (b - a) := by |
rw [← neg_sub, toIocMod_neg, neg_zero]
|
/-
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.Ring.Action.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.GroupTheory.Congruence.Basic
#align_import... | Mathlib/RingTheory/Congruence.lean | 466 | 467 | theorem coe_sInf (S : Set (RingCon R)) : ⇑(sInf S) = sInf ((⇑) '' S) := by |
ext; simp only [sInf_image, iInf_apply, iInf_Prop_eq]; rfl
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 1,298 | 1,361 | theorem contDiffOn_convolution_right_with_param {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
{g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
(hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) :
ContDiffOn 𝕜 n (... |
/- The result is known when all the universes are the same, from
`contDiffOn_convolution_right_with_param_aux`. We reduce to this situation by pushing
everything through `ULift` continuous linear equivalences. -/
let eG : Type max uG uE' uF uP := ULift.{max uE' uF uP} G
borelize eG
let eE' : Type max u... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 718 | 720 | theorem floor_nonpos (ha : a ≤ 0) : ⌊a⌋ ≤ 0 := by |
rw [← @cast_le α, Int.cast_zero]
exact (floor_le a).trans ha
|
/-
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 | 87 | 88 | theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by |
rw [← map_one (C R), derivativeFun_C (1 : R)]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 458 | 460 | theorem tendsto_norm_atTop_iff_cobounded' {f : α → E} {l : Filter α} :
Tendsto (‖f ·‖) l atTop ↔ Tendsto f l (cobounded E) := by |
rw [← comap_norm_atTop', tendsto_comap_iff]; rfl
|
/-
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.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"... | Mathlib/Probability/Martingale/Centering.lean | 174 | 183 | theorem martingalePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ∀ i, |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R) := by |
filter_upwards [hbdd, predictablePart_bdd_difference ℱ hbdd] with ω hω₁ hω₂ i
simp only [two_mul, martingalePart, Pi.sub_apply]
have : |f (i + 1) ω - predictablePart f ℱ μ (i + 1) ω - (f i ω - predictablePart f ℱ μ i ω)| =
|f (i + 1) ω - f i ω - (predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω)... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 475 | 476 | theorem map_atTop (e : α ≃o β) : map (e : α → β) atTop = atTop := by |
rw [← e.comap_atTop, map_comap_of_surjective e.surjective]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Init.Align
import Mathlib.Data.Fintype.Order
import Mathlib.Algebra.DirectLimit
import Mathlib.ModelTheory.Quotients
import Mathlib.ModelT... | Mathlib/ModelTheory/DirectLimit.lean | 349 | 358 | theorem exists_fg_substructure_in_Sigma (S : L.Substructure (DirectLimit G f)) (S_fg : S.FG) :
∃ i, ∃ T : L.Substructure (G i), T.map (of L ι G f i).toHom = S := by |
let ⟨A, A_closure⟩ := S_fg
let ⟨i, y, eq_y⟩ := exists_quotient_mk'_sigma_mk'_eq G _ (fun a : A ↦ a.1)
use i
use Substructure.closure L (range y)
rw [Substructure.map_closure]
simp only [Embedding.coe_toHom, of_apply]
rw [← image_univ, image_image, image_univ, ← eq_y,
Subtype.range_coe_subtype, Finset... |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 206 | 210 | theorem noncommProd_eq_pow_card (s : Multiset α) (comm) (m : α) (h : ∀ x ∈ s, x = m) :
s.noncommProd comm = m ^ Multiset.card s := by |
induction s using Quotient.inductionOn
simp only [quot_mk_to_coe, noncommProd_coe, coe_card, mem_coe] at *
exact List.prod_eq_pow_card _ m h
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
/-!
# Lemmas about booleans
These are the lemmas about booleans w... | Mathlib/Init/Data/Bool/Lemmas.lean | 76 | 76 | theorem not_eq_true_eq_eq_false (a : Bool) : (not a = true) = (a = false) := by | cases a <;> simp
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topolog... | Mathlib/Topology/Algebra/UniformRing.lean | 77 | 84 | theorem continuous_mul : Continuous fun p : Completion α × Completion α => p.1 * p.2 := by |
let m := (AddMonoidHom.mul : α →+ α →+ α).compr₂ toCompl
have : Continuous fun p : α × α => m p.1 p.2 := by
apply (continuous_coe α).comp _
simp only [AddMonoidHom.coe_mul, AddMonoidHom.coe_mulLeft]
exact _root_.continuous_mul
have di : DenseInducing (toCompl : α → Completion α) := denseInducing_coe
... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 398 | 400 | theorem node4L_size {l x m y r} (hm : Sized m) :
size (@node4L α l x m y r) = size l + size m + size r + 2 := by |
cases m <;> simp [node4L, node3L, node'] <;> [abel; (simp [size, hm.1]; abel)]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 651 | 651 | theorem bihimp_bihimp_cancel_right : b ⇔ a ⇔ a = b := by | simp [bihimp_assoc]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 464 | 465 | theorem spanExt_app_right : (spanExt iX iY iZ wf wg).app WalkingSpan.right = iZ := by |
dsimp [spanExt]
|
/-
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
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 104 | 105 | theorem edgeFinset_sdiff [DecidableEq V] :
(G₁ \ G₂).edgeFinset = G₁.edgeFinset \ G₂.edgeFinset := by | simp [edgeFinset]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 1,145 | 1,154 | theorem exists_superset_card_eq [Infinite α] (s : Finset α) (n : ℕ) (hn : s.card ≤ n) :
∃ t : Finset α, s ⊆ t ∧ t.card = n := by |
induction' n with n IH generalizing s
· exact ⟨s, subset_refl _, Nat.eq_zero_of_le_zero hn⟩
· rcases hn.eq_or_lt with hn' | hn'
· exact ⟨s, subset_refl _, hn'⟩
obtain ⟨t, hs, ht⟩ := IH _ (Nat.le_of_lt_succ hn')
obtain ⟨x, hx⟩ := exists_not_mem_finset t
refine ⟨Finset.cons x t hx, hs.trans (Finset... |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,573 | 1,577 | theorem map_top_of_surjective (f : G →* N) (h : Function.Surjective f) : Subgroup.map f ⊤ = ⊤ := by |
rw [eq_top_iff]
intro x _
obtain ⟨y, hy⟩ := h x
exact ⟨y, trivial, hy⟩
|
/-
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.ContMDiff.Defs
/-!
## Basic properties of smooth functions between manifolds
In this file, we show that sta... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 52 | 77 | theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
(hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g ∘ f) s x := by |
rw [contMDiffWithinAt_iff] at hg hf ⊢
refine ⟨hg.1.comp hf.1 st, ?_⟩
set e := extChartAt I x
set e' := extChartAt I' (f x)
have : e' (f x) = (writtenInExtChartAt I I' x f) (e x) := by simp only [e, e', mfld_simps]
rw [this] at hg
have A : ∀ᶠ y in 𝓝[e.symm ⁻¹' s ∩ range I] e x, f (e.symm y) ∈ t ∧ f (e.sy... |
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