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) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Order.Group.Abs
#align_import data.int.order.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f3... | Mathlib/Algebra/Order/Group/Int.lean | 115 | 116 | theorem emod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a % b < |b| := by |
rw [← emod_abs]; exact emod_lt_of_pos _ (abs_pos.2 H)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Po... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 35 | 44 | theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M → N) (p : M → Prop)
(h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
(g : ι → M) (s : Finset ι) (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (g i)) :
f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) := by |
refine le_trans
(Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ ?_ ?_) ?_
· simp [hs_nonempty.ne_empty]
· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanp... | Mathlib/Data/Int/Cast/Basic.lean | 93 | 98 | theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by |
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e, neg_sub]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Matrix.CharP
#align_import l... | Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean | 47 | 50 | theorem ZMod.charpoly_pow_card {p : ℕ} [Fact p.Prime] (M : Matrix n n (ZMod p)) :
(M ^ p).charpoly = M.charpoly := by |
have h := FiniteField.Matrix.charpoly_pow_card M
rwa [ZMod.card] at h
|
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.i... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 104 | 107 | theorem IsDiag.smul [Monoid R] [AddMonoid α] [DistribMulAction R α] (k : R) {A : Matrix n n α}
(ha : A.IsDiag) : (k • A).IsDiag := by |
intro i j h
simp [ha h]
|
/-
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.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-com... | Mathlib/SetTheory/Game/Birthday.lean | 122 | 129 | theorem toPGame_birthday (o : Ordinal) : o.toPGame.birthday = o := by |
induction' o using Ordinal.induction with o IH
rw [toPGame_def, PGame.birthday]
simp only [lsub_empty, max_zero_right]
-- Porting note: was `nth_rw 1 [← lsub_typein o]`
conv_rhs => rw [← lsub_typein o]
congr with x
exact IH _ (typein_lt_self x)
|
/-
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.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 85 | 92 | theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i) := by |
rintro x hx y hy hxy a b ha hb hab
rw [mem_iUnion] at hx hy
obtain ⟨i, hx⟩ := hx
obtain ⟨j, hy⟩ := hy
obtain ⟨k, hik, hjk⟩ := hdir i j
exact interior_mono (subset_iUnion s k) (hs (hik hx) (hjk hy) hxy ha hb hab)
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
#align_import number_theory.primes_congruent_one from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be5... | Mathlib/NumberTheory/PrimesCongruentOne.lean | 60 | 64 | theorem frequently_atTop_modEq_one {k : ℕ} (hk0 : k ≠ 0) :
∃ᶠ p in atTop, Nat.Prime p ∧ p ≡ 1 [MOD k] := by |
refine frequently_atTop.2 fun n => ?_
obtain ⟨p, hp⟩ := exists_prime_gt_modEq_one n hk0
exact ⟨p, ⟨hp.2.1.le, hp.1, hp.2.2⟩⟩
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import... | Mathlib/Data/Real/Irrational.lean | 38 | 40 | theorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by |
rintro ⟨a, rfl⟩
exact tr (isAlgebraic_algebraMap a)
|
/-
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 | 101 | 102 | theorem mem_iff {A : Matrix (Sum l l) (Sum l l) R} :
A ∈ symplecticGroup l R ↔ A * J l R * Aᵀ = J l R := by | simp [symplecticGroup]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 54 | 74 | theorem borel_eq_generateFrom_Iio : borel α = .generateFrom (range Iio) := by |
refine le_antisymm ?_ (generateFrom_le ?_)
· rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals α _ _ _)]
letI : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
have H : ∀ a : α, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ ⟨_, rfl⟩
refine... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathl... | Mathlib/LinearAlgebra/Basic.lean | 73 | 80 | theorem isLinearMap_add [Semiring R] [AddCommMonoid M] [Module R M] :
IsLinearMap R fun x : M × M => x.1 + x.2 := by |
apply IsLinearMap.mk
· intro x y
simp only [Prod.fst_add, Prod.snd_add]
abel -- Porting Note: was cc
· intro x y
simp [smul_add]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.exterior_algebra.grading from "leanprover-com... | Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean | 52 | 54 | theorem GradedAlgebra.ι_sq_zero (m : M) : GradedAlgebra.ι R M m * GradedAlgebra.ι R M m = 0 := by |
rw [GradedAlgebra.ι_apply, DirectSum.of_mul_of]
exact DFinsupp.single_eq_zero.mpr (Subtype.ext <| ExteriorAlgebra.ι_sq_zero _)
|
/-
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.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.modeq from "leanprover-communit... | Mathlib/Order/Filter/ModEq.lean | 33 | 34 | theorem frequently_even : ∃ᶠ m : ℕ in atTop, Even m := by |
simpa only [even_iff] using frequently_mod_eq zero_lt_two
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.CondDistrib
#align_import probability.kernel.condexp from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
/-... | Mathlib/Probability/Kernel/Condexp.lean | 44 | 49 | theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id [TopologicalSpace F]
(hm : m ≤ mΩ) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x : Ω × Ω => f x.2)
(@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) := by |
rw [← aestronglyMeasurable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf
simp_rw [id] at hf ⊢
exact hf
|
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bra... | Mathlib/GroupTheory/Commutator.lean | 31 | 32 | theorem commutatorElement_eq_one_iff_mul_comm : ⁅g₁, g₂⁆ = 1 ↔ g₁ * g₂ = g₂ * g₁ := by |
rw [commutatorElement_def, mul_inv_eq_one, mul_inv_eq_iff_eq_mul]
|
/-
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.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c066... | Mathlib/Algebra/GCDMonoid/Finset.lean | 92 | 92 | theorem normalize_lcm : normalize (s.lcm f) = s.lcm f := by | simp [lcm_def]
|
/-
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.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 73 | 76 | theorem injective (f : α → β) (hf : Injective f) : Injective (mapFun f : 𝕎 α → 𝕎 β) := by |
intros _ _ h
ext p
exact hf (congr_arg (fun x => coeff x p) h : _)
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.ContinuousOn
#align_import topology.algebra.order.left_right from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e... | Mathlib/Topology/Order/LeftRight.lean | 95 | 97 | theorem continuousWithinAt_Ioi_iff_Ici {a : α} {f : α → β} :
ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a := by |
simp only [← Ici_diff_left, continuousWithinAt_diff_self]
|
/-
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.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
/-!
# ... | Mathlib/Topology/Baire/Lemmas.lean | 85 | 88 | theorem eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by |
simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
tauto
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jujian Zhang
-/
import Mathlib.RingTheory.Noetherian
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.Module.Proj... | Mathlib/RingTheory/Flat/Basic.lean | 117 | 119 | theorem iff_lTensor_injective' :
Module.Flat R M ↔ ∀ (I : Ideal R), Function.Injective (lTensor M I.subtype) := by |
simpa [← comm_comp_rTensor_comp_comm_eq] using Module.Flat.iff_rTensor_injective' R M
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Cases
import Mathlib.Algebra.NeZero
import Mathlib.Logic.Function.Basic
#align_import algebra.char_zero.defs ... | Mathlib/Algebra/CharZero/Defs.lean | 92 | 92 | theorem cast_eq_one {n : ℕ} : (n : R) = 1 ↔ n = 1 := by | rw [← cast_one, cast_inj]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"... | Mathlib/Analysis/NormedSpace/Ray.lean | 91 | 94 | theorem sameRay_iff_inv_norm_smul_eq : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y := by |
rcases eq_or_ne x 0 with (rfl | hx); · simp [SameRay.zero_left]
rcases eq_or_ne y 0 with (rfl | hy); · simp [SameRay.zero_right]
simp only [sameRay_iff_inv_norm_smul_eq_of_ne hx hy, *, false_or_iff]
|
/-
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
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 126 | 127 | theorem WalkingParallelPairHom.id.sizeOf_spec' (X : WalkingParallelPair) :
(WalkingParallelPairHom._sizeOf_inst X X).sizeOf (𝟙 X) = 1 + sizeOf X := by | cases X <;> rfl
|
/-
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.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathli... | Mathlib/Probability/StrongLaw.lean | 82 | 85 | theorem _root_.MeasureTheory.AEStronglyMeasurable.truncation (hf : AEStronglyMeasurable f μ)
{A : ℝ} : AEStronglyMeasurable (truncation f A) μ := by |
apply AEStronglyMeasurable.comp_aemeasurable _ hf.aemeasurable
exact (stronglyMeasurable_id.indicator measurableSet_Ioc).aestronglyMeasurable
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Hom
#align_import algebra.group.prod from "leanprover-communi... | Mathlib/Algebra/Group/Prod.lean | 79 | 81 | theorem one_mk_mul_one_mk [Monoid M] [Mul N] (b₁ b₂ : N) :
((1 : M), b₁) * (1, b₂) = (1, b₁ * b₂) := by |
rw [mk_mul_mk, mul_one]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.RingTheory.Nilpotent.Defs
#align_import algebra.char_p.basic from "leanprover-community/mathl... | Mathlib/Algebra/CharP/Reduced.lean | 46 | 50 | theorem ExpChar.pow_prime_pow_mul_eq_one_iff (p k m : ℕ) [ExpChar R p] (x : R) :
x ^ (p ^ k * m) = 1 ↔ x ^ m = 1 := by |
rw [pow_mul']
convert ← (iterateFrobenius_inj R p k).eq_iff
apply map_one
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 85 | 86 | theorem preimage_coe_Ioo_top : (some : α → WithTop α) ⁻¹' Ioo a ⊤ = Ioi a := by |
simp [← Ioi_inter_Iio]
|
/-
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.PseudoMetric
#align_import topology.metric_space.basic fr... | Mathlib/Topology/MetricSpace/Basic.lean | 121 | 126 | theorem subsingleton_closedBall (x : γ) {r : ℝ} (hr : r ≤ 0) : (closedBall x r).Subsingleton := by |
rcases hr.lt_or_eq with (hr | rfl)
· rw [closedBall_eq_empty.2 hr]
exact subsingleton_empty
· rw [closedBall_zero]
exact subsingleton_singleton
|
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
#align_import analysis.complex.arg from "leanprover-community/ma... | Mathlib/Analysis/Complex/Arg.lean | 31 | 38 | theorem sameRay_iff : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg := by |
rcases eq_or_ne x 0 with (rfl | hx)
· simp
rcases eq_or_ne y 0 with (rfl | hy)
· simp
simp only [hx, hy, false_or_iff, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]
field_simp [hx, hy]
rw [mul_comm, eq_comm]
|
/-
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.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#alig... | Mathlib/Analysis/PSeries.lean | 71 | 76 | theorem le_sum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : Monotone u) (n : ℕ) :
(∑ k ∈ range (u n), f k) ≤
∑ k ∈ range (u 0), f k + ∑ k ∈ range n, (u (k + 1) - u k) • f (u k) := by |
convert add_le_add_left (le_sum_schlomilch' hf h_pos hu n) (∑ k ∈ range (u 0), f k)
rw [← sum_range_add_sum_Ico _ (hu n.zero_le)]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection
#align_import order.modular_lattice from "leanpr... | Mathlib/Order/ModularLattice.lean | 103 | 105 | theorem covBy_sup_of_inf_covBy_of_inf_covBy_right : a ⊓ b ⋖ a → a ⊓ b ⋖ b → b ⋖ a ⊔ b := by |
rw [inf_comm, sup_comm]
exact fun ha hb => covBy_sup_of_inf_covBy_of_inf_covBy_left hb ha
|
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | Mathlib/Algebra/Lie/Free.lean | 91 | 92 | theorem Rel.neg {a b : lib R X} (h : Rel R X a b) : Rel R X (-a) (-b) := by |
simpa only [neg_one_smul] using h.smul (-1)
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Cont... | Mathlib/Data/Real/Cardinality.lean | 105 | 110 | theorem cantorFunction_le (h1 : 0 ≤ c) (h2 : c < 1) (h3 : ∀ n, f n → g n) :
cantorFunction c f ≤ cantorFunction c g := by |
apply tsum_le_tsum _ (summable_cantor_function f h1 h2) (summable_cantor_function g h1 h2)
intro n; cases h : f n
· simp [h, cantorFunctionAux_nonneg h1]
replace h3 : g n = true := h3 n h; simp [h, h3]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 84 | 84 | theorem volume_Icc {a b : ℝ} : volume (Icc a b) = ofReal (b - a) := by | simp [volume_val]
|
/-
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 | 130 | 134 | theorem vieta_formula_quadratic {b c x : α} (h : x * x - b * x + c = 0) :
∃ y : α, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c := by |
have : c = x * (b - x) := (eq_neg_of_add_eq_zero_right h).trans (by simp [mul_sub, mul_comm])
refine ⟨b - x, ?_, by simp, by rw [this]⟩
rw [this, sub_add, ← sub_mul, sub_self]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.NumberTheory.NumberField.Embeddings
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb1... | Mathlib/NumberTheory/NumberField/Units/Basic.lean | 54 | 57 | theorem NumberField.isUnit_iff_norm [NumberField K] {x : 𝓞 K} :
IsUnit x ↔ |(RingOfIntegers.norm ℚ x : ℚ)| = 1 := by |
convert (RingOfIntegers.isUnit_norm ℚ (F := K)).symm
rw [← abs_one, abs_eq_abs, ← Rat.RingOfIntegers.isUnit_iff]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 63 | 69 | theorem sum_smul_index_linearMap' [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂]
[Module R M₂] {v : α →₀ M} {c : R} {h : α → M →ₗ[R] M₂} :
((c • v).sum fun a => h a) = c • v.sum fun a => h a := by |
rw [Finsupp.sum_smul_index', Finsupp.smul_sum]
· simp only [map_smul]
· intro i
exact (h i).map_zero
|
/-
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.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# The... | Mathlib/CategoryTheory/Sites/Plus.lean | 90 | 95 | theorem diagramNatTrans_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) (X : C) :
J.diagramNatTrans (η ≫ γ) X = J.diagramNatTrans η X ≫ J.diagramNatTrans γ X := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 52 | 52 | theorem map_apply : (map f p) b = ∑' a, if b = f a then p a else 0 := by | simp [map]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Cases
#align_import ... | Mathlib/Logic/Equiv/Option.lean | 89 | 91 | theorem removeNone_aux_some {x : α} (h : ∃ x', e (some x) = some x') :
some (removeNone_aux e x) = e (some x) := by |
simp [removeNone_aux, Option.isSome_iff_exists.mpr h]
|
/-
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.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 136 | 138 | theorem IsClosable.closure_isClosed {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure.IsClosed := by |
rw [IsClosed, ← hf.graph_closure_eq_closure_graph]
exact f.graph.isClosed_topologicalClosure
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Category.TopCat.Adjunctions
#align_import topology.category.Top.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e62... | Mathlib/Topology/Category/TopCat/EpiMono.lean | 27 | 34 | theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by |
suffices Epi f ↔ Epi ((forget TopCat).map f) by
rw [this, CategoryTheory.epi_iff_surjective]
rfl
constructor
· intro
infer_instance
· apply Functor.epi_of_epi_map
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | Mathlib/Topology/Connected/LocallyConnected.lean | 104 | 115 | theorem locallyConnectedSpace_iff_connected_subsets :
LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U := by |
constructor
· rw [locallyConnectedSpace_iff_open_connected_subsets]
intro h x U hxU
rcases h x U hxU with ⟨V, hVU, hV₁, hxV, hV₂⟩
exact ⟨V, hV₁.mem_nhds hxV, hV₂.isPreconnected, hVU⟩
· rw [locallyConnectedSpace_iff_connectedComponentIn_open]
refine fun h U hU x _ => isOpen_iff_mem_nhds.mpr fun y ... |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprove... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 98 | 102 | theorem comp {f : X → Y} {g : Y → Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x) :
ConformalAt (g ∘ f) x := by |
rcases hf with ⟨f', hf₁, cf⟩
rcases hg with ⟨g', hg₁, cg⟩
exact ⟨g'.comp f', hg₁.comp x hf₁, cg.comp cf⟩
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Instances.Real
import M... | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | 60 | 62 | theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n ↦ dist (f n) (f n.succ))
(ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by |
simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0
|
/-
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.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category fro... | Mathlib/CategoryTheory/PathCategory.lean | 93 | 100 | theorem lift_spec {C} [Category C] (φ : V ⥤q C) : of ⋙q (lift φ).toPrefunctor = φ := by |
fapply Prefunctor.ext
· rintro X
rfl
· rintro X Y f
rcases φ with ⟨φo, φm⟩
dsimp [lift, Quiver.Hom.toPath]
simp only [Category.id_comp]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 133 | 137 | theorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v)
(hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :
(p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Locus of une... | Mathlib/Data/Finsupp/NeLocus.lean | 93 | 98 | theorem zipWith_neLocus_eq_left [DecidableEq N] [Zero M] [DecidableEq P] [Zero P] [Zero N]
{F : M → N → P} (F0 : F 0 0 = 0) (f : α →₀ M) (g₁ g₂ : α →₀ N)
(hF : ∀ f, Function.Injective fun g => F f g) :
(zipWith F F0 f g₁).neLocus (zipWith F F0 f g₂) = g₁.neLocus g₂ := by |
ext
simpa only [mem_neLocus] using (hF _).ne_iff
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Uni Marx
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
i... | Mathlib/CategoryTheory/Category/RelCat.lean | 62 | 63 | theorem rel_id_apply₂ (X : RelCat) (x y : X) : (𝟙 X) x y ↔ x = y := by |
rw [RelCat.Hom.rel_id]
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "lean... | Mathlib/Algebra/CharZero/Lemmas.lean | 100 | 102 | theorem zero_eq_bit0 {a : R} : 0 = bit0 a ↔ a = 0 := by |
rw [eq_comm]
exact bit0_eq_zero
|
/-
Copyright (c) 2022 Jesse Reimann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jesse Reimann, Kalle Kytölä
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.integral.riesz_markov_kakutani from... | Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean | 81 | 89 | theorem exists_lt_rieszContentAux_add_pos (K : Compacts X) {ε : ℝ≥0} (εpos : 0 < ε) :
∃ f : X →ᵇ ℝ≥0, (∀ x ∈ K, (1 : ℝ≥0) ≤ f x) ∧ Λ f < rieszContentAux Λ K + ε := by |
--choose a test function `f` s.t. `Λf = α < λ(K) + ε`
obtain ⟨α, ⟨⟨f, f_hyp⟩, α_hyp⟩⟩ :=
exists_lt_of_csInf_lt (rieszContentAux_image_nonempty Λ K)
(lt_add_of_pos_right (rieszContentAux Λ K) εpos)
refine ⟨f, f_hyp.left, ?_⟩
rw [f_hyp.right]
exact α_hyp
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 78 | 79 | theorem countP_eq_zero : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by |
simp only [countP_eq_length_filter, length_eq_zero, filter_eq_nil]
|
/-
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.Algebra.Category.GroupCat.Colimits
import Mathlib.Algebra.Category.GroupCat.FilteredColimits
import Mathlib.Algebra.Category.GroupCat.Kernels
import Math... | Mathlib/Algebra/Category/GroupCat/Abelian.lean | 51 | 57 | theorem exact_iff : Exact f g ↔ f.range = g.ker := by |
rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm
((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),
fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,
(QuotientAddGroup.ker_le_range_iff ... |
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Richard Hill
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.Ri... | Mathlib/Algebra/Polynomial/Derivation.lean | 67 | 67 | theorem mkDerivation_X (a : A) : mkDerivation R a X = a := by | simp [mkDerivation_apply]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"... | Mathlib/Algebra/Ring/Center.lean | 46 | 67 | theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by | rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_natCast, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev, add_mul, add_mul, add_mul,
neg_mul, one_mul, neg_mul 1, one_mul, ← neg_mul, ad... |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal approximation theory o... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 87 | 90 | theorem le_lfpApprox {a : Ordinal} : x ≤ lfpApprox f x a := by |
unfold lfpApprox
apply le_sSup
simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, true_or]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 70 | 73 | theorem rank_zero_iff_forall_zero :
Module.rank R M = 0 ↔ ∀ x : M, x = 0 := by |
simp_rw [rank_eq_zero_iff, smul_eq_zero, and_or_left, not_and_self_iff, false_or,
exists_and_right, and_iff_right (exists_ne (0 : R))]
|
/-
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.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.RegularMon... | Mathlib/CategoryTheory/Limits/MonoCoprod.lean | 63 | 69 | theorem binaryCofan_inr {A B : C} [MonoCoprod C] (c : BinaryCofan A B) (hc : IsColimit c) :
Mono c.inr := by |
haveI hc' : IsColimit (BinaryCofan.mk c.inr c.inl) :=
BinaryCofan.IsColimit.mk _ (fun f₁ f₂ => hc.desc (BinaryCofan.mk f₂ f₁))
(by aesop_cat) (by aesop_cat)
(fun f₁ f₂ m h₁ h₂ => BinaryCofan.IsColimit.hom_ext hc (by aesop_cat) (by aesop_cat))
exact binaryCofan_inl _ hc'
|
/-
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.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monoton... | Mathlib/Data/Nat/Totient.lean | 84 | 109 | theorem Ico_filter_coprime_le {a : ℕ} (k n : ℕ) (a_pos : 0 < a) :
((Ico k (k + n)).filter (Coprime a)).card ≤ totient a * (n / a + 1) := by |
conv_lhs => rw [← Nat.mod_add_div n a]
induction' n / a with i ih
· rw [← filter_coprime_Ico_eq_totient a k]
simp only [add_zero, mul_one, mul_zero, le_of_lt (mod_lt n a_pos),
Nat.zero_eq, zero_add]
-- Porting note: below line was `mono`
refine Finset.card_mono ?_
refine monotone_filter_lef... |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 84 | 86 | theorem Sublist.countP_le (s : l₁ <+ l₂) : countP p l₁ ≤ countP p l₂ := by |
simp only [countP_eq_length_filter]
apply s.filter _ |>.length_le
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
[`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/d... | Mathlib/Data/Finset/Sym.lean | 62 | 65 | theorem sym2_univ [Fintype α] (inst : Fintype (Sym2 α) := Sym2.instFintype) :
(univ : Finset α).sym2 = univ := by |
ext
simp only [mem_sym2_iff, mem_univ, implies_true]
|
/-
Copyright (c) 2021 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import dat... | Mathlib/Data/Nat/Choose/Central.lean | 72 | 81 | theorem succ_mul_centralBinom_succ (n : ℕ) :
(n + 1) * centralBinom (n + 1) = 2 * (2 * n + 1) * centralBinom n :=
calc
(n + 1) * (2 * (n + 1)).choose (n + 1) = (2 * n + 2).choose (n + 1) * (n + 1) := mul_comm _ _
_ = (2 * n + 1).choose n * (2 * n + 2) := by | rw [choose_succ_right_eq, choose_mul_succ_eq]
_ = 2 * ((2 * n + 1).choose n * (n + 1)) := by ring
_ = 2 * ((2 * n + 1).choose n * (2 * n + 1 - n)) := by rw [two_mul n, add_assoc,
Nat.add_sub_cancel_left]
_ = 2 * ((2 * n).choose n * (2 * n + 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.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 40 | 41 | theorem Multipliable.of_inv (hf : Multipliable fun b ↦ (f b)⁻¹) : Multipliable f := by |
simpa only [inv_inv] using hf.inv
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 44 | 52 | theorem eq_zero_of_degree_lt_of_eval_finset_eq_zero (degree_f_lt : f.degree < s.card)
(eval_f : ∀ x ∈ s, f.eval x = 0) : f = 0 := by |
rw [← mem_degreeLT] at degree_f_lt
simp_rw [eval_eq_sum_degreeLTEquiv degree_f_lt] at eval_f
rw [← degreeLTEquiv_eq_zero_iff_eq_zero degree_f_lt]
exact
Matrix.eq_zero_of_forall_index_sum_mul_pow_eq_zero
(Injective.comp (Embedding.subtype _).inj' (equivFinOfCardEq (card_coe _)).symm.injective)
f... |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
/-!
#... | Mathlib/Control/Bitraversable/Lemmas.lean | 72 | 75 | theorem comp_tfst {α₀ α₁ α₂ β} (f : α₀ → F α₁) (f' : α₁ → G α₂) (x : t α₀ β) :
Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x := by |
rw [← comp_bitraverse]
simp only [Function.comp, tfst, map_pure, Pure.pure]
|
/-
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.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
im... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 103 | 105 | theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 := by |
convert Iff.rfl
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Cont... | Mathlib/Data/Real/Cardinality.lean | 78 | 79 | theorem cantorFunctionAux_eq (h : f n = g n) :
cantorFunctionAux c f n = cantorFunctionAux c g n := by | simp [cantorFunctionAux, h]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.GroupTheory.Perm.Cycle.Basic
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef94... | Mathlib/GroupTheory/Perm/Closure.lean | 111 | 122 | theorem closure_prime_cycle_swap {σ τ : Perm α} (h0 : (Fintype.card α).Prime) (h1 : IsCycle σ)
(h2 : σ.support = Finset.univ) (h3 : IsSwap τ) : closure ({σ, τ} : Set (Perm α)) = ⊤ := by |
obtain ⟨x, y, h4, h5⟩ := h3
obtain ⟨i, hi⟩ :=
h1.exists_pow_eq (mem_support.mp ((Finset.ext_iff.mp h2 x).mpr (Finset.mem_univ x)))
(mem_support.mp ((Finset.ext_iff.mp h2 y).mpr (Finset.mem_univ y)))
rw [h5, ← hi]
refine closure_cycle_coprime_swap
(Nat.Coprime.symm (h0.coprime_iff_not_dvd.mpr fun ... |
/-
Copyright (c) 2018 Mitchell Rowett. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Rowett, Scott Morrison
-/
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Group... | Mathlib/GroupTheory/Coset.lean | 105 | 106 | theorem leftCoset_assoc (s : Set α) (a b : α) : a • (b • s) = (a * b) • s := by |
simp [← image_smul, (image_comp _ _ _).symm, Function.comp, mul_assoc]
|
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
... | Mathlib/Data/Nat/Dist.lean | 63 | 63 | theorem dist_tri_right' (n m : ℕ) : n ≤ m + dist n m := by | rw [dist_comm]; apply dist_tri_right
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-com... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 74 | 74 | theorem dist_self {v : V} : dist G v v = 0 := by | simp
|
/-
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 | 94 | 97 | theorem eq_loopyOn_iff : M = loopyOn E ↔ M.E = E ∧ ∀ X ⊆ M.E, M.Indep X → X = ∅ := by |
simp only [eq_iff_indep_iff_indep_forall, loopyOn_ground, loopyOn_indep_iff, and_congr_right_iff]
rintro rfl
refine ⟨fun h I hI ↦ (h I hI).1, fun h I hIE ↦ ⟨h I hIE, by rintro rfl; simp⟩⟩
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 64 | 65 | theorem map₂_mk_eq_prod (f : Filter α) (g : Filter β) : map₂ Prod.mk f g = f ×ˢ g := by |
simp only [← map_prod_eq_map₂, map_id']
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 56 | 56 | theorem χ₄_nat_mod_four (n : ℕ) : χ₄ n = χ₄ (n % 4 : ℕ) := by | rw [← ZMod.natCast_mod n 4]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 120 | 122 | theorem splitUpper_eq_bot {i x} : I.splitUpper i x = ⊥ ↔ I.upper i ≤ x := by |
rw [splitUpper, mk'_eq_bot, exists_update_iff I.lower fun j y => I.upper j ≤ y]
simp [(I.lower_lt_upper _).not_le]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib... | Mathlib/CategoryTheory/Comma/Over.lean | 67 | 67 | theorem over_right (U : Over X) : U.right = ⟨⟨⟩⟩ := by | simp only
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 52 | 59 | theorem fromBlocks_eq_of_invertible₁₁ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix l m α)
(D : Matrix l n α) [Invertible A] :
fromBlocks A B C D =
fromBlocks 1 0 (C * ⅟ A) 1 * fromBlocks A 0 0 (D - C * ⅟ A * B) *
fromBlocks 1 (⅟ A * B) 0 1 := by |
simp only [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add,
Matrix.one_mul, Matrix.mul_one, invOf_mul_self, Matrix.mul_invOf_self_assoc,
Matrix.mul_invOf_mul_self_cancel, Matrix.mul_assoc, add_sub_cancel]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 81 | 85 | theorem weightedTotalDegree'_eq_bot_iff (w : σ → M) (p : MvPolynomial σ R) :
weightedTotalDegree' w p = ⊥ ↔ p = 0 := by |
simp only [weightedTotalDegree', Finset.sup_eq_bot_iff, mem_support_iff, WithBot.coe_ne_bot,
MvPolynomial.eq_zero_iff]
exact forall_congr' fun _ => Classical.not_not
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.deriv.zpow from "leanpro... | Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 106 | 113 | theorem iter_deriv_zpow' (m : ℤ) (k : ℕ) :
(deriv^[k] fun x : 𝕜 => x ^ m) =
fun x => (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k) := by |
induction' k with k ihk
· simp only [Nat.zero_eq, one_mul, Int.ofNat_zero, id, sub_zero, Finset.prod_range_zero,
Function.iterate_zero]
· simp only [Function.iterate_succ_apply', ihk, deriv_const_mul_field', deriv_zpow',
Finset.prod_range_succ, Int.ofNat_succ, ← sub_sub, Int.cast_sub, Int.cast_natCas... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb... | Mathlib/Algebra/CharP/Two.lean | 134 | 139 | theorem orderOf_neg_one [Nontrivial R] : orderOf (-1 : R) = if ringChar R = 2 then 1 else 2 := by |
split_ifs with h
· rw [neg_one_eq_one_iff.2 h, orderOf_one]
apply orderOf_eq_prime
· simp
simpa [neg_one_eq_one_iff] using h
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 111 | 112 | theorem polar_comm (f : M → R) (x y : M) : polar f x y = polar f y x := by |
rw [polar, polar, add_comm, sub_sub, sub_sub, add_comm (f x) (f y)]
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 112 | 114 | theorem prod_top : f ×ˢ (⊤ : Filter β) = f.comap Prod.fst := by |
dsimp only [SProd.sprod]
rw [Filter.prod, comap_top, inf_top_eq]
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute... | Mathlib/Algebra/Field/Basic.lean | 37 | 37 | theorem same_add_div (h : b ≠ 0) : (b + a) / b = 1 + a / b := by | rw [← div_self h, add_div]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#al... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 114 | 116 | theorem dist_vadd_left (v : V) (x : P) : dist (v +ᵥ x) x = ‖v‖ := by |
-- porting note (#10745): was `simp [dist_eq_norm_vsub V _ x]`
rw [dist_eq_norm_vsub V _ x, vadd_vsub]
|
/-
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.Filter.Basic
import Mathlib.Data.Set.Countable
#align_import order.filter.countable_Inter from "leanprover-community/mathlib"@"b9e46fe10... | Mathlib/Order/Filter/CountableInter.lean | 58 | 62 | theorem countable_bInter_mem {ι : Type*} {S : Set ι} (hS : S.Countable) {s : ∀ i ∈ S, Set α} :
(⋂ i, ⋂ hi : i ∈ S, s i ‹_›) ∈ l ↔ ∀ i, ∀ hi : i ∈ S, s i ‹_› ∈ l := by |
rw [biInter_eq_iInter]
haveI := hS.toEncodable
exact countable_iInter_mem.trans Subtype.forall
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_the... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 99 | 106 | theorem degree_trunc_lt (f : R⟦X⟧) (n) : (trunc n f).degree < n := by |
rw [degree_lt_iff_coeff_zero]
intros
rw [coeff_trunc]
split_ifs with h
· rw [← not_le] at h
contradiction
· rfl
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "lea... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 26 | 33 | theorem snorm'_add_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(hq1 : 1 ≤ q) : snorm' (f + g) q μ ≤ snorm' f q μ + snorm' g q μ :=
calc
(∫⁻ a, (‖(f + g) a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) ≤
(∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0∞)) a ^ q ∂μ) ^ (1 / ... |
gcongr with a
simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
_ ≤ snorm' f q μ + snorm' g q μ := ENNReal.lintegral_Lp_add_le hf.ennnorm hg.ennnorm hq1
|
/-
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.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | Mathlib/RingTheory/Localization/InvSubmonoid.lean | 77 | 81 | theorem smul_toInvSubmonoid (m : M) : m • (toInvSubmonoid M S m : S) = 1 := by |
convert mul_toInvSubmonoid M S m
ext
rw [← Algebra.smul_def]
rfl
|
/-
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.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 97 | 100 | theorem contDiffWithinAtProp_self_source {f : E → H'} {s : Set E} {x : E} :
ContDiffWithinAtProp 𝓘(𝕜, E) I' n f s x ↔ ContDiffWithinAt 𝕜 n (I' ∘ f) s x := by |
simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq]
|
/-
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.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 138 | 140 | theorem transReflReparamAux_mem_I (t : I) : transReflReparamAux t ∈ I := by |
unfold transReflReparamAux
split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d449... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 61 | 64 | theorem AffineEquiv.pointReflection_midpoint_left (x y : P) :
pointReflection R (midpoint R x y) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 116 | 118 | theorem measure_iUnion_null_iff {ι : Sort*} [Countable ι] {s : ι → Set α} :
μ (⋃ i, s i) = 0 ↔ ∀ i, μ (s i) = 0 := by |
rw [← sUnion_range, measure_sUnion_null_iff (countable_range s), forall_mem_range]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 94 | 96 | theorem AffineTargetMorphismProperty.toProperty_apply (P : AffineTargetMorphismProperty)
{X Y : Scheme} (f : X ⟶ Y) [i : IsAffine Y] : P.toProperty f ↔ P f := by |
delta AffineTargetMorphismProperty.toProperty; 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.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb66... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 78 | 85 | theorem union_left {s'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s' t) :
IsMetricSeparated (s ∪ s') t := by |
rcases h, h' with ⟨⟨r, r0, hr⟩, ⟨r', r0', hr'⟩⟩
refine ⟨min r r', ?_, fun x hx y hy => hx.elim ?_ ?_⟩
· rw [← pos_iff_ne_zero] at r0 r0' ⊢
exact lt_min r0 r0'
· exact fun hx => (min_le_left _ _).trans (hr _ hx _ hy)
· exact fun hx => (min_le_right _ _).trans (hr' _ hx _ hy)
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valu... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 97 | 99 | theorem int_valuation_ne_zero (x : R) (hx : x ≠ 0) : v.intValuationDef x ≠ 0 := by |
rw [intValuationDef, if_neg hx]
exact WithZero.coe_ne_zero
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 75 | 79 | theorem pNilradical_le_nilradical {R : Type*} [CommSemiring R] {p : ℕ} :
pNilradical R p ≤ nilradical R := by |
by_cases hp : 1 < p
· rw [pNilradical, if_pos hp]
simp_rw [pNilradical, if_neg hp, bot_le]
|
/-
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 | 80 | 81 | theorem gcd_add_self_left (m n : ℕ) : gcd (m + n) n = gcd m n := by |
rw [gcd_comm, gcd_add_self_right, gcd_comm]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import ring_theory.integ... | Mathlib/RingTheory/IntegralDomain.lean | 111 | 113 | theorem Finite.isField_of_domain (R) [CommRing R] [IsDomain R] [Finite R] : IsField R := by |
cases nonempty_fintype R
exact @Field.toIsField R (@Fintype.fieldOfDomain R _ _ (Classical.decEq R) _)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.Algebra.Module.Prod
#align_import algebra.algebra.prod from "leanprover-community/mathlib"@"28aa996fc6fb4317... | Mathlib/Algebra/Algebra/Prod.lean | 87 | 87 | theorem fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (fst R B C).comp (prod f g) = f := by | ext; rfl
|
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