Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
/--
A `Heap` is the nodes of the pairing heap.
Each node have two pointers: `child` going to the first c... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 123 | 127 | theorem Heap.size_merge (le) {s₁ s₂ : Heap α} (h₁ : s₁.NoSibling) (h₂ : s₂.NoSibling) :
(merge le s₁ s₂).size = s₁.size + s₂.size := by |
match h₁, h₂ with
| .nil, .nil | .nil, .node _ _ | .node _ _, .nil => simp [size]
| .node _ _, .node _ _ => unfold merge; dsimp; split <;> simp_arith [size]
|
/-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import ... | Mathlib/GroupTheory/Archimedean.lean | 60 | 87 | theorem AddSubgroup.exists_isLeast_pos {H : AddSubgroup G} (hbot : H ≠ ⊥) {a : G} (h₀ : 0 < a)
(hd : Disjoint (H : Set G) (Ioo 0 a)) : ∃ b, IsLeast { g : G | g ∈ H ∧ 0 < g } b := by |
-- todo: move to a lemma?
have hex : ∀ g > 0, ∃ n : ℕ, g ∈ Ioc (n • a) ((n + 1) • a) := fun g hg => by
rcases existsUnique_add_zsmul_mem_Ico h₀ 0 (g - a) with ⟨m, ⟨hm, hm'⟩, -⟩
simp only [zero_add, sub_le_iff_le_add, sub_add_cancel, ← add_one_zsmul] at hm hm'
lift m to ℕ
· rw [← Int.lt_add_one_iff,... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat ... | Mathlib/Data/UnionFind.lean | 103 | 112 | theorem set {arr : Array α} {n} {m : Fin n → β} (H : Agrees arr f m)
{i : Fin arr.size} {x} {m' : Fin n → β}
(hm₁ : ∀ (j : Fin n), j.1 ≠ i → m' j = m j)
(hm₂ : ∀ (h : i < n), f x = m' ⟨i, h⟩) : Agrees (arr.set i x) f m' := by |
cases H
refine mk' (by simp) fun j hj₁ hj₂ ↦ ?_
suffices f (Array.set arr i x)[j] = m' ⟨j, hj₂⟩ by simp_all [Array.get_set]
by_cases h : i = j
· subst h; rw [Array.get_set_eq, ← hm₂]
· rw [arr.get_set_ne _ _ _ h, hm₁ ⟨j, _⟩ (Ne.symm h)]; rfl
|
/-
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.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f70... | Mathlib/Data/PFun.lean | 80 | 80 | theorem mem_dom (f : α →. β) (x : α) : x ∈ Dom f ↔ ∃ y, y ∈ f x := by | simp [Dom, Part.dom_iff_mem]
|
/-
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.Data.DFinsupp.Interval
import Mathlib.Data.DFinsupp.Multiset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.multiset.interval from "leanprover-... | Mathlib/Data/Multiset/Interval.lean | 72 | 74 | theorem card_Ioo :
(Finset.Ioo s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (t.count i + 1 - s.count i) - 2 := by |
rw [Finset.card_Ioo_eq_card_Icc_sub_two, card_Icc]
|
/-
Copyright (c) 2024 Thomas Browning, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
/-!
# Subgroups gen... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 74 | 88 | theorem swap_mem_closure_isSwap {S : Set (Perm α)} (hS : ∀ f ∈ S, f.IsSwap) {x y : α} :
swap x y ∈ closure S ↔ x ∈ orbit (closure S) y := by |
refine ⟨fun h ↦ ⟨⟨swap x y, h⟩, swap_apply_right x y⟩, fun hf ↦ ?_⟩
by_contra h
have := exists_smul_not_mem_of_subset_orbit_closure S {x | swap x y ∈ closure S}
(fun f hf ↦ ?_) (fun z hz ↦ ?_) h ⟨y, ?_⟩
· obtain ⟨σ, hσ, a, ha, hσa⟩ := this
obtain ⟨z, w, hzw, rfl⟩ := hS σ hσ
have := ne_of_mem_of_not... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 92 | 93 | theorem wellFoundedOn_univ : (univ : Set α).WellFoundedOn r ↔ WellFounded r := by |
simp [wellFoundedOn_iff]
|
/-
Copyright (c) 2022 Matej Penciak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matej Penciak, Moritz Doll, Fabien Clery
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 137 | 139 | theorem neg_mem (h : A ∈ symplecticGroup l R) : -A ∈ symplecticGroup l R := by |
rw [mem_iff] at h ⊢
simp [h]
|
/-
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.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#al... | Mathlib/Algebra/Order/UpperLower.lean | 70 | 72 | theorem IsUpperSet.mul_right (hs : IsUpperSet s) : IsUpperSet (s * t) := by |
rw [mul_comm]
exact hs.mul_left
|
/-
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.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 72 | 86 | theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
spectralRadius ℂ a = ‖a‖₊ := by |
refine (ENNReal.pow_strictMono two_ne_zero).injective ?_
have heq :
(fun n : ℕ => (‖(a⋆ * a) ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) =
(fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ) := by
funext n
rw [Function.comp_apply, ← rpow_natCast, ← rpow_mul, mul_comm, rpow_mul, rpow_natCast, ←
... |
/-
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.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f7... | Mathlib/Algebra/Polynomial/Monomial.lean | 28 | 32 | theorem monomial_one_eq_iff [Nontrivial R] {i j : ℕ} :
(monomial i 1 : R[X]) = monomial j 1 ↔ i = j := by |
-- Porting note: `ofFinsupp.injEq` is required.
simp_rw [← ofFinsupp_single, ofFinsupp.injEq]
exact AddMonoidAlgebra.of_injective.eq_iff
|
/-
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 | 107 | 107 | theorem mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c := by | rw [mul_comm, inv_mul_le_iff h]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "lea... | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 85 | 90 | theorem expMapCircle_eq_expMapCircle {x y : ℝ} :
expMapCircle x = expMapCircle y ↔ ∃ m : ℤ, x = y + m * (2 * π) := by |
rw [Subtype.ext_iff, expMapCircle_apply, expMapCircle_apply, exp_eq_exp_iff_exists_int]
refine exists_congr fun n => ?_
rw [← mul_assoc, ← add_mul, mul_left_inj' I_ne_zero]
norm_cast
|
/-
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.Order.Filter.ENNReal
#align_import measure_theory.function.ess_sup from "leanprover-community/... | Mathlib/MeasureTheory/Function/EssSup.lean | 131 | 135 | theorem meas_lt_essInf
(hf : IsBoundedUnder (· ≥ ·) (ae μ) f := by | isBoundedDefault) :
μ { y | f y < essInf f μ } = 0 := by
simp_rw [← not_le]
exact ae_essInf_le hf
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 137 | 139 | theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ := by |
rw [← image_inv]
exact ((OrderIso.inv α).map_csSup' hs₀ hs₁).symm
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 68 | 69 | theorem logb_neg_eq_logb (x : ℝ) : logb b (-x) = logb b x := by |
rw [← logb_abs x, ← logb_abs (-x), abs_neg]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 50 | 57 | 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_sq, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]
· congr 1
simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj,
Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
ring
all_goals positivity
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.StoneCech
#align_import topology.extremally_disconnected from "leanprover-community/mathlib"@"7e281deff072... | Mathlib/Topology/ExtremallyDisconnected.lean | 83 | 92 | theorem StoneCech.projective [DiscreteTopology X] : CompactT2.Projective (StoneCech X) := by |
intro Y Z _tsY _tsZ _csY _t2Y _csZ _csZ f g hf hg g_sur
let s : Z → Y := fun z => Classical.choose <| g_sur z
have hs : g ∘ s = id := funext fun z => Classical.choose_spec (g_sur z)
let t := s ∘ f ∘ stoneCechUnit
have ht : Continuous t := continuous_of_discreteTopology
let h : StoneCech X → Y := stoneCechE... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
/-!
# Games d... | Mathlib/SetTheory/Game/State.lean | 57 | 61 | theorem turnBound_ne_zero_of_right_move {s t : S} (m : t ∈ r s) : turnBound s ≠ 0 := by |
intro h
have t := right_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 61 | 65 | theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ := by |
induction' q with q hq
· rfl
· simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 96 | 104 | theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
(hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂ := by |
simp only [Finset.weightedVSubOfPoint_apply]
congr
ext i
rcases eq_or_ne i j with h | h
· simp [h]
· simp [hw i h]
|
/-
Copyright (c) 2022 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Batteries.Data.UInt
@[ext] theorem Char.ext : {a b : Char} → a.val = b.val → a = b
| ⟨_,_⟩, ⟨_,_⟩, rfl => rfl
theorem Char.ext_iff {x y : Char} : x = y ↔ x... | .lake/packages/batteries/Batteries/Data/Char.lean | 30 | 31 | theorem csize_pos (c) : 0 < csize c := by |
rcases csize_eq c with _|_|_|_ <;> simp_all (config := {decide := true})
|
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Filtered.Basic
/-!
# A functor from a small category to a filtered category factors throug... | Mathlib/CategoryTheory/Filtered/Small.lean | 97 | 115 | theorem small_fullSubcategory_filteredClosure :
Small.{max v w} (FullSubcategory (FilteredClosure f)) := by |
refine small_of_injective_of_exists (FilteredClosureSmall.abstractFilteredClosureRealization f)
FullSubcategory.ext ?_
rintro ⟨j, h⟩
induction h with
| base x => exact ⟨⟨0, ⟨x⟩⟩, rfl⟩
| max hj₁ hj₂ ih ih' =>
rcases ih with ⟨⟨n, x⟩, rfl⟩
rcases ih' with ⟨⟨m, y⟩, rfl⟩
refine ⟨⟨(Max.max n m).suc... |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 66 | 68 | theorem gauge_def' : gauge s x = sInf {r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s} := by |
congrm sInf {r | ?_}
exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.Group.Fin
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Fintype.Card
#align_import data.zmod.defs from "lean... | Mathlib/Data/ZMod/Defs.lean | 124 | 127 | theorem card (n : ℕ) [Fintype (ZMod n)] : Fintype.card (ZMod n) = n := by |
cases n with
| zero => exact (not_finite (ZMod 0)).elim
| succ n => convert Fintype.card_fin (n + 1) using 2
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathl... | Mathlib/SetTheory/Surreal/Basic.lean | 123 | 131 | theorem lf_asymm {x y : PGame} (ox : Numeric x) (oy : Numeric y) : x ⧏ y → ¬y ⧏ x := by |
refine numeric_rec (C := fun x => ∀ z (_oz : Numeric z), x ⧏ z → ¬z ⧏ x)
(fun xl xr xL xR hx _oxl _oxr IHxl IHxr => ?_) x ox y oy
refine numeric_rec fun yl yr yL yR hy oyl oyr _IHyl _IHyr => ?_
rw [mk_lf_mk, mk_lf_mk]; rintro (⟨i, h₁⟩ | ⟨j, h₁⟩) (⟨i, h₂⟩ | ⟨j, h₂⟩)
· exact IHxl _ _ (oyl _) (h₁.moveLeft_lf ... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.N... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 118 | 119 | theorem mul_inverse_cancel_right (x y : M₀) (h : IsUnit x) : y * x * inverse x = y := by |
rw [mul_assoc, mul_inverse_cancel x h, mul_one]
|
/-
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.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathli... | Mathlib/CategoryTheory/Extensive.lean | 102 | 112 | theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
(c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by |
let X := F.obj ⟨WalkingPair.left⟩
let Y := F.obj ⟨WalkingPair.right⟩
have : F = pair X Y := by
apply Functor.hext
· rintro ⟨⟨⟩⟩ <;> rfl
· rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp
clear_value X Y
subst this
exact FinitaryExtensive.van_kampen' c hc
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 103 | 105 | theorem traverse_id : traverse pure = (pure : t α → Id (t α)) := by |
ext
exact id_traverse _
|
/-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
/-!
# Identities of ModularForms and Slash... | Mathlib/NumberTheory/ModularForms/Identities.lean | 34 | 37 | theorem T_zpow_width_invariant (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f (((ModularGroup.T ^ (N * n))) • z) = f z := by |
rw [modular_T_zpow_smul z (N * n)]
simpa only [Int.cast_mul, Int.cast_natCast] using vAdd_width_periodic N k n f z
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Al... | Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | 84 | 87 | theorem ZeroAtFilter.boundedAtFilter [NormedAddCommGroup β] {l : Filter α} {f : α → β}
(hf : ZeroAtFilter l f) : BoundedAtFilter l f := by |
rw [ZeroAtFilter, ← Asymptotics.isLittleO_const_iff (one_ne_zero' ℝ)] at hf
exact hf.isBigO
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Algebra.Ring.Basic
import Mathlib.Algebra.Field.MinimalAxioms
/-!
# T... | Mathlib/ModelTheory/Algebra/Field/Basic.lean | 81 | 86 | theorem FieldAxiom.realize_toSentence_iff_toProp {K : Type*}
[Add K] [Mul K] [Neg K] [Zero K] [One K] [CompatibleRing K]
(ax : FieldAxiom) :
(K ⊨ (ax.toSentence : Sentence Language.ring)) ↔ ax.toProp K := by |
cases ax <;>
simp [Sentence.Realize, Formula.Realize, Fin.snoc]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.monotone_convergence from "leanprover-community/mathlib"@"4c19a16e4b705bf... | Mathlib/Topology/Order/MonotoneConvergence.lean | 96 | 100 | theorem tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atTop (𝓝 a) := by |
suffices Tendsto (rangeFactorization f) atTop atTop from
(SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this
exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathl... | Mathlib/SetTheory/Surreal/Basic.lean | 71 | 75 | theorem numeric_def {x : PGame} :
Numeric x ↔
(∀ i j, x.moveLeft i < x.moveRight j) ∧
(∀ i, Numeric (x.moveLeft i)) ∧ ∀ j, Numeric (x.moveRight j) := by |
cases x; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Compactness.Compact
/-!
# Locally compact spaces
We define the following classes of topological spaces:
* `W... | Mathlib/Topology/Compactness/LocallyCompact.lean | 40 | 45 | theorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :
∃ K', IsCompact K' ∧ K ⊆ interior K' := by |
choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x
rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩
refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩
exact iUnion₂_subset fun x hx ↦ interior_mono <| subset_iUnion₂ (s := fun x _ ↦ s x) x hx
|
/-
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.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f13... | Mathlib/Data/Sigma/Basic.lean | 70 | 71 | theorem ext_iff {x₀ x₁ : Sigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by |
cases x₀; cases x₁; exact Sigma.mk.inj_iff
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 115 | 117 | theorem rpow_def_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℝ) :
x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * π) := by |
split_ifs with h <;> simp [rpow_def, *]; exact rpow_def_of_neg (lt_of_le_of_ne hx h) _
|
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.Cat... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 75 | 80 | theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') :
NatTrans.Equifibered α := by |
rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩
all_goals
dsimp; simp only [Discrete.functor_map_id]
exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccb... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 137 | 139 | theorem submodule_eq (v : ℙ K V) : v.submodule = K ∙ v.rep := by |
conv_lhs => rw [← v.mk_rep]
rfl
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.Pointwise
#align_import group_theory.submonoid.inverses from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782b... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 73 | 76 | theorem leftInv_leftInv_le : S.leftInv.leftInv ≤ S := by |
rintro x ⟨⟨y, z, h₁⟩, h₂ : x * y = 1⟩
convert z.prop
rw [← mul_one x, ← h₁, ← mul_assoc, h₂, one_mul]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 94 | 94 | theorem p_comm {P Q : Karoubi C} (f : Hom P Q) : P.p ≫ f.f = f.f ≫ Q.p := by | rw [p_comp, comp_p]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 81 | 87 | theorem submatrix {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :
(M.submatrix e e).PosSemidef := by |
classical
rw [(by simp : M = 1 * M * 1), submatrix_mul (he₂ := Function.bijective_id),
submatrix_mul (he₂ := Function.bijective_id), submatrix_id_id]
simpa only [conjTranspose_submatrix, conjTranspose_one] using
conjTranspose_mul_mul_same hM (Matrix.submatrix 1 id e)
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Control.EquivFunctor
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Whiskering
import Mathlib.CategoryTheory.Types
#align_import... | Mathlib/CategoryTheory/Core.lean | 52 | 53 | theorem id_hom (X : C) : Iso.hom (coreCategory.id X) = @CategoryStruct.id C _ X := by |
rfl
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
/-!
# Relations hol... | Mathlib/Data/Finset/Pairwise.lean | 86 | 89 | theorem pairwise_iff_coe_toFinset_pairwise (hn : l.Nodup) (hs : Symmetric r) :
(l.toFinset : Set α).Pairwise r ↔ l.Pairwise r := by |
letI : IsSymm α r := ⟨hs⟩
rw [coe_toFinset, hn.pairwise_coe]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 87 | 89 | theorem toFinsupp_toFreeAbelianGroup (f : X →₀ ℤ) :
FreeAbelianGroup.toFinsupp (Finsupp.toFreeAbelianGroup f) = f := by |
rw [← AddMonoidHom.comp_apply, toFinsupp_comp_toFreeAbelianGroup, AddMonoidHom.id_apply]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import... | Mathlib/Topology/List.lean | 28 | 66 | theorem nhds_list (as : List α) : 𝓝 as = traverse 𝓝 as := by |
refine nhds_mkOfNhds _ _ ?_ ?_
· intro l
induction l with
| nil => exact le_rfl
| cons a l ih =>
suffices List.cons <$> pure a <*> pure l ≤ List.cons <$> 𝓝 a <*> traverse 𝓝 l by
simpa only [functor_norm] using this
exact Filter.seq_mono (Filter.map_mono <| pure_le_nhds a) ih
· i... |
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 61 | 64 | theorem isIntegrallyClosed_eq_field_fractions' [IsDomain S] [Algebra K S] [IsScalarTower R K S]
{s : S} (hs : IsIntegral R s) : minpoly K s = (minpoly R s).map (algebraMap R K) := by |
let L := FractionRing S
rw [← isIntegrallyClosed_eq_field_fractions K L hs, algebraMap_eq (IsFractionRing.injective S L)]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 104 | 107 | theorem comp_differentiableAt_iff {f : G → E} {x : G} :
DifferentiableAt 𝕜 (iso ∘ f) x ↔ DifferentiableAt 𝕜 f x := by |
rw [← differentiableWithinAt_univ, ← differentiableWithinAt_univ,
iso.comp_differentiableWithinAt_iff]
|
/-
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.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 86 | 88 | theorem memberSubfamily_union (a : α) (𝒜 ℬ : Finset (Finset α)) :
(𝒜 ∪ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∪ ℬ.memberSubfamily a := by |
simp_rw [memberSubfamily, filter_union, image_union]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.... | Mathlib/Analysis/Convex/Topology.lean | 124 | 132 | theorem Convex.combo_interior_closure_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
(ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • interior s + b • closure s ⊆ interior s :=
interior_smul₀ ha.ne' s ▸
calc
interior (a • s) + b • closure s ⊆ interior (a • s) + closure (b • s) :=
add... | rw [isOpen_interior.add_closure (b • s)]
_ ⊆ interior (a • s + b • s) := subset_interior_add_left
_ ⊆ interior s := interior_mono <| hs.set_combo_subset ha.le hb hab
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecf... | Mathlib/Data/PNat/Interval.lean | 113 | 114 | theorem card_fintype_Ico : Fintype.card (Set.Ico a b) = b - a := by |
rw [← card_Ico, Fintype.card_ofFinset]
|
/-
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.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 113 | 114 | theorem mgf_zero_fun : mgf 0 μ t = (μ Set.univ).toReal := by |
simp only [mgf, Pi.zero_apply, mul_zero, exp_zero, integral_const, smul_eq_mul, mul_one]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.S... | Mathlib/AlgebraicGeometry/Scheme.lean | 155 | 157 | theorem congr_app {X Y : Scheme} {f g : X ⟶ Y} (e : f = g) (U) :
f.val.c.app U = g.val.c.app U ≫ X.presheaf.map (eqToHom (by subst e; rfl)) := by |
subst e; dsimp; simp
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# A... | Mathlib/Algebra/Homology/Augment.lean | 92 | 94 | theorem augment_d_succ_succ (C : ChainComplex V ℕ) {X : V} (f : C.X 0 ⟶ X) (w : C.d 1 0 ≫ f = 0)
(i j : ℕ) : (augment C f w).d (i + 1) (j + 1) = C.d i j := by |
cases i <;> rfl
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kyle Miller
-/
import Mathlib.Data.Int.GCD
import Mathlib.Tactic.NormNum
/-! # `norm_num` extensions for GCD-adjacent functions
This module defines some `norm_num` ex... | Mathlib/Tactic/NormNum/GCD.lean | 22 | 28 | theorem int_gcd_helper' {d : ℕ} {x y : ℤ} (a b : ℤ) (h₁ : (d : ℤ) ∣ x) (h₂ : (d : ℤ) ∣ y)
(h₃ : x * a + y * b = d) : Int.gcd x y = d := by |
refine Nat.dvd_antisymm ?_ (Int.natCast_dvd_natCast.1 (Int.dvd_gcd h₁ h₂))
rw [← Int.natCast_dvd_natCast, ← h₃]
apply dvd_add
· exact Int.gcd_dvd_left.mul_right _
· exact Int.gcd_dvd_right.mul_right _
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 109 | 111 | theorem mem_iff_eq_smul_generator (S : Submodule R M) [S.IsPrincipal] {x : M} :
x ∈ S ↔ ∃ s : R, x = s • generator S := by |
simp_rw [@eq_comm _ x, ← mem_span_singleton, span_singleton_generator]
|
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Nick Kuhn
-/
import Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves
/-!
# Description of the covering sieves of the coherent topology
This file characterises the coveri... | Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean | 82 | 99 | theorem coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily (S : Sieve X) :
(S ∈ GrothendieckTopology.sieves (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 Y T hS Y Y R S _ _ a b
· 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⟩
· exact ⟨Unit, inferInstance, fun _ => Y, fun _ => ... |
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
/... | Mathlib/Probability/Kernel/Invariance.lean | 57 | 60 | theorem const_bind_eq_comp_const (κ : kernel α β) (μ : Measure α) :
const α (μ.bind κ) = κ ∘ₖ const α μ := by |
ext a s hs
simp_rw [comp_apply' _ _ _ hs, const_apply, Measure.bind_apply hs (kernel.measurable _)]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanpro... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 57 | 65 | theorem parallelepiped_basis_eq (b : Basis ι ℝ E) :
parallelepiped b = {x | ∀ i, b.repr x i ∈ Set.Icc 0 1} := by |
classical
ext x
simp_rw [mem_parallelepiped_iff, mem_setOf_eq, b.ext_elem_iff, _root_.map_sum,
_root_.map_smul, Finset.sum_apply', Basis.repr_self, Finsupp.smul_single, smul_eq_mul,
mul_one, Finsupp.single_apply, Finset.sum_ite_eq', Finset.mem_univ, ite_true, mem_Icc,
Pi.le_def, Pi.zero_apply, Pi.one... |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f... | Mathlib/Data/Multiset/Fintype.lean | 122 | 126 | theorem Multiset.toEnumFinset_mono {m₁ m₂ : Multiset α} (h : m₁ ≤ m₂) :
m₁.toEnumFinset ⊆ m₂.toEnumFinset := by |
intro p
simp only [Multiset.mem_toEnumFinset]
exact gt_of_ge_of_gt (Multiset.le_iff_count.mp h p.1)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768ef... | Mathlib/Data/Set/Pairwise/Lattice.lean | 89 | 101 | theorem PairwiseDisjoint.prod_left {f : ι × ι' → α}
(hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i'))
(ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')) :
(s ×ˢ t : Set (ι × ι')).PairwiseDisjoint f := by |
rintro ⟨i, i'⟩ hi ⟨j, j'⟩ hj h
rw [mem_prod] at hi hj
obtain rfl | hij := eq_or_ne i j
· refine (ht hi.2 hj.2 <| (Prod.mk.inj_left _).ne_iff.1 h).mono ?_ ?_
· convert le_iSup₂ (α := α) i hi.1; rfl
· convert le_iSup₂ (α := α) i hj.1; rfl
· refine (hs hi.1 hj.1 hij).mono ?_ ?_
· convert le_iSup₂ (α... |
/-
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.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 117 | 133 | theorem degreeLT_eq_span_X_pow [DecidableEq R] {n : ℕ} :
degreeLT R n = Submodule.span R ↑((Finset.range n).image fun n => X ^ n : Finset R[X]) := by |
apply le_antisymm
· intro p hp
replace hp := mem_degreeLT.1 hp
rw [← Polynomial.sum_monomial_eq p, Polynomial.sum]
refine Submodule.sum_mem _ fun k hk => ?_
have := WithBot.coe_lt_coe.1 ((Finset.sup_lt_iff <| WithBot.bot_lt_coe n).1 hp k hk)
rw [← C_mul_X_pow_eq_monomial, C_mul']
refine
... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, Robert Y. Lewis
-/
import Mathlib.Algebra.Group.Defs
#align_import group_theory.eckmann_hilton from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4f... | Mathlib/GroupTheory/EckmannHilton.lean | 64 | 69 | theorem mul : m₁ = m₂ := by |
funext a b
calc
m₁ a b = m₁ (m₂ a e₁) (m₂ e₁ b) := by
{ simp only [one h₁ h₂ distrib, h₁.left_id, h₁.right_id, h₂.left_id, h₂.right_id] }
_ = m₂ a b := by simp only [distrib, h₁.left_id, h₁.right_id, h₂.left_id, h₂.right_id]
|
/-
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.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of options
... | Mathlib/Data/Option/NAry.lean | 109 | 110 | theorem map_uncurry (f : α → β → γ) (x : Option (α × β)) :
x.map (uncurry f) = map₂ f (x.map Prod.fst) (x.map Prod.snd) := by | cases x <;> rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import ring_theory.roots_of_unity.complex from "leanprover-c... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 33 | 50 | theorem isPrimitiveRoot_exp_of_coprime (i n : ℕ) (h0 : n ≠ 0) (hi : i.Coprime n) :
IsPrimitiveRoot (exp (2 * π * I * (i / n))) n := by |
rw [IsPrimitiveRoot.iff_def]
simp only [← exp_nat_mul, exp_eq_one_iff]
have hn0 : (n : ℂ) ≠ 0 := mod_cast h0
constructor
· use i
field_simp [hn0, mul_comm (i : ℂ), mul_comm (n : ℂ)]
· simp only [hn0, mul_right_comm _ _ ↑n, mul_left_inj' two_pi_I_ne_zero, Ne, not_false_iff,
mul_comm _ (i : ℂ), ← m... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Properties of LUB and GLB in an order topology
-/
open Set Filter TopologicalSpa... | Mathlib/Topology/Order/IsLUB.lean | 77 | 80 | theorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :
IsLUB s a := by |
rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
exact isLUB_of_mem_nhds hsa (mem_principal_self s)
|
/-
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 | 53 | 55 | theorem newtonMap_apply_of_isUnit (h : IsUnit <| aeval x (derivative P)) :
P.newtonMap x = x - h.unit⁻¹ * aeval x P := by |
simp [newtonMap_apply, Ring.inverse, h]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.e... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 104 | 108 | theorem PreservesEqualizer.iso_inv_ι :
(PreservesEqualizer.iso G f g).inv ≫ G.map (equalizer.ι f g) =
equalizer.ι (G.map f) (G.map g) := by |
rw [← Iso.cancel_iso_hom_left (PreservesEqualizer.iso G f g), ← Category.assoc, Iso.hom_inv_id]
simp
|
/-
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.Data.Finset.Sigma
import Mathlib.Data.Fintype.Card
#align_import data.finset.pi_induction from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299... | Mathlib/Data/Finset/PiInduction.lean | 37 | 63 | theorem induction_on_pi_of_choice (r : ∀ i, α i → Finset (α i) → Prop)
(H_ex : ∀ (i) (s : Finset (α i)), s.Nonempty → ∃ x ∈ s, r i x (s.erase x))
{p : (∀ i, Finset (α i)) → Prop} (f : ∀ i, Finset (α i)) (h0 : p fun _ ↦ ∅)
(step :
∀ (g : ∀ i, Finset (α i)) (i : ι) (x : α i),
r i x (g i) → p g →... |
cases nonempty_fintype ι
induction' hs : univ.sigma f using Finset.strongInductionOn with s ihs generalizing f; subst s
rcases eq_empty_or_nonempty (univ.sigma f) with he | hne
· convert h0 using 1
simpa [funext_iff] using he
· rcases sigma_nonempty.1 hne with ⟨i, -, hi⟩
rcases H_ex i (f i) hi with ⟨... |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebr... | Mathlib/Algebra/Ring/BooleanRing.lean | 101 | 101 | theorem sub_eq_add : a - b = a + b := by | rw [sub_eq_add_neg, add_right_inj, neg_eq]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 114 | 115 | theorem card_Ioc : (Ioc a b).card = b - a := by |
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# ... | Mathlib/CategoryTheory/Sums/Basic.lean | 62 | 63 | theorem hom_inl_inr_false {X : C} {Y : D} (f : Sum.inl X ⟶ Sum.inr Y) : False := by |
cases f
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.categor... | Mathlib/CategoryTheory/Functor/Category.lean | 125 | 125 | theorem id_hcomp_app {H : E ⥤ C} (α : F ⟶ G) (X : E) : (𝟙 H ◫ α).app X = α.app _ := by | simp
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
/-!
# The structure... | Mathlib/Data/Fintype/Fin.lean | 61 | 64 | theorem card_filter_univ_succ' (p : Fin (n + 1) → Prop) [DecidablePred p] :
(univ.filter p).card = ite (p 0) 1 0 + (univ.filter (p ∘ Fin.succ)).card := by |
rw [Fin.univ_succ, filter_cons, card_disjUnion, filter_map, card_map]
split_ifs <;> simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"... | Mathlib/Data/Sum/Basic.lean | 54 | 55 | theorem eq_left_iff_getLeft_eq {a : α} : x = inl a ↔ ∃ h, x.getLeft h = a := by |
cases x <;> simp
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#a... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 67 | 72 | theorem hasStrictFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 := by |
convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt
ext x
change x = expSeries 𝕂 𝔸 1 fun _ => x
simp [expSeries_apply_eq, Nat.factorial]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.image... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 62 | 63 | theorem inv_comp_image_ι_map {X Y : A} (f : X ⟶ Y) :
(iso L f).inv ≫ image.ι (L.map f) = L.map (image.ι f) := by | simp
|
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Homotopy.Equiv
import Mathlib.CategoryTheory.Equivalence
import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product
#align_import a... | Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean | 96 | 97 | theorem heq_path_of_eq_image : HEq ((πₘ f).map ⟦p⟧) ((πₘ g).map ⟦q⟧) := by |
simp only [map_eq, ← Path.Homotopic.map_lift]; apply Path.Homotopic.hpath_hext; exact hfg
|
/-
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 | 91 | 103 | theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (f : ι → S) :
∃ b : M, ∀ i ∈ s, IsLocalization.IsInteger R ((b : R) • f i) := by |
haveI := Classical.propDecidable
refine ⟨∏ i ∈ s, (sec M (f i)).2, fun i hi => ⟨?_, ?_⟩⟩
· exact (∏ j ∈ s.erase i, (sec M (f j)).2) * (sec M (f i)).1
rw [RingHom.map_mul, sec_spec', ← mul_assoc, ← (algebraMap R S).map_mul, ← Algebra.smul_def]
congr 2
refine _root_.trans ?_ (map_prod (Submonoid.subtype M) _... |
/-
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.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.or... | Mathlib/Algebra/Order/Group/MinMax.lean | 69 | 70 | theorem min_div_div_left' (a b c : α) : min (a / b) (a / c) = a / max b c := by |
simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cf... | Mathlib/Data/Nat/Pairing.lean | 93 | 100 | theorem unpair_lt {n : ℕ} (n1 : 1 ≤ n) : (unpair n).1 < n := by |
let s := sqrt n
simp only [unpair, ge_iff_le, Nat.sub_le_iff_le_add]
by_cases h : n - s * s < s <;> simp [h]
· exact lt_of_lt_of_le h (sqrt_le_self _)
· simp at h
have s0 : 0 < s := sqrt_pos.2 n1
exact lt_of_le_of_lt h (Nat.sub_lt n1 (Nat.mul_pos s0 s0))
|
/-
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
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 56 | 59 | theorem norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two (x y : V) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2 := by |
rw [norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero]
exact inner_eq_zero_iff_angle_eq_pi_div_two x y
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
/-!
# Sober spaces
A quasi... | Mathlib/Topology/Sober.lean | 107 | 111 | theorem isGenericPoint_iff_forall_closed (hS : IsClosed S) (hxS : x ∈ S) :
IsGenericPoint x S ↔ ∀ Z : Set α, IsClosed Z → x ∈ Z → S ⊆ Z := by |
have : closure {x} ⊆ S := closure_minimal (singleton_subset_iff.2 hxS) hS
simp_rw [IsGenericPoint, subset_antisymm_iff, this, true_and_iff, closure, subset_sInter_iff,
mem_setOf_eq, and_imp, singleton_subset_iff]
|
/-
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.GroupWithZero.Units.Basic
import Mathli... | Mathlib/Algebra/GroupWithZero/Divisibility.lean | 122 | 124 | theorem ne_zero_of_dvd_ne_zero {p q : α} (h₁ : q ≠ 0) (h₂ : p ∣ q) : p ≠ 0 := by |
rcases h₂ with ⟨u, rfl⟩
exact left_ne_zero_of_mul h₁
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8... | Mathlib/LinearAlgebra/SModEq.lean | 53 | 54 | theorem bot : x ≡ y [SMOD (⊥ : Submodule R M)] ↔ x = y := by |
rw [SModEq.def, Submodule.Quotient.eq, mem_bot, sub_eq_zero]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 57 | 59 | theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s := by |
simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 76 | 81 | theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by |
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
|
/-
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.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 96 | 100 | theorem lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N' := by |
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩
rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩
exact h x hx m hm
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-comm... | Mathlib/Data/Nat/Log.lean | 108 | 111 | theorem pow_le_of_le_log {b x y : ℕ} (hy : y ≠ 0) (h : x ≤ log b y) : b ^ x ≤ y := by |
refine (le_or_lt b 1).elim (fun hb => ?_) fun hb => (pow_le_iff_le_log hb hy).2 h
rw [log_of_left_le_one hb, Nat.le_zero] at h
rwa [h, Nat.pow_zero, one_le_iff_ne_zero]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 53 | 56 | theorem tendstoUniformly_tsum {f : α → β → F} (hu : Summable u) (hfu : ∀ n x, ‖f n x‖ ≤ u n) :
TendstoUniformly (fun t : Finset α => fun x => ∑ n ∈ t, f n x)
(fun x => ∑' n, f n x) atTop := by |
rw [← tendstoUniformlyOn_univ]; exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanpro... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 67 | 71 | theorem image_parallelepiped (f : E →ₗ[ℝ] F) (v : ι → E) :
f '' parallelepiped v = parallelepiped (f ∘ v) := by |
simp only [parallelepiped, ← image_comp]
congr 1 with t
simp only [Function.comp_apply, _root_.map_sum, LinearMap.map_smulₛₗ, RingHom.id_apply]
|
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
#align_import topology.homotopy.path from "... | Mathlib/Topology/Homotopy/Path.lean | 83 | 85 | theorem eval_zero (F : Homotopy p₀ p₁) : F.eval 0 = p₀ := by |
ext t
simp [eval]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tacti... | Mathlib/RingTheory/Ideal/Quotient.lean | 137 | 138 | theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by |
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 116 | 119 | theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n := by |
rw [wittPolynomial, map_sum, wittPolynomial]
refine sum_congr rfl fun i _ => ?_
rw [map_monomial, RingHom.map_pow, map_natCast]
|
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
/-!
# Infinite sums and products over `ℕ` and `ℤ`
This file contains lemma... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 62 | 65 | theorem prod_range_mul {f : ℕ → M} {k : ℕ} (h : HasProd (fun n ↦ f (n + k)) m) :
HasProd f ((∏ i ∈ range k, f i) * m) := by |
refine ((range k).hasProd f).mul_compl ?_
rwa [← (notMemRangeEquiv k).symm.hasProd_iff]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e31... | Mathlib/Data/QPF/Multivariate/Basic.lean | 126 | 138 | theorem liftP_iff {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (x : F α) :
LiftP p x ↔ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ i j, p (f i j) := by |
constructor
· rintro ⟨y, hy⟩
cases' h : repr y with a f
use a, fun i j => (f i j).val
constructor
· rw [← hy, ← abs_repr y, h, ← abs_map]; rfl
intro i j
apply (f i j).property
rintro ⟨a, f, h₀, h₁⟩
use abs ⟨a, fun i j => ⟨f i j, h₁ i j⟩⟩
rw [← abs_map, h₀]; rfl
|
/-
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.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
/-!
# N-ary images of s... | Mathlib/Data/Set/NAry.lean | 72 | 73 | theorem image2_subset_iff_right : image2 f s t ⊆ u ↔ ∀ b ∈ t, (fun a => f a b) '' s ⊆ u := by |
simp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage, @forall₂_swap α]
|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheor... | Mathlib/Algebra/Module/Torsion.lean | 110 | 123 | theorem CompleteLattice.Independent.linear_independent' {ι R M : Type*} {v : ι → M} [Ring R]
[AddCommGroup M] [Module R M] (hv : CompleteLattice.Independent fun i => R ∙ v i)
(h_ne_zero : ∀ i, Ideal.torsionOf R M (v i) = ⊥) : LinearIndependent R v := by |
refine linearIndependent_iff_not_smul_mem_span.mpr fun i r hi => ?_
replace hv := CompleteLattice.independent_def.mp hv i
simp only [iSup_subtype', ← Submodule.span_range_eq_iSup (ι := Subtype _), disjoint_iff] at hv
have : r • v i ∈ (⊥ : Submodule R M) := by
rw [← hv, Submodule.mem_inf]
refine ⟨Submod... |
/-
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.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "lea... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 68 | 76 | theorem liftAlternating_ι (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (m : M) :
liftAlternating (R := R) (M := M) (N := N) f (ι R m) = f 1 ![m] := by |
dsimp [liftAlternating]
rw [foldl_ι, LinearMap.mk₂_apply, AlternatingMap.curryLeft_apply_apply]
congr
-- Porting note: In Lean 3, `congr` could use the `[Subsingleton (Fin 0 → M)]` instance to finish
-- the proof. Here, the instance can be synthesized but `congr` does not use it so the following
-- line is... |
/-
Copyright (c) 2022 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.small_sets from "leanprover-commu... | Mathlib/Order/Filter/SmallSets.lean | 47 | 51 | theorem bind_smallSets_gc :
GaloisConnection (fun L : Filter (Set α) ↦ L.bind principal) smallSets := by |
intro L l
simp_rw [smallSets_eq_generate, le_generate_iff, image_subset_iff]
rfl
|
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