Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,152 | 1,154 | theorem addValuation.apply {x : ℚ_[p]} (hx : x ≠ 0) :
Padic.addValuation x = (x.valuation : WithTop ℤ) := by |
simp only [Padic.addValuation, AddValuation.of_apply, addValuationDef, if_neg hx]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 127 | 132 | theorem norm_le_half_period {x : AddCircle p} (hp : p ≠ 0) : ‖x‖ ≤ |p| / 2 := by |
obtain ⟨x⟩ := x
change ‖(x : AddCircle p)‖ ≤ |p| / 2
rw [norm_eq, ← mul_le_mul_left (abs_pos.mpr (inv_ne_zero hp)), ← abs_mul, mul_sub, mul_left_comm,
← mul_div_assoc, ← abs_mul, inv_mul_cancel hp, mul_one, abs_one]
exact abs_sub_round (p⁻¹ * x)
|
/-
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.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 92 | 93 | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := by |
rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub]
|
/-
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 | 93 | 94 | theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 362 | 366 | theorem natSepDegree_pow {n : ℕ} : (f ^ n).natSepDegree = if n = 0 then 0 else f.natSepDegree := by |
simp only [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_pow]
by_cases h : n = 0
· simp only [h, zero_smul, Multiset.toFinset_zero, Finset.card_empty, ite_true]
simp only [h, Multiset.toFinset_nsmul _ n h, ite_false]
|
/-
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 | 282 | 300 | theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
[IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
gauge (a • s) = a⁻¹ • gauge s := by |
obtain rfl | ha' := ha.eq_or_lt
· rw [inv_zero, zero_smul, gauge_of_subset_zero (zero_smul_set_subset _)]
ext x
rw [gauge_def', Pi.smul_apply, gauge_def', ← Real.sInf_smul_of_nonneg (inv_nonneg.2 ha)]
congr 1
ext r
simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
constructor
· rintro ⟨hr, y, hy, h⟩
si... |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 148 | 161 | theorem normed_le_div_measure_closedBall_rOut [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
(x : E) :
f.normed μ x ≤ K ^ finrank ℝ E / (μ (closedBall c f.rOut)).toReal := by |
have K_pos : 0 < K := by
simpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)
have : f x / ∫ y, f y ∂μ ≤ 1 / ∫ y, f y ∂μ := by
gcongr
· exact f.integral_pos.le
· exact f.le_one
apply this.trans
rw [div_le_div_iff f.integral_pos, one_mul, ← div_le_iff' (pow_pos K_... |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Algebra.Ring.NegOnePow
/-!
# Miscellaneous results about determinant
In this file, we collect var... | Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean | 21 | 47 | theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : ℕ}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : ∑ j, M j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.submatrix (Fin.succAbove j₁) id).det =
Int.negOnePow (j₁ - j₂) • (M.submatrix (Fin.succAbove j₂) id).det := by |
suffices ∀ j, (M.submatrix (Fin.succAbove j) id).det =
Int.negOnePow j • (M.submatrix (Fin.succAbove 0) id).det by
rw [this j₁, this j₂, smul_smul, ← Int.negOnePow_add, sub_add_cancel]
intro j
induction j using Fin.induction with
| zero => rw [Fin.val_zero, Nat.cast_zero, Int.negOnePow_zero, one_smul... |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 311 | 320 | theorem convex_Iio (r : β) : Convex 𝕜 (Iio r) := by |
intro x hx y hy a b ha hb hab
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_add] at hab
rwa [zero_smul, zero_add, hab, one_smul]
rw [mem_Iio] at hx hy
calc
a • x + b • y < a • r + b • r := add_lt_add_of_lt_of_le
(smul_lt_smul_of_pos_left hx ha') (smul_le_smul_of_nonneg_left hy.le hb)
_ = r :... |
/-
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 | 243 | 251 | theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
calc
dist (log z.im) (log w.im) = dist (mk ⟨0, z.im⟩ z.im_pos) (mk ⟨0, w.im⟩ w.im_pos) :=
Eq.symm <| dist_of_re_eq rfl
_ ≤ dist z w := by |
simp_rw [dist_eq]
dsimp only [coe_mk, mk_im]
gcongr
simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 184 | 185 | theorem Ioo_subset_Ioo (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := by |
simpa [← coe_subset] using Set.Ioo_subset_Ioo ha hb
|
/-
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
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 898 | 899 | theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by |
simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,193 | 1,201 | theorem lintegral_coe_eq_integral (f : α → ℝ≥0) (hfi : Integrable (fun x => (f x : ℝ)) μ) :
∫⁻ a, f a ∂μ = ENNReal.ofReal (∫ a, f a ∂μ) := by |
simp_rw [integral_eq_lintegral_of_nonneg_ae (eventually_of_forall fun x => (f x).coe_nonneg)
hfi.aestronglyMeasurable, ← ENNReal.coe_nnreal_eq]
rw [ENNReal.ofReal_toReal]
rw [← lt_top_iff_ne_top]
convert hfi.hasFiniteIntegral
-- Porting note: `convert` no longer unfolds `HasFiniteIntegral`
simp_rw [H... |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
import... | Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 189 | 190 | theorem im_inv_neg_coe_pos (z : ℍ) : 0 < (-z : ℂ)⁻¹.im := by |
simpa using div_pos z.property (normSq_pos z)
|
/-
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.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/... | Mathlib/Probability/IdentDistrib.lean | 162 | 168 | theorem aestronglyMeasurable_snd [TopologicalSpace γ] [MetrizableSpace γ] [BorelSpace γ]
(h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν := by |
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩
rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩
refine ⟨closure t, t_sep.closure, ?_⟩
apply h.ae_mem_snd isClosed_closure.measurableSet
filter_upwards [ht] with x hx using subset_closure 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.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,398 | 1,400 | theorem iSup_split_single (f : β → α) (i₀ : β) : ⨆ i, f i = f i₀ ⊔ ⨆ (i) (_ : i ≠ i₀), f i := by |
convert iSup_split f (fun i => i = i₀)
simp
|
/-
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
-/
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Algebra.Order.Ring.Cano... | Mathlib/Algebra/Order/Ring/WithTop.lean | 89 | 91 | theorem mul_lt_top' [LT α] {a b : WithTop α} (ha : a < ⊤) (hb : b < ⊤) : a * b < ⊤ := by |
rw [WithTop.lt_top_iff_ne_top] at *
simp only [Ne, mul_eq_top_iff, *, and_false, false_and, or_self, not_false_eq_true]
|
/-
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.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#a... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 897 | 900 | theorem IsCycleOn.pow_apply_eq_pow_apply {s : Finset α} (hf : f.IsCycleOn s) (ha : a ∈ s)
{m n : ℕ} : (f ^ m) a = (f ^ n) a ↔ m ≡ n [MOD s.card] := by |
rw [Nat.modEq_iff_dvd, ← hf.zpow_apply_eq ha]
simp [sub_eq_neg_add, zpow_add, eq_inv_iff_eq, eq_comm]
|
/-
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 | 85 | 85 | theorem gcd_self_add_left (m n : ℕ) : gcd (m + n) m = gcd n m := by | rw [add_comm, gcd_add_self_left]
|
/-
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.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 81 | 89 | theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R}
(hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]
apply le_of_eq_of_le _ (Set.IsWF.min_le_min_of_subset (support_add_subset (x := x) (y := y)))
· simp
· simp [hy]
· exact (Set.IsWF.min_union _ _ _ _).symm
|
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 602 | 610 | theorem Collinear.collinear_insert_iff_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
(hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₃ : p₂ ≠ p₃) :
Collinear k (insert p₁ s) ↔ Collinear k ({p₁, p₂, p₃} : Set P) := by |
have hv : vectorSpan k (insert p₁ s) = vectorSpan k ({p₁, p₂, p₃} : Set P) := by
-- Porting note: Original proof used `conv_lhs` and `conv_rhs`, but these tactics timed out.
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan]
symm
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, h.... |
/-
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.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 258 | 261 | theorem StarConvex.preimage_add_left (hs : StarConvex 𝕜 (x + z) s) :
StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s) := by |
rw [add_comm] at hs
simpa only [add_comm] using hs.preimage_add_right
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 573 | 592 | theorem isNormal_map (hφ : Function.Injective φ.obj) (hφ' : im φ hφ = ⊤) (Sn : S.IsNormal) :
(map φ hφ S).IsNormal :=
{ wide := fun d => by
obtain ⟨c, rfl⟩ := obj_surjective_of_im_eq_top φ hφ hφ' d
change Map.Arrows φ hφ S _ _ (𝟙 _); rw [← Functor.map_id]
constructor; exact Sn.wide c
conj :... | rw [hφ']; apply mem_top_objs
rw [mem_im_objs_iff] at this
obtain ⟨c', rfl⟩ := this
have : g ∈ (im φ hφ).arrows (φ.obj c) (φ.obj c') := by rw [hφ']; trivial
rw [mem_im_iff] at this
obtain ⟨b, b', f, hb, hb', _, hf⟩ := this; cases hφ hb; cases hφ hb'
change Map.Arrows φ hφ S (φ.obj c'... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathli... | Mathlib/Probability/Martingale/Basic.lean | 326 | 333 | theorem Submartingale.condexp_sub_nonneg {f : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) {i j : ι}
(hij : i ≤ j) : 0 ≤ᵐ[μ] μ[f j - f i|ℱ i] := by |
by_cases h : SigmaFinite (μ.trim (ℱ.le i))
swap; · rw [condexp_of_not_sigmaFinite (ℱ.le i) h]
refine EventuallyLE.trans ?_ (condexp_sub (hf.integrable _) (hf.integrable _)).symm.le
rw [eventually_sub_nonneg,
condexp_of_stronglyMeasurable (ℱ.le _) (hf.adapted _) (hf.integrable _)]
exact hf.2.1 i j hij
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Logic.Equiv.Embedding
#align_import data.fintype.card_embedding from "leanprover-community/mathlib"@"98e83c3d... | Mathlib/Data/Fintype/CardEmbedding.lean | 36 | 50 | theorem card_embedding_eq {α β : Type*} [Fintype α] [Fintype β] [emb : Fintype (α ↪ β)] :
‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖ := by |
rw [Subsingleton.elim emb Embedding.fintype]
refine Fintype.induction_empty_option (P := fun t ↦ ‖t ↪ β‖ = ‖β‖.descFactorial ‖t‖)
(fun α₁ α₂ h₂ e ih ↦ ?_) (?_) (fun γ h ih ↦ ?_) α <;> dsimp only <;> clear! α
· letI := Fintype.ofEquiv _ e.symm
rw [← card_congr (Equiv.embeddingCongr e (Equiv.refl β)), ... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,777 | 1,784 | theorem count_of_coprime [DecidableEq (Associates α)] {a : Associates α} (ha : a ≠ 0)
{b : Associates α} (hb : b ≠ 0) (hab : ∀ d, d ∣ a → d ∣ b → ¬Prime d) {p : Associates α}
(hp : Irreducible p) : count p a.factors = 0 ∨ count p b.factors = 0 := by |
rw [or_iff_not_imp_left, ← Ne]
intro hca
contrapose! hab with hcb
exact ⟨p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb,
UniqueFactorizationMonoid.irreducible_iff_prime.mp hp⟩
|
/-
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.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 119 | 119 | theorem cylinder_zero (x : ∀ n, E n) : cylinder x 0 = univ := by | simp [cylinder_eq_pi]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 358 | 360 | theorem ball_sub_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
ball a ε - ball b δ = ball (a - b) (ε + δ) := by |
simp_rw [sub_eq_add_neg, neg_ball, ball_add_ball hε hδ]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 620 | 644 | theorem tfae_modEq :
TFAE
[a ≡ b [PMOD p], ∀ z : ℤ, b - z • p ∉ Set.Ioo a (a + p), toIcoMod hp a b ≠ toIocMod hp a b,
toIcoMod hp a b + p = toIocMod hp a b] := by |
rw [modEq_iff_toIcoMod_eq_left hp]
tfae_have 3 → 2
· rw [← not_exists, not_imp_not]
exact fun ⟨i, hi⟩ =>
((toIcoMod_eq_iff hp).2 ⟨Set.Ioo_subset_Ico_self hi, i, (sub_add_cancel b _).symm⟩).trans
((toIocMod_eq_iff hp).2 ⟨Set.Ioo_subset_Ioc_self hi, i, (sub_add_cancel b _).symm⟩).symm
tfae_have... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,764 | 1,782 | theorem biprod.map_eq_map' {W X Y Z : C} [HasBinaryBiproduct W X] [HasBinaryBiproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : biprod.map f g = biprod.map' f g := by |
ext
· simp only [mapPair_left, IsColimit.ι_map, IsLimit.map_π, biprod.inl_fst_assoc,
Category.assoc, ← BinaryBicone.toCone_π_app_left, ← BinaryBiproduct.bicone_fst, ←
BinaryBicone.toCocone_ι_app_left, ← BinaryBiproduct.bicone_inl];
dsimp; simp
· simp only [mapPair_left, IsColimit.ι_map, IsLimit.m... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 305 | 306 | theorem inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ := by |
simp only [inner_sub_left, inner_sub_right]; ring
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce608... | Mathlib/MeasureTheory/Group/AddCircle.lean | 34 | 48 | theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =ᵐ[volume] ball x ε := by |
rcases le_or_lt ε 0 with hε | hε
· rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closedBall,
min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ENNReal.ofReal_eq_zero]
exact mul_nonpos_of_nonneg_of_nonpos zero_le_two hε
· suffices volume (closedBall x ε) ≤ volume (ball x ε) by
exact (ae_eq_of_subse... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 838 | 860 | theorem skew_product [SFinite μa] [SFinite μc] {f : α → β} (hf : MeasurePreserving f μa μb)
{g : α → γ → δ} (hgm : Measurable (uncurry g)) (hg : ∀ᵐ x ∂μa, map (g x) μc = μd) :
MeasurePreserving (fun p : α × γ => (f p.1, g p.1 p.2)) (μa.prod μc) (μb.prod μd) := by |
classical
have : Measurable fun p : α × γ => (f p.1, g p.1 p.2) := (hf.1.comp measurable_fst).prod_mk hgm
/- if `μa = 0`, then the lemma is trivial, otherwise we can use `hg`
to deduce `SFinite μd`. -/
rcases eq_or_ne μa 0 with (rfl | ha)
· rw [← hf.map_eq, zero_prod, Measure.map_zero, zero_prod]
exa... |
/-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
impo... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 146 | 159 | theorem cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt [Nontrivial R]
(hv : IsTranscendenceBasis R v) (hR : #R ≤ ℵ₀) (hK : ℵ₀ < #K) : #K = #ι :=
have : ℵ₀ ≤ #ι := le_of_not_lt fun h => not_le_of_gt hK <|
calc
#K ≤ max (max #R #ι) ℵ₀ := cardinal_le_max_transcendence_basis v hv
_ ≤ _ := max_l... |
rw [max_eq_left, max_eq_right]
· exact le_trans hR this
· exact le_max_of_le_right this)
(mk_le_of_injective (show Function.Injective v from hv.1.injective))
|
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 181 | 184 | theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=
calc
coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _
_ = if m = n then a else 0 := by | simp only [Finsupp.unique_single_eq_iff]
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 385 | 388 | theorem subset_generated (c d : C) : X c d ⊆ (generated X).arrows c d := by |
dsimp only [generated, sInf]
simp only [subset_iInter₂_iff]
exact fun S hS f fS => hS _ _ fS
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
/-!
# Lemmas for Red-black trees
The main theorem in this file is `WF_def`, which shows that the ... | .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 51 | 52 | theorem reverse_eq_iff {t t' : RBNode α} : t.reverse = t' ↔ t = t'.reverse := by |
constructor <;> rintro rfl <;> simp
|
/-
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.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 790 | 792 | theorem continuous_iInf_rng {t₁ : TopologicalSpace α} {t₂ : ι → TopologicalSpace β} :
Continuous[t₁, iInf t₂] f ↔ ∀ i, Continuous[t₁, t₂ i] f := by |
simp only [continuous_iff_coinduced_le, le_iInf_iff]
|
/-
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.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
imp... | Mathlib/RingTheory/LaurentSeries.lean | 369 | 373 | theorem coe_C (r : F) : ((RatFunc.C r : RatFunc F) : LaurentSeries F) = HahnSeries.C r := by |
rw [coe_num_denom, num_C, denom_C, Polynomial.coe_C, -- Porting note: removed `coe_C`
Polynomial.coe_one,
PowerSeries.coe_one, div_one]
simp only [algebraMap_eq_C, ofPowerSeries_C, C_apply] -- Porting note: added
|
/-
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.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 443 | 444 | theorem coeff_zero_X_mul (φ : MvPowerSeries σ R) (s : σ) : coeff R (0 : σ →₀ ℕ) (X s * φ) = 0 := by |
rw [← (φ.commute_X s).eq, coeff_zero_mul_X]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 553 | 575 | theorem _root_.NumberField.is_primitive_element_of_infinitePlace_lt {x : 𝓞 K}
{w : InfinitePlace K} (h₁ : x ≠ 0) (h₂ : ∀ ⦃w'⦄, w' ≠ w → w' x < 1)
(h₃ : IsReal w ∨ |(w.embedding x).re| < 1) : ℚ⟮(x : K)⟯ = ⊤ := by |
rw [Field.primitive_element_iff_algHom_eq_of_eval ℚ ℂ ?_ _ w.embedding.toRatAlgHom]
· intro ψ hψ
have h : 1 ≤ w x := one_le_of_lt_one h₁ h₂
have main : w = InfinitePlace.mk ψ.toRingHom := by
erw [← norm_embedding_eq, hψ] at h
contrapose! h
exact h₂ h.symm
rw [(mk_embedding w).symm, mk... |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 1,022 | 1,026 | theorem mk'_eq_mk'_iff (m₁ m₂ : M) (s₁ s₂ : S) :
mk' f m₁ s₁ = mk' f m₂ s₂ ↔ ∃ s : S, s • s₁ • m₂ = s • s₂ • m₁ := by |
delta mk'
rw [(fromLocalizedModule.inj S f).eq_iff, LocalizedModule.mk_eq]
simp_rw [eq_comm]
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,266 | 1,267 | theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by |
simp [deleteVerts]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 87 | 88 | theorem gold_pow_sub_gold_pow (n : ℕ) : φ ^ (n + 2) - φ ^ (n + 1) = φ ^ n := by |
rw [goldenRatio]; ring_nf; norm_num; ring
|
/-
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, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 251 | 252 | theorem finrank_matrix (m n : Type*) [Fintype m] [Fintype n] :
finrank R (Matrix m n R) = card m * card n := by | simp [finrank]
|
/-
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.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 95 | 100 | theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
(Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom := by |
change (N₁Γ₀.app K).inv = _
simp only [N₁Γ₀_app]
rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Subse... | Mathlib/Algebra/Group/Submonoid/Basic.lean | 575 | 577 | theorem iSup_eq_closure {ι : Sort*} (p : ι → Submonoid M) :
⨆ i, p i = Submonoid.closure (⋃ i, (p i : Set M)) := by |
simp_rw [Submonoid.closure_iUnion, Submonoid.closure_eq]
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 818 | 819 | theorem le_toNNReal_iff_coe_le {r : ℝ≥0} {p : ℝ} (hp : 0 ≤ p) : r ≤ Real.toNNReal p ↔ ↑r ≤ p := by |
rw [← NNReal.coe_le_coe, Real.coe_toNNReal p hp]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Init.Align
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.Algebra.Category.ModuleCat.EpiMono
#align... | Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | 359 | 387 | theorem pseudo_exact_of_exact {P Q R : C} {f : P ⟶ Q} {g : Q ⟶ R} (h : Exact f g) :
(∀ a, g (f a) = 0) ∧ ∀ b, g b = 0 → ∃ a, f a = b :=
⟨fun a => by
rw [← comp_apply, h.w]
exact zero_apply _ _, fun b' =>
Quotient.inductionOn b' fun b hb => by
have hb' : b.hom ≫ g = 0 := (pseudoZero_iff _).1 hb
... |
rw [Abelian.image.fac]
_ = (pullback.snd ≫ c) ≫ kernel.ι (cokernel.π f) := by
rw [← Category.assoc, pullback.condition]
_ = pullback.snd ≫ b.hom := by
rw [Category.assoc]
congr
⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,075 | 1,077 | theorem tendsto_sin_pi_div_two : Tendsto sin (𝓝[<] (π / 2)) (𝓝 1) := by |
convert continuous_sin.continuousWithinAt.tendsto
simp
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,162 | 1,169 | theorem continuousOn_writtenInExtend_iff {f' : PartialHomeomorph M' H'} {g : M → M'}
(hs : s ⊆ f.source) (hmaps : MapsTo g s f'.source) :
ContinuousOn (f'.extend I' ∘ g ∘ (f.extend I).symm) (f.extend I '' s) ↔ ContinuousOn g s := by |
refine forall_mem_image.trans <| forall₂_congr fun x hx ↦ ?_
refine (continuousWithinAt_congr_nhds ?_).trans
(continuousWithinAt_writtenInExtend_iff _ _ _ (hs hx) (hmaps hx) hmaps)
rw [← map_extend_nhdsWithin_eq_image_of_subset, ← map_extend_nhdsWithin]
exacts [hs hx, hs hx, hs]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,587 | 3,593 | theorem subgroupOf_sup (A A' B : Subgroup G) (hA : A ≤ B) (hA' : A' ≤ B) :
(A ⊔ A').subgroupOf B = A.subgroupOf B ⊔ A'.subgroupOf B := by |
refine
map_injective_of_ker_le B.subtype (ker_le_comap _ _)
(le_trans (ker_le_comap B.subtype _) le_sup_left) ?_
simp only [subgroupOf, map_comap_eq, map_sup, subtype_range]
rw [inf_of_le_right (sup_le hA hA'), inf_of_le_right hA', inf_of_le_right hA]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 390 | 391 | theorem casesOn'_none_coe (f : Option α → β) (o : Option α) :
casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by | cases o <;> rfl
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,087 | 1,088 | theorem iUnion_Icc_zsmul : ⋃ n : ℤ, Icc (n • p) ((n + 1) • p) = univ := by |
simpa only [zero_add] using iUnion_Icc_add_zsmul hp 0
|
/-
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.Data.Nat.ModEq
import Mathlib.Tactic.Abel
import Mathlib.Tactic.GCongr.Core
#align_import data.int.modeq from "leanprover-community/mathlib"@"47a1a73351de... | Mathlib/Data/Int/ModEq.lean | 293 | 300 | theorem mod_coprime {a b : ℕ} (hab : Nat.Coprime a b) : ∃ y : ℤ, a * y ≡ 1 [ZMOD b] :=
⟨Nat.gcdA a b,
have hgcd : Nat.gcd a b = 1 := Nat.Coprime.gcd_eq_one hab
calc
↑a * Nat.gcdA a b ≡ ↑a * Nat.gcdA a b + ↑b * Nat.gcdB a b [ZMOD ↑b] :=
ModEq.symm <| modEq_add_fac _ <| ModEq.refl _
_ ≡ 1 [Z... | rw [← Nat.gcd_eq_gcd_ab, hgcd]; rfl
⟩
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,343 | 1,350 | theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
HasFTaylorSeriesUpTo (n + 1 : ℕ) f p ↔
(∀ x, (p x 0).uncurry0 = f x) ∧
(∀ x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
HasFTaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
(p x).sh... |
simp only [hasFTaylorSeriesUpToOn_succ_iff_right, ← hasFTaylorSeriesUpToOn_univ_iff, mem_univ,
forall_true_left, hasFDerivWithinAt_univ]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 91 | 95 | theorem bernoulli'_spec' (n : ℕ) :
(∑ k ∈ antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by |
refine ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans ?_).trans (bernoulli'_spec n)
refine sum_congr rfl fun x hx => ?_
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mat... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 217 | 220 | theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
(H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p := by |
rw [trailingDegree_eq_natTrailingDegree hp] at H
exact WithTop.coe_le_coe.mp H
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
/-!
# zip & u... | Mathlib/Data/List/Zip.lean | 143 | 145 | theorem zip_of_prod {l : List α} {l' : List β} {lp : List (α × β)} (hl : lp.map Prod.fst = l)
(hr : lp.map Prod.snd = l') : lp = l.zip l' := by |
rw [← hl, ← hr, ← zip_unzip lp, ← unzip_left, ← unzip_right, zip_unzip, zip_unzip]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 129 | 130 | theorem LTCmp.cmp_iff_gt [LT α] [LTCmp (α := α) cmp] : cmp x y = .gt ↔ y < x := by |
rw [OrientedCmp.cmp_eq_gt, LTCmp.cmp_iff_lt]
|
/-
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 | 74 | 78 | theorem memberSubfamily_inter (a : α) (𝒜 ℬ : Finset (Finset α)) :
(𝒜 ∩ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∩ ℬ.memberSubfamily a := by |
unfold memberSubfamily
rw [filter_inter_distrib, image_inter_of_injOn _ _ ((erase_injOn' _).mono _)]
simp
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 92 | 93 | theorem mul_star_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (x * star x) := by |
simpa only [star_star] using star_mul_self (star x)
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 677 | 681 | theorem realize_relabel_sum_inr (φ : L.Formula (Fin n)) {v : Empty → M} {x : Fin n → M} :
(BoundedFormula.relabel Sum.inr φ).Realize v x ↔ φ.Realize x := by |
rw [BoundedFormula.realize_relabel, Formula.Realize, Sum.elim_comp_inr, Fin.castAdd_zero,
cast_refl, Function.comp_id,
Subsingleton.elim (x ∘ (natAdd n : Fin 0 → Fin n)) default]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import M... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 864 | 883 | theorem exists_disjoint_closedBall_covering_ae (μ : Measure α) [SigmaFinite μ] (f : α → Set ℝ)
(s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) (R : α → ℝ)
(hR : ∀ x ∈ s, 0 < R x) :
∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧
(∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧ μ (s \ ⋃ x ∈ t, close... |
let g x := f x ∩ Ioo 0 (R x)
have hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := fun x hx δ δpos ↦ by
rcases hf x hx (min δ (R x)) (lt_min δpos (hR x hx)) with ⟨r, hr⟩
exact ⟨r, ⟨⟨hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)⟩,
⟨hr.2.1, hr.2.2.trans_le (min_le_left _ _)⟩⟩⟩
rcases exists_disj... |
/-
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.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 146 | 165 | theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
(hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
(lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
condexpIndL1Fin hm hs h... |
ext1
have hμst :=
((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_add _ _).symm
have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 355 | 356 | theorem _root_.Wbtw.angle₃₁₂_eq_zero_of_ne {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) (hp₂p₁ : p₂ ≠ p₁) :
∠ p₃ p₁ p₂ = 0 := by | rw [← h.angle₂₁₃_eq_zero_of_ne hp₂p₁, angle_comm]
|
/-
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.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.Unifo... | Mathlib/Topology/Instances/Real.lean | 77 | 78 | theorem Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop := by |
simp only [← comap_dist_right_atTop (0 : ℝ), Real.dist_eq, sub_zero, comap_abs_atTop]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 742 | 743 | theorem Iic_eq_cons_Iio (b : α) : Iic b = (Iio b).cons b not_mem_Iio_self := by |
classical rw [cons_eq_insert, Iio_insert]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,621 | 2,622 | theorem filter_congr_decidable (s : Finset α) (p : α → Prop) (h : DecidablePred p)
[DecidablePred p] : @filter α p h s = s.filter p := by | congr
|
/-
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.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 236 | 238 | theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by |
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
|
/-
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 | 87 | 89 | theorem add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by |
rw [SModEq.def] at hxy₁ hxy₂ ⊢
simp_rw [Quotient.mk_add, hxy₁, hxy₂]
|
/-
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.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 936 | 940 | theorem num_div_dvd (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
num (algebraMap _ _ p / algebraMap _ _ q) ∣ p := by |
rw [num_div _ q, C_mul_dvd]
· exact EuclideanDomain.div_dvd_of_dvd (gcd_dvd_left p q)
· simpa only [Ne, inv_eq_zero, Polynomial.leadingCoeff_eq_zero] using right_div_gcd_ne_zero hq
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 206 | 209 | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by |
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 137 | 138 | theorem neg {x : R} (hx : IsSelfAdjoint x) : IsSelfAdjoint (-x) := by |
simp only [isSelfAdjoint_iff, star_neg, hx.star_eq]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a2... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 186 | 188 | theorem ContMDiffOn.div {f g : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s)
(hg : ContMDiffOn I' I n g s) : ContMDiffOn I' I n (fun x => f x / g x) s := by |
simp_rw [div_eq_mul_inv]; exact hf.mul hg.inv
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Heather Macbeth
-/
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Math... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 124 | 134 | theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by |
rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.sub_mem
(A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)
(A.le_topologicalClosure g.property))
?_)
_
exact mod_cast abs_mem_subalgebra_closure A _
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 903 | 910 | theorem quotientEquivQuotientMinpolyMap_apply_mk (pb : PowerBasis R S) (I : Ideal R) (g : R[X]) :
pb.quotientEquivQuotientMinpolyMap I (Ideal.Quotient.mk (I.map (algebraMap R S))
(aeval pb.gen g)) = Ideal.Quotient.mk
(Ideal.span ({(minpoly R pb.gen).map (Ideal.Quotient.mk I)} : Set (Polynomial (R ⧸ I)... |
rw [PowerBasis.quotientEquivQuotientMinpolyMap, AlgEquiv.trans_apply, AlgEquiv.ofRingEquiv_apply,
quotientEquiv_mk, AlgEquiv.coe_ringEquiv', AdjoinRoot.equiv'_symm_apply, PowerBasis.lift_aeval,
AdjoinRoot.aeval_eq, AdjoinRoot.quotEquivQuotMap_apply_mk]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 469 | 470 | theorem inf_sdiff_distrib_right (a b c : α) : a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c) := by |
simp_rw [inf_comm _ c, inf_sdiff_distrib_left]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 588 | 592 | theorem cons_eq_smul {i} {m : M i} {ls h1 h2} :
cons m ls h1 h2 = of m • ls := by |
rw [of_smul_def, equivPair_eq_of_fstIdx_ne _]
· simp [cons, rcons, h2]
· exact h1
|
/-
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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 812 | 812 | theorem le_nhds_iff {f} : f ≤ 𝓝 x ↔ ∀ s : Set X, x ∈ s → IsOpen s → s ∈ f := by | simp [nhds_def]
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 1,096 | 1,100 | theorem finprod_mem_biUnion {I : Set ι} {t : ι → Set α} (h : I.PairwiseDisjoint t) (hI : I.Finite)
(ht : ∀ i ∈ I, (t i).Finite) : ∏ᶠ a ∈ ⋃ x ∈ I, t x, f a = ∏ᶠ i ∈ I, ∏ᶠ j ∈ t i, f j := by |
haveI := hI.fintype
rw [biUnion_eq_iUnion, finprod_mem_iUnion, ← finprod_set_coe_eq_finprod_mem]
exacts [fun x y hxy => h x.2 y.2 (Subtype.coe_injective.ne hxy), fun b => ht b b.2]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 622 | 631 | theorem inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) :
⟪x, y⟫ = ‖x‖ * ‖y‖ * Real.Angle.cos (o.oangle x y) := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
have : ‖x‖ ≠ 0 := by simpa using hx
have : ‖y‖ ≠ 0 := by simpa using hy
rw [oangle, Real.Angle.cos_coe, Complex.cos_arg, o.abs_kahler]
· simp only [kahler_apply_apply, real_smul, add_re, ofReal_re, mul_re, I_re, ofReal_im]
field_simp
·... |
/-
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.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 56 | 59 | theorem restrict_toOuterMeasure_eq_toOuterMeasure_restrict (h : MeasurableSet s) :
(μ.restrict s).toOuterMeasure = OuterMeasure.restrict s μ.toOuterMeasure := by |
simp_rw [restrict, restrictₗ, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,
toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, μ.trimmed]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,227 | 1,230 | theorem finrank_dualCoannihilator_eq {Φ : Subspace K (Module.Dual K V)} :
finrank K Φ.dualCoannihilator = finrank K Φ.dualAnnihilator := by |
rw [Submodule.dualCoannihilator, ← Module.evalEquiv_toLinearMap]
exact LinearEquiv.finrank_eq (LinearEquiv.ofSubmodule' _ _)
|
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Scott Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanpr... | Mathlib/Data/Matrix/Basis.lean | 241 | 244 | theorem diag_eq_of_commute_stdBasisMatrix {i j : n} {M : Matrix n n α}
(hM : Commute (stdBasisMatrix i j 1) M) : M i i = M j j := by |
have := ext_iff.mpr hM i j
aesop
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 2,000 | 2,004 | theorem isLeftRegular_of_le_isCancelMulZero (f : LocalizationWithZeroMap S N)
[IsCancelMulZero M] (h : ∀ ⦃x⦄, x ∈ S → IsRegular x) : IsCancelMulZero N := by |
have : IsLeftCancelMulZero N :=
leftCancelMulZero_of_le_isLeftRegular f (fun x h' => (h h').left)
exact IsLeftCancelMulZero.to_isCancelMulZero
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathl... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 313 | 324 | theorem sum_lt {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α} :
s.Nonempty → (∀ i ∈ s, padicNorm p (F i) < t) → padicNorm p (∑ i ∈ s, F i) < t := by |
classical
refine s.induction_on (by rintro ⟨-, ⟨⟩⟩) ?_
rintro a S haS IH - ht
by_cases hs : S.Nonempty
· rw [Finset.sum_insert haS]
exact
lt_of_le_of_lt padicNorm.nonarchimedean
(max_lt (ht a (Finset.mem_insert_self a S))
(IH hs fun b hb ↦ ht b (Finset.mem_insert_o... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
/-!
# Functions invariant under (quasi)ergodic map
In this file we prove tha... | Mathlib/Dynamics/Ergodic/Function.lean | 27 | 35 | theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
{s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
(h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
(hg_eq : g ∘ f =ᵐ[μ] g) :
∃ c, g =ᵐ[μ] const α c := by |
refine exists_eventuallyEq_const_of_eventually_mem_of_forall_separating MeasurableSet hs ?_
refine fun U hU ↦ h.ae_mem_or_ae_nmem₀ (s := g ⁻¹' U) (hgm hU) ?_b
refine (hg_eq.mono fun x hx ↦ ?_).set_eq
rw [← preimage_comp, mem_preimage, mem_preimage, hx]
|
/-
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 | 252 | 253 | theorem lift_of_apply (f : X → L) (x) : lift R f (of R x) = f x := by |
rw [← @Function.comp_apply _ _ _ (lift R f) (of R) x, of_comp_lift]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 503 | 506 | theorem Wbtw.trans_left_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w y z)
(h₂ : Wbtw R w x y) (h : y ≠ z) : x ≠ z := by |
rintro rfl
exact h (h₁.swap_right_iff.1 h₂)
|
/-
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.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 295 | 296 | theorem inv_hom_id_assoc (f : X ⟶ Y) [I : IsIso f] {Z} (g : Y ⟶ Z) : inv f ≫ f ≫ g = g := by |
simp [← Category.assoc]
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 658 | 660 | theorem finprod_mem_of_eqOn_one (hf : s.EqOn f 1) : ∏ᶠ i ∈ s, f i = 1 := by |
rw [← finprod_mem_one s]
exact finprod_mem_congr rfl hf
|
/-
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 | 148 | 163 | theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
{ x | gauge s x ≤ a } = ⋂ (r : ℝ) (_ : a < r), r • s := by |
ext x
simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
refine ⟨fun h r hr => ?_, fun h => le_of_forall_pos_lt_add fun ε hε => ?_⟩
· have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Module.Hom
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.NonUnitalS... | Mathlib/Algebra/Ring/CentroidHom.lean | 436 | 438 | theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by |
simp only [Function.comp_apply]
rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 682 | 685 | theorem prod_apply_symm {s : Set (α × β)} (hs : MeasurableSet s) :
μ.prod ν s = ∫⁻ y, μ ((fun x => (x, y)) ⁻¹' s) ∂ν := by |
rw [← prod_swap, map_apply measurable_swap hs, prod_apply (measurable_swap hs)]
rfl
|
/-
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 | 308 | 311 | theorem sin_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖ := by |
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg, sin_angle_add_mul_norm_of_inner_eq_zero h, norm_neg]
|
/-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Projectivization/Subspace.lean | 210 | 216 | theorem span_eq_sInf {S : Set (ℙ K V)} : span S = sInf { W : Subspace K V| S ⊆ W } := by |
ext x
simp_rw [mem_carrier_iff, mem_span x]
refine ⟨fun hx => ?_, fun hx W hW => ?_⟩
· rintro W ⟨T, hT, rfl⟩
exact hx T hT
· exact (@sInf_le _ _ { W : Subspace K V | S ⊆ ↑W } W hW) hx
|
/-
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.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417... | Mathlib/Algebra/Polynomial/Lifts.lean | 257 | 258 | theorem lifts_iff_liftsRing (p : S[X]) : p ∈ lifts f ↔ p ∈ liftsRing f := by |
simp only [lifts, liftsRing, RingHom.mem_range, RingHom.mem_rangeS]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 474 | 476 | theorem addHaar_ball [Nontrivial E] (x : E) {r : ℝ} (hr : 0 ≤ r) :
μ (ball x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) := by |
rw [← addHaar_ball_mul μ x hr, mul_one]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Support
import Mathlib.Topology.Order... | Mathlib/Topology/Algebra/Order/Compact.lean | 391 | 396 | theorem IsCompact.bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
BddBelow s := by |
rcases s.eq_empty_or_nonempty with rfl | hne
· exact bddBelow_empty
· obtain ⟨a, -, has⟩ := hs.exists_isLeast hne
exact ⟨a, has⟩
|
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