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 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 54 | 54 | theorem choose_zero_right (n : ℕ) : choose n 0 = 1 := by | cases n <;> rfl
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 679 | 713 | theorem Nonneg.add {a b : ℤ√d} (ha : Nonneg a) (hb : Nonneg b) : Nonneg (a + b) := by |
rcases nonneg_cases ha with ⟨x, y, rfl | rfl | rfl⟩ <;>
rcases nonneg_cases hb with ⟨z, w, rfl | rfl | rfl⟩
· trivial
· refine nonnegg_cases_right fun i h => sqLe_of_le ?_ ?_ (nonnegg_pos_neg.1 hb)
· dsimp only at h
exact Int.ofNat_le.1 (le_of_neg_le_neg (Int.le.intro y (by simp [add_comm, *])))
... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 401 | 404 | theorem finRotate_last' : finRotate (n + 1) ⟨n, by omega⟩ = ⟨0, Nat.zero_lt_succ _⟩ := by |
dsimp [finRotate_succ]
rw [finAddFlip_apply_mk_right le_rfl]
simp
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.Topology.Algebra.Module.Cardinality
/-!
# Connectedn... | Mathlib/Analysis/NormedSpace/Connected.lean | 164 | 168 | theorem isPreconnected_sphere (h : 1 < Module.rank ℝ E) (x : E) (r : ℝ) :
IsPreconnected (sphere x r) := by |
rcases le_or_lt 0 r with hr|hr
· exact (isConnected_sphere h x hr).isPreconnected
· simpa [hr] using isPreconnected_empty
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Generator.lean | 257 | 265 | theorem isSeparating_iff_epi (𝒢 : Set C)
[∀ A : C, HasCoproduct fun f : ΣG : 𝒢, (G : C) ⟶ A => (f.1 : C)] :
IsSeparating 𝒢 ↔ ∀ A : C, Epi (Sigma.desc (@Sigma.snd 𝒢 fun G => (G : C) ⟶ A)) := by |
refine ⟨fun h A => ⟨fun u v huv => h _ _ fun G hG f => ?_⟩, fun h X Y f g hh => ?_⟩
· simpa using Sigma.ι (fun f : ΣG : 𝒢, (G : C) ⟶ A => (f.1 : C)) ⟨⟨G, hG⟩, f⟩ ≫= huv
· haveI := h X
refine
(cancel_epi (Sigma.desc (@Sigma.snd 𝒢 fun G => (G : C) ⟶ X))).1 (colimit.hom_ext fun j => ?_)
simpa using ... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 803 | 812 | theorem eq_σ_of_epi {n : ℕ} (θ : mk (n + 1) ⟶ mk n) [Epi θ] : ∃ i : Fin (n + 1), θ = σ i := by |
rcases eq_σ_comp_of_not_injective θ (by
by_contra h
simpa using le_of_mono (mono_iff_injective.mpr h)) with ⟨i, θ', h⟩
use i
haveI : Epi (σ i ≫ θ') := by
rw [← h]
infer_instance
haveI := CategoryTheory.epi_of_epi (σ i) θ'
rw [h, eq_id_of_epi θ', Category.comp_id]
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,079 | 1,080 | theorem mk_embedding_eq_zero_iff_lift_lt : #(α ↪ β') = 0 ↔ lift.{u} #β' < lift.{v} #α := by |
rw [mk_eq_zero_iff, ← not_nonempty_iff, ← lift_mk_le', not_le]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 586 | 589 | theorem card_support_ne_one (f : Perm α) : f.support.card ≠ 1 := by |
by_cases h : f = 1
· exact ne_of_eq_of_ne (card_support_eq_zero.mpr h) zero_ne_one
· exact ne_of_gt (one_lt_card_support_of_ne_one 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.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 | 438 | 465 | theorem isCycle_swap_mul_aux₂ {α : Type*} [DecidableEq α] :
∀ (n : ℤ) {b x : α} {f : Perm α} (_ : (swap x (f x) * f) b ≠ b) (_ : (f ^ n) (f x) = b),
∃ i : ℤ, ((swap x (f x) * f) ^ i) (f x) = b := by |
intro n
induction' n with n n
· exact isCycle_swap_mul_aux₁ n
· intro b x f hb h
exact if hfbx' : f x = b then ⟨0, hfbx'⟩
else
have : f b ≠ b ∧ b ≠ x := ne_and_ne_of_swap_mul_apply_ne_self hb
have hb : (swap x (f⁻¹ x) * f⁻¹) (f⁻¹ b) ≠ f⁻¹ b := by
rw [mul_apply, swap_apply_de... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 56 | 57 | theorem nhds_zero : 𝓝 (0 : Γ₀) = ⨅ γ ≠ 0, 𝓟 (Iio γ) := by |
rw [nhds_eq_update, update_same]
|
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 752 | 762 | theorem sign_eq_of_affineCombination_mem_affineSpan_pair {p : ι → P} (h : AffineIndependent k p)
{w w₁ w₂ : ι → k} {s : Finset ι} (hw : ∑ i ∈ s, w i = 1) (hw₁ : ∑ i ∈ s, w₁ i = 1)
(hw₂ : ∑ i ∈ s, w₂ i = 1)
(hs :
s.affineCombination k p w ∈ line[k, s.affineCombination k p w₁, s.affineCombination k p w₂... |
rw [affineCombination_mem_affineSpan_pair h hw hw₁ hw₂] at hs
rcases hs with ⟨r, hr⟩
rw [hr i hi, hr j hj, hi0, hj0, add_zero, add_zero, sub_zero, sub_zero, sign_mul, sign_mul, hij]
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | Mathlib/Analysis/NormedSpace/Dual.lean | 181 | 185 | theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) := by |
dsimp only [NormedSpace.polar]
simp only [LinearMap.polar_eq_iInter, LinearMap.flip_apply]
refine isClosed_biInter fun z _ => ?_
exact isClosed_Iic.preimage (ContinuousLinearMap.apply 𝕜 𝕜 z).continuous.norm
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
#align_i... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 290 | 294 | theorem rightUnitor_naturality {X₁ X₂ : C} (f : X₁ ⟶ X₂) :
tensorHom ℬ f (𝟙 𝒯.cone.pt) ≫ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X₂ 𝒯.cone.pt).isLimit).hom =
(BinaryFan.rightUnitor 𝒯.isLimit (ℬ X₁ 𝒯.cone.pt).isLimit).hom ≫ f := by |
dsimp [tensorHom]
simp
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.pfunctor.multivariate.W from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e3... | Mathlib/Data/PFunctor/Multivariate/W.lean | 305 | 306 | theorem wDest'_wMk {α : TypeVec n} (a : P.A) (f' : P.drop.B a ⟹ α) (f : P.last.B a → P.W α) :
P.wDest' (P.wMk a f' f) = ⟨a, splitFun f' f⟩ := by | rw [wDest', wRec_eq]
|
/-
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.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
/-!
# ... | Mathlib/SetTheory/Cardinal/ToNat.lean | 189 | 191 | theorem toNat_lift_add_lift {a : Cardinal.{u}} {b : Cardinal.{v}} (ha : a < ℵ₀) (hb : b < ℵ₀) :
toNat (lift.{v} a + lift.{u} b) = toNat a + toNat b := by |
simp [*]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 979 | 981 | theorem smul_leftMulMatrix_algebraMap_ne (x : S) (i j) {k k'} (h : k ≠ k') :
leftMulMatrix (b.smul c) (algebraMap _ _ x) (i, k) (j, k') = 0 := by |
rw [smul_leftMulMatrix_algebraMap, blockDiagonal_apply_ne _ _ _ h]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 187 | 189 | theorem length_le : c.length ≤ n := by |
conv_rhs => rw [← c.blocks_sum]
exact length_le_sum_of_one_le _ fun i hi => c.one_le_blocks hi
|
/-
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.Analysis.SpecialFunctions.Bernstein
import Mathlib.Topology.Algebra.Algebra
#align_import topology.continuous_function.weierstrass from "leanprover-co... | Mathlib/Topology/ContinuousFunction/Weierstrass.lean | 114 | 122 | theorem exists_polynomial_near_of_continuousOn (a b : ℝ) (f : ℝ → ℝ)
(c : ContinuousOn f (Set.Icc a b)) (ε : ℝ) (pos : 0 < ε) :
∃ p : ℝ[X], ∀ x ∈ Set.Icc a b, |p.eval x - f x| < ε := by |
let f' : C(Set.Icc a b, ℝ) := ⟨fun x => f x, continuousOn_iff_continuous_restrict.mp c⟩
obtain ⟨p, b⟩ := exists_polynomial_near_continuousMap a b f' ε pos
use p
rw [norm_lt_iff _ pos] at b
intro x m
exact b ⟨x, m⟩
|
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 178 | 196 | theorem contMDiffWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x₀ : M} :
ContMDiffWithinAt IM (IB.prod 𝓘(𝕜, F)) n f s x₀ ↔
ContMDiffWithinAt IM IB n (fun x => (f x).proj) s x₀ ∧
ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun x ↦ (trivializationAt F E (f x₀).proj (f x)).2) s x₀ := by |
simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]
rw [and_and_and_comm, ← FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]
intro hf
simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp,
PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEqu... |
/-
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.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 878 | 880 | theorem foldr_hom (f : β₁ → β₂) (g₁ : α → β₁ → β₁) (g₂ : α → β₂ → β₂) (l : List α) (init : β₁)
(H : ∀ x y, g₂ x (f y) = f (g₁ x y)) : l.foldr g₂ (f init) = f (l.foldr g₁ init) := by |
induction l <;> simp [*, H]
|
/-
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 | 218 | 225 | theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ) :
Convex ℝ { x | gauge s x ≤ a } := by |
by_cases ha : 0 ≤ a
· rw [gauge_le_eq hs h₀ absorbs ha]
exact convex_iInter fun i => convex_iInter fun _ => hs.smul _
· -- Porting note: `convert` needed help
convert convex_empty (𝕜 := ℝ) (E := E)
exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx
|
/-
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.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compac... | Mathlib/Topology/PartitionOfUnity.lean | 387 | 406 | theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X → ℝ) → Prop)
(h01 : ∀ s t, IsClosed s → IsClosed t → Disjoint s t →
∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
(hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U)
(... |
rcases exists_subset_iUnion_closure_subset hs ho (fun x _ => hf.point_finite x) hU with
⟨V, hsV, hVo, hVU⟩
have hVU' : ∀ i, V i ⊆ U i := fun i => Subset.trans subset_closure (hVU i)
rcases exists_subset_iUnion_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x)
hsV with
⟨W, hsW, hWo,... |
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
/-!
# `L^p` distance on products of two metric space... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 390 | 410 | theorem prod_lipschitzWith_equiv_aux [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
LipschitzWith 1 (WithLp.equiv p (α × β)) := by |
intro x y
rcases p.dichotomy with (rfl | h)
· simp [edist]
· have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel₀ 1 (zero_lt_one.trans_le h).ne'
rw [prod_edist_eq_add (zero_lt_one.trans_le h)]
simp only [edist, forall_prop_of_true, one_mul, ENNReal.coe_one, ge_iff_le, sup_le_iff]
construc... |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,880 | 1,924 | theorem PartialHomeomorph.contDiffAt_symm [CompleteSpace E] (f : PartialHomeomorph E F)
{f₀' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target)
(hf₀' : HasFDerivAt f (f₀' : E →L[𝕜] F) (f.symm a)) (hf : ContDiffAt 𝕜 n f (f.symm a)) :
ContDiffAt 𝕜 n f.symm a := by |
-- We prove this by induction on `n`
induction' n using ENat.nat_induction with n IH Itop
· rw [contDiffAt_zero]
exact ⟨f.target, IsOpen.mem_nhds f.open_target ha, f.continuousOn_invFun⟩
· obtain ⟨f', ⟨u, hu, hff'⟩, hf'⟩ := contDiffAt_succ_iff_hasFDerivAt.mp hf
rw [contDiffAt_succ_iff_hasFDerivAt]
... |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 317 | 322 | theorem compl_mem_measurableCylinders (hs : s ∈ measurableCylinders α) :
sᶜ ∈ measurableCylinders α := by |
rw [mem_measurableCylinders] at hs ⊢
obtain ⟨s, S, hS, rfl⟩ := hs
refine ⟨s, Sᶜ, hS.compl, ?_⟩
rw [compl_cylinder]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Gr... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 595 | 597 | theorem mem_fundamentalInterior :
x ∈ fundamentalInterior G s ↔ x ∈ s ∧ ∀ g : G, g ≠ 1 → x ∉ g • s := by |
simp [fundamentalInterior]
|
/-
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 | 49 | 55 | theorem affineSegment_comm (x y : P) : affineSegment R x y = affineSegment R y x := by |
refine Set.ext fun z => ?_
constructor <;>
· rintro ⟨t, ht, hxy⟩
refine ⟨1 - t, ?_, ?_⟩
· rwa [Set.sub_mem_Icc_iff_right, sub_self, sub_zero]
· rwa [lineMap_apply_one_sub]
|
/-
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 | 850 | 854 | theorem integral_zero : ∫ _ : α, (0 : G) ∂μ = 0 := by |
by_cases hG : CompleteSpace G
· simp only [integral, hG, L1.integral]
exact setToFun_zero (dominatedFinMeasAdditive_weightedSMul μ)
· simp [integral, hG]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 665 | 669 | theorem Tendsto.isCompact_insert_range_of_cofinite {f : ι → X} {x} (hf : Tendsto f cofinite (𝓝 x)) :
IsCompact (insert x (range f)) := by |
letI : TopologicalSpace ι := ⊥; haveI h : DiscreteTopology ι := ⟨rfl⟩
rw [← cocompact_eq_cofinite ι] at hf
exact hf.isCompact_insert_range_of_cocompact continuous_of_discreteTopology
|
/-
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, Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 442 | 454 | theorem minFac_eq_two_iff (n : ℕ) : minFac n = 2 ↔ 2 ∣ n := by |
constructor
· intro h
rw [← h]
exact minFac_dvd n
· intro h
have ub := minFac_le_of_dvd (le_refl 2) h
have lb := minFac_pos n
refine ub.eq_or_lt.resolve_right fun h' => ?_
have := le_antisymm (Nat.succ_le_of_lt lb) (Nat.lt_succ_iff.mp h')
rw [eq_comm, Nat.minFac_eq_one_iff] at this
... |
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
/-!
# Integration with respect to a finite product of measures
... | Mathlib/MeasureTheory/Integral/Pi.lean | 65 | 84 | theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by |
induction n with
| zero =>
simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
| succ n n_ih =>
calc
_ = ∫ x : E 0 × ((i : Fin n) → E (Fin.succ i)),
f 0 x.1... |
/-
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.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 761 | 765 | theorem exists_insert_toList_zoom_node {t : RBNode α} (ht : Balanced t c n)
(e : zoom (cmp v) t = (node c' l v' r, p)) :
∃ L R, t.toList = L ++ v' :: R ∧ (t.insert cmp v).toList = L ++ v :: R := by |
refine ⟨p.listL ++ l.toList, r.toList ++ p.listR, ?_⟩
simp [← zoom_toList e, insert_toList_zoom_node ht e]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 278 | 283 | theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
{k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) := by |
simp_rw [← continuousWithinAt_univ] at hf
simpa only [inter_univ] using
exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt hk hf
|
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#a... | Mathlib/Geometry/Euclidean/Triangle.lean | 198 | 202 | theorem sin_angle_add_angle_sub_add_angle_sub_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0 := by |
rw [add_assoc, Real.sin_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,
sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy]
ring
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 75 | 80 | theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ)
(hh : AEStronglyMeasurable h μ) :
(∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ := by |
rw [← lintegral_add_left' (hf.edist hh)]
refine lintegral_mono fun a => ?_
apply edist_triangle_right
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 552 | 554 | theorem norm_nonneg' (a : E) : 0 ≤ ‖a‖ := by |
rw [← dist_one_right]
exact dist_nonneg
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,426 | 1,432 | theorem Iio_union_Ico' (h₁ : c ≤ b) : Iio b ∪ Ico c d = Iio (max b d) := by |
ext1 x
simp_rw [mem_union, mem_Iio, mem_Ico, lt_max_iff]
by_cases hc : c ≤ x
· simp only [hc, true_and] -- Porting note: restore `tauto`
· have hxb : x < b := (lt_of_not_ge hc).trans_le h₁
simp only [hxb, true_or] -- Porting note: restore `tauto`
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
... | Mathlib/Data/PEquiv.lean | 388 | 391 | theorem single_subsingleton_eq_refl [Subsingleton α] (a b : α) : single a b = PEquiv.refl α := by |
ext i j
dsimp [single]
rw [if_pos (Subsingleton.elim i a), Subsingleton.elim i j, Subsingleton.elim b j]
|
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.h... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 86 | 89 | theorem coeff_hermite_succ_succ (n k : ℕ) : coeff (hermite (n + 1)) (k + 1) =
coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2) := by |
rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm]
norm_cast
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 93 | 94 | theorem T_sub_two (n : ℤ) : T R (n - 2) = 2 * X * T R (n - 1) - T R n := by |
linear_combination (norm := ring_nf) T_add_two R (n - 2)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Neighborhoods to the left and to the right on an ... | Mathlib/Topology/Order/LeftRightNhds.lean | 337 | 342 | theorem Filter.Tendsto.add_atTop {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :
Tendsto (fun x => f x + g x) l atTop := by |
nontriviality α
obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C
refine tendsto_atTop_add_left_of_le' _ C' ?_ hg
exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt
|
/-
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.Data.Rat.Encodable
import Mathlib.Data.Real.EReal
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Topology.Order.MonotoneContinuity
#al... | Mathlib/Topology/Instances/EReal.lean | 226 | 238 | theorem continuousAt_add {p : EReal × EReal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) p := by |
rcases p with ⟨x, y⟩
induction x <;> induction y
· exact continuousAt_add_bot_bot
· exact continuousAt_add_bot_coe _
· simp at h'
· exact continuousAt_add_coe_bot _
· exact continuousAt_add_coe_coe _ _
· exact continuousAt_add_coe_top _
· simp at h
· exact continuousAt_add_top_coe _
· exact conti... |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.L... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 422 | 437 | theorem inner_mul_areaForm_sub' (a x : E) : ⟪a, x⟫ • ω a - ω a x • innerₛₗ ℝ a = ‖a‖ ^ 2 • ω x := by |
by_cases ha : a = 0
· simp [ha]
apply (o.basisRightAngleRotation a ha).ext
intro i
fin_cases i
· simp only [o.areaForm_swap a x, neg_smul, sub_neg_eq_add, Fin.mk_zero,
coe_basisRightAngleRotation, Matrix.cons_val_zero, LinearMap.add_apply, LinearMap.smul_apply,
areaForm_apply_self, smul_eq_mul,... |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 160 | 162 | theorem tsub_tsub_tsub_le_tsub : c - a - (c - b) ≤ b - a := by |
rw [tsub_le_iff_left, tsub_le_iff_left, add_left_comm]
exact le_tsub_add.trans (add_le_add_left le_add_tsub _)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 854 | 858 | theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1,
∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by |
rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists,
← isCoprime_iff_sup_eq]
simp
|
/-
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, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,024 | 1,114 | theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (hab : a ≤ b)
(hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x)
(φint : IntegrableOn φ (Icc a b)) (hφg : ∀ x ∈ Ico a b, g' x ≤ φ x) :
g b - g a ≤ ∫ y in a..b, φ y := by |
refine le_of_forall_pos_le_add fun ε εpos => ?_
-- Bound from above `g'` by a lower-semicontinuous function `G'`.
rcases exists_lt_lowerSemicontinuous_integral_lt φ φint εpos with
⟨G', f_lt_G', G'cont, G'int, G'lt_top, hG'⟩
-- we will show by "induction" that `g t - g a ≤ ∫ u in a..t, G' u` for all `t ∈ [a... |
/-
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, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 725 | 727 | theorem preimage_const_mul_Iio_of_neg (a : α) {c : α} (h : c < 0) :
(c * ·) ⁻¹' Iio a = Ioi (a / c) := by |
simpa only [mul_comm] using preimage_mul_const_Iio_of_neg a h
|
/-
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.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,148 | 1,157 | theorem suffix_cons_iff : l₁ <:+ a :: l₂ ↔ l₁ = a :: l₂ ∨ l₁ <:+ l₂ := by |
constructor
· rintro ⟨⟨hd, tl⟩, hl₃⟩
· exact Or.inl hl₃
· simp only [cons_append] at hl₃
injection hl₃ with _ hl₄
exact Or.inr ⟨_, hl₄⟩
· rintro (rfl | hl₁)
· exact (a :: l₂).suffix_refl
· exact hl₁.trans (l₂.suffix_cons _)
|
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import... | Mathlib/GroupTheory/PGroup.lean | 348 | 358 | theorem disjoint_of_ne (p₁ p₂ : ℕ) [hp₁ : Fact p₁.Prime] [hp₂ : Fact p₂.Prime] (hne : p₁ ≠ p₂)
(H₁ H₂ : Subgroup G) (hH₁ : IsPGroup p₁ H₁) (hH₂ : IsPGroup p₂ H₂) : Disjoint H₁ H₂ := by |
rw [Subgroup.disjoint_def]
intro x hx₁ hx₂
obtain ⟨n₁, hn₁⟩ := iff_orderOf.mp hH₁ ⟨x, hx₁⟩
obtain ⟨n₂, hn₂⟩ := iff_orderOf.mp hH₂ ⟨x, hx₂⟩
rw [Subgroup.orderOf_mk] at hn₁ hn₂
have : p₁ ^ n₁ = p₂ ^ n₂ := by rw [← hn₁, ← hn₂]
rcases n₁.eq_zero_or_pos with (rfl | hn₁)
· simpa using hn₁
· exact absurd (e... |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 47 | 51 | theorem le_min {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) : c ≤ min a b := by |
-- Porting note: no `min_tac` tactic
if h : a ≤ b
then simp [min_def, if_pos h]; exact h₁
else simp [min_def, if_neg h]; exact h₂
|
/-
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 | 164 | 169 | theorem finrank_vectorSpan_range_le [Fintype ι] (p : ι → P) {n : ℕ} (hc : Fintype.card ι = n + 1) :
finrank k (vectorSpan k (Set.range p)) ≤ n := by |
classical
rw [← Set.image_univ, ← Finset.coe_univ, ← Finset.coe_image]
rw [← Finset.card_univ] at hc
exact finrank_vectorSpan_image_finset_le _ _ _ hc
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 494 | 499 | theorem rootSet_prod [CommRing S] [IsDomain S] [Algebra R S] {ι : Type*} (f : ι → R[X])
(s : Finset ι) (h : s.prod f ≠ 0) : (s.prod f).rootSet S = ⋃ i ∈ s, (f i).rootSet S := by |
classical
simp only [rootSet, aroots, ← Finset.mem_coe]
rw [Polynomial.map_prod, roots_prod, Finset.bind_toFinset, s.val_toFinset, Finset.coe_biUnion]
rwa [← Polynomial.map_prod, Ne, map_eq_zero]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 136 | 138 | theorem inv_pos_le_iff_one_le_mul' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b := by |
rw [inv_eq_one_div]
exact div_le_iff' ha
|
/-
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.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 803 | 812 | theorem perfectField_of_perfectClosure_eq_bot [h : PerfectField E] (eq : perfectClosure F E = ⊥) :
PerfectField F := by |
let p := ringExpChar F
haveI := expChar_of_injective_algebraMap (algebraMap F E).injective p
haveI := PerfectRing.ofSurjective F p fun x ↦ by
obtain ⟨y, h⟩ := surjective_frobenius E p (algebraMap F E x)
have : y ∈ perfectClosure F E := ⟨1, x, by rw [← h, pow_one, frobenius_def, ringExpChar.eq F p]⟩
o... |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 902 | 909 | theorem toReal_neg_iff_sign_neg {θ : Angle} : θ.toReal < 0 ↔ θ.sign = -1 := by |
rw [sign, ← sin_toReal, sign_eq_neg_one_iff]
rcases lt_trichotomy θ.toReal 0 with (h | h | h)
· exact ⟨fun _ => Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_toReal θ), fun _ => h⟩
· simp [h]
· exact
⟨fun hn => False.elim (h.asymm hn), fun hn =>
False.elim (hn.not_le (sin_nonneg_of_nonneg_of_le... |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 130 | 138 | theorem zero_memℓp : Memℓp (0 : ∀ i, E i) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp
· apply memℓp_infty
simp only [norm_zero, Pi.zero_apply]
exact bddAbove_singleton.mono Set.range_const_subset
· apply memℓp_gen
simp [Real.zero_rpow hp.ne', summable_zero]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 934 | 936 | theorem ncard_add_ncard_compl (s : Set α) (hs : s.Finite := by | toFinite_tac)
(hsc : sᶜ.Finite := by toFinite_tac) : s.ncard + sᶜ.ncard = Nat.card α := by
rw [← ncard_univ, ← ncard_union_eq (@disjoint_compl_right _ _ s) hs hsc, union_compl_self]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr... | Mathlib/Order/Hom/Basic.lean | 1,118 | 1,121 | theorem ofRelIsoLT_toRelIsoLT {α β} [PartialOrder α] [PartialOrder β] (e : α ≃o β) :
ofRelIsoLT (toRelIsoLT e) = e := by |
ext
simp
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Categ... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 234 | 235 | theorem prodIsoProd_inv_snd (X Y : TopCat.{u}) :
(prodIsoProd X Y).inv ≫ Limits.prod.snd = prodSnd := by | simp [Iso.inv_comp_eq]
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Ba... | Mathlib/RepresentationTheory/Basic.lean | 106 | 107 | theorem asAlgebraHom_single (g : G) (r : k) : asAlgebraHom ρ (Finsupp.single g r) = r • ρ g := by |
simp only [asAlgebraHom_def, MonoidAlgebra.lift_single]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 592 | 597 | theorem LinearIndependent.image_of_comp {ι ι'} (s : Set ι) (f : ι → ι') (g : ι' → M)
(hs : LinearIndependent R fun x : s => g (f x)) :
LinearIndependent R fun x : f '' s => g x := by |
nontriviality R
have : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp
exact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 278 | 287 | theorem ge_logGammaSeq (hf_conv : ConvexOn ℝ (Ioi 0) f)
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) (hn : n ≠ 0) :
f 1 + logGammaSeq x n ≤ f x := by |
dsimp [logGammaSeq]
rw [← add_sub_assoc, sub_le_iff_le_add, ← f_add_nat_eq (@hf_feq) hx, add_comm x _]
refine le_trans (le_of_eq ?_) (f_add_nat_ge hf_conv @hf_feq ?_ hx)
· rw [f_nat_eq @hf_feq, Nat.add_sub_cancel, Nat.cast_add_one, add_sub_cancel_right]
· ring
· exact Nat.succ_ne_zero _
· omega
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset... | Mathlib/GroupTheory/Exponent.lean | 344 | 364 | theorem exponent_ne_zero_iff_range_orderOf_finite (h : ∀ g : G, 0 < orderOf g) :
exponent G ≠ 0 ↔ (Set.range (orderOf : G → ℕ)).Finite := by |
refine ⟨fun he => ?_, fun he => ?_⟩
· by_contra h
obtain ⟨m, ⟨t, rfl⟩, het⟩ := Set.Infinite.exists_gt h (exponent G)
exact pow_ne_one_of_lt_orderOf' he het (pow_exponent_eq_one t)
· lift Set.range (orderOf (G := G)) to Finset ℕ using he with t ht
have htpos : 0 < t.prod id := by
refine Finset.p... |
/-
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.Restrict
/-!
# Classes of measures
We introduce the following typeclasses for measures:
* `IsProbabilityMeasur... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,506 | 1,508 | theorem measure_zero_of_nhdsWithin (hs : IsCompact s) :
(∀ a ∈ s, ∃ t ∈ 𝓝[s] a, μ t = 0) → μ s = 0 := by |
simpa only [← compl_mem_ae_iff] using hs.compl_mem_sets_of_nhdsWithin
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 543 | 546 | theorem pentagon_hom_inv_inv_inv_inv :
W ◁ (α_ X Y Z).hom ≫ (α_ W X (Y ⊗ Z)).inv ≫ (α_ (W ⊗ X) Y Z).inv =
(α_ W (X ⊗ Y) Z).inv ≫ (α_ W X Y).inv ▷ Z := by |
simp [← cancel_epi (W ◁ (α_ X Y Z).inv)]
|
/-
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.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import prob... | Mathlib/Probability/Martingale/BorelCantelli.lean | 295 | 300 | theorem martingalePart_process_ae_eq (ℱ : Filtration ℕ m0) (μ : Measure Ω) (s : ℕ → Set Ω) (n : ℕ) :
martingalePart (process s) ℱ μ n =
∑ k ∈ Finset.range n, ((s (k + 1)).indicator 1 - μ[(s (k + 1)).indicator 1|ℱ k]) := by |
simp only [martingalePart_eq_sum, process_zero, zero_add]
refine Finset.sum_congr rfl fun k _ => ?_
simp only [process, Finset.sum_range_succ_sub_sum]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 262 | 264 | theorem coeFn_smul [IsScalarTower R ℝ≥0 ℝ≥0] (c : R) (μ : FiniteMeasure Ω) :
(⇑(c • μ) : Set Ω → ℝ≥0) = c • (⇑μ : Set Ω → ℝ≥0) := by |
funext; simp [← ENNReal.coe_inj, ENNReal.coe_smul]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 176 | 181 | theorem exists_int_eq_of_part_denom {b : K}
(nth_part_denom_eq : (of v).partialDenominators.get? n = some b) : ∃ z : ℤ, b = (z : K) := by |
obtain ⟨gp, nth_s_eq, gp_b_eq_b_n⟩ : ∃ gp, (of v).s.get? n = some gp ∧ gp.b = b :=
exists_s_b_of_part_denom nth_part_denom_eq
have : ∃ z : ℤ, gp.b = (z : K) := (of_part_num_eq_one_and_exists_int_part_denom_eq nth_s_eq).right
rwa [gp_b_eq_b_n] at this
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "... | Mathlib/NumberTheory/NumberField/Norm.lean | 65 | 69 | theorem norm_algebraMap [IsSeparable K L] (x : 𝓞 K) :
norm K (algebraMap (𝓞 K) (𝓞 L) x) = x ^ finrank K L := by |
rw [RingOfIntegers.ext_iff, RingOfIntegers.coe_eq_algebraMap,
RingOfIntegers.algebraMap_norm_algebraMap, Algebra.norm_algebraMap,
RingOfIntegers.coe_eq_algebraMap, map_pow]
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 78 | 83 | theorem IsLindelof.inter_right (hs : IsLindelof s) (ht : IsClosed t) : IsLindelof (s ∩ t) := by |
intro f hnf _ hstf
rw [← inf_principal, le_inf_iff] at hstf
obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f := hs hstf.1
have hxt : x ∈ t := ht.mem_of_nhdsWithin_neBot <| hx.mono hstf.2
exact ⟨x, ⟨hsx, hxt⟩, hx⟩
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 953 | 955 | theorem map_id : map (f₂ := f₁) (g₂ := g₁) (id V₁) (id W₁) rfl rfl = id (f₁.equalizer g₁) := by |
ext
rfl
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216... | Mathlib/Data/Bool/Basic.lean | 277 | 278 | theorem apply_apply_apply (f : Bool → Bool) (x : Bool) : f (f (f x)) = f x := by |
cases x <;> cases h₁ : f true <;> cases h₂ : f false <;> simp only [h₁, h₂]
|
/-
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
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 412 | 420 | theorem count_neg_zpow (n : ℤ) (I : FractionalIdeal R⁰ K) :
count K v (I ^ (-n)) = - count K v (I ^ n) := by |
by_cases hI : I = 0
· by_cases hn : n = 0
· rw [hn, neg_zero, zpow_zero, count_one, neg_zero]
· rw [hI, zero_zpow n hn, zero_zpow (-n) (neg_ne_zero.mpr hn), count_zero, neg_zero]
· rw [eq_neg_iff_add_eq_zero, ← count_mul K v (zpow_ne_zero _ hI) (zpow_ne_zero _ hI),
← zpow_add₀ hI, neg_add_self, zpo... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Degrees
import ... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 408 | 429 | theorem eval_eq_eval_mv_eval' (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) :
eval (Fin.cons y s : Fin (n + 1) → R) f =
Polynomial.eval y (Polynomial.map (eval s) (finSuccEquiv R n f)) := by |
-- turn this into a def `Polynomial.mapAlgHom`
let φ : (MvPolynomial (Fin n) R)[X] →ₐ[R] R[X] :=
{ Polynomial.mapRingHom (eval s) with
commutes' := fun r => by
convert Polynomial.map_C (eval s)
exact (eval_C _).symm }
show
aeval (Fin.cons y s : Fin (n + 1) → R) f =
(Polynomial... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.LinearM... | Mathlib/Algebra/Module/Equiv.lean | 1,316 | 1,321 | theorem funLeft_injective_of_surjective (f : m → n) (hf : Surjective f) :
Injective (funLeft R M f) := by |
obtain ⟨g, hg⟩ := hf.hasRightInverse
suffices LeftInverse (funLeft R M g) (funLeft R M f) by exact this.injective
intro x
rw [← LinearMap.comp_apply, ← funLeft_comp, hg.id, funLeft_id]
|
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 118 | 121 | theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt 𝕜 f x) :
HasGradientAt f (∇ f x) x := by |
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual 𝕜 F).apply_symm_apply (fderiv 𝕜 f x)]
exact h.hasFDerivAt
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_im... | Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 | theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by |
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exac... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 736 | 754 | theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by |
refine fun h => le_antisymm (le_sInf ?_) ?_
· intro j hj y hy
cases' (mem_map_iff_of_surjective f hf).1 hy with x hx
cases' (Set.mem_image _ _ _).mp hj with J hJ
rw [← hJ.right, ← hx.right]
exact mem_map_of_mem f (sInf_le_of_le hJ.left (le_of_eq rfl) hx.left)
· intro y hy
cases' hf y with x h... |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,188 | 1,189 | theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by |
simp only [eq_univ_iff_forall, mem_sUnion]
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/... | Mathlib/Algebra/Polynomial/Mirror.lean | 66 | 72 | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by |
by_cases hp : p = 0
· rw [hp, mirror_zero]
nontriviality R
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]
rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero]
|
/-
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 | 968 | 970 | theorem mk'_smul (r : R) (m : M) (s : S) : mk' f (r • m) s = r • mk' f m s := by |
delta mk'
rw [← LocalizedModule.smul'_mk, LinearMap.map_smul]
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix f... | Mathlib/Control/LawfulFix.lean | 71 | 91 | theorem mem_iff (a : α) (b : β a) : b ∈ Part.fix f a ↔ ∃ i, b ∈ approx f i a := by |
by_cases h₀ : ∃ i : ℕ, (approx f i a).Dom
· simp only [Part.fix_def f h₀]
constructor <;> intro hh
· exact ⟨_, hh⟩
have h₁ := Nat.find_spec h₀
rw [dom_iff_mem] at h₁
cases' h₁ with y h₁
replace h₁ := approx_mono' f _ _ h₁
suffices y = b by
subst this
exact h₁
cases' hh w... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 685 | 686 | theorem bit1_comp : comp (bit1 p : R[X]) q = bit1 (p.comp q) := by |
simp only [bit1, add_comp, bit0_comp, one_comp]
|
/-
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.DivergenceTheorem
import Mathlib.Analysis.BoxIntegral.Integrability
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.... | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | 320 | 359 | theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F] [PartialOrder F] [MeasureSpace F] [BorelSpace F]
(eL : F ≃L[ℝ] ℝⁿ⁺¹) (he_ord : ∀ x y, eL x ≤ eL y ↔ x ≤ y)
(he_vol : MeasurePreserving eL volume volume) (f : Fin (n + 1) → F → E)
... |
ext1 x; simp only [Set.mem_preimage, Set.mem_Icc, he_ord]
have hIcc' : Icc (eL a) (eL b) = eL.symm ⁻¹' Icc a b := by rw [← hIcc, eL.symm_preimage_preimage]
calc
∫ x in Icc a b, DF x = ∫ x in Icc a b, ∑ i, f' i x (eL.symm <| e i) := by simp only [hDF]
_ = ∫ x in Icc (eL a) (eL b), ∑ i, f' i (eL.symm x) ... |
/-
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 | 294 | 294 | theorem sinh_neg : sinh (-x) = -sinh x := by | simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 594 | 598 | theorem card_roots_of_disc_ne_zero [DecidableEq K] (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z})
(hd : P.disc ≠ 0) : (map φ P).roots.toFinset.card = 3 := by |
rw [toFinset_card_of_nodup <| (disc_ne_zero_iff_roots_nodup ha h3).mp hd,
← splits_iff_card_roots ha, splits_iff_roots_eq_three ha]
exact ⟨x, ⟨y, ⟨z, h3⟩⟩⟩
|
/-
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 | 723 | 725 | theorem toIcoMod_eq_self : toIcoMod hp a b = b ↔ b ∈ Set.Ico a (a + p) := by |
rw [toIcoMod_eq_iff, and_iff_left]
exact ⟨0, by simp⟩
|
/-
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 | 192 | 193 | theorem sup_span {S : Set (ℙ K V)} {W : Subspace K V} : W ⊔ span S = span (W ∪ S) := by |
rw [span_union, span_coe]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 176 | 183 | theorem rotate_perm (l : List α) (n : ℕ) : l.rotate n ~ l := by |
rw [rotate_eq_rotate']
induction' n with n hn generalizing l
· simp
· cases' l with hd tl
· simp
· rw [rotate'_cons_succ]
exact (hn _).trans (perm_append_singleton _ _)
|
/-
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 | 606 | 614 | theorem dense_iff_inter_open :
Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty := by |
constructor <;> intro h
· rintro U U_op ⟨x, x_in⟩
exact mem_closure_iff.1 (h _) U U_op x_in
· intro x
rw [mem_closure_iff]
intro U U_op x_in
exact h U U_op ⟨_, x_in⟩
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sh... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 262 | 265 | theorem hom_ext {α β : F ⟶ F'.1} (he : ∀ i, α.app (op (B i)) = β.app (op (B i))) : α = β := by |
apply (restrictHomEquivHom F F' h).symm.injective
ext i
exact he i.unop
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Ma... | Mathlib/CategoryTheory/Limits/Final.lean | 763 | 766 | theorem initial_of_initial_comp [Initial F] [Initial (F ⋙ G)] : Initial G := by |
suffices Final G.op from initial_of_final_op _
have : Final (F.op ⋙ G.op) := show Final (F ⋙ G).op from inferInstance
exact final_of_final_comp F.op G.op
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
#align_import data.finset.sups from "leanprover-community/mathlib"@"8818fde... | Mathlib/Data/Finset/Sups.lean | 561 | 562 | theorem disjSups_inter_subset_left : (s₁ ∩ s₂) ○ t ⊆ s₁ ○ t ∩ s₂ ○ t := by |
simpa only [disjSups, inter_product, filter_inter_distrib] using image_inter_subset _ _ _
|
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,298 | 1,299 | theorem rTensor_comp_apply (x : N ⊗[R] M) :
(g.comp f).rTensor M x = (g.rTensor M) ((f.rTensor M) x) := by | rw [rTensor_comp, coe_comp]; rfl
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.inner_product_space.linear_pmap from "leanprover-comm... | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 186 | 188 | theorem adjoint_apply_of_dense (y : T†.domain) : T† y = adjointAux hT y := by |
change (if hT : Dense (T.domain : Set E) then adjointAux hT else 0) y = _
simp only [hT, dif_pos, LinearMap.coe_mk]
|
/-
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 | 73 | 84 | theorem Finset.exists_eq_pow_of_mul_eq_pow_of_coprime {ι R : Type*} [CommSemiring R] [IsDomain R]
[GCDMonoid R] [Unique Rˣ] {n : ℕ} {c : R} {s : Finset ι} {f : ι → R}
(h : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → IsCoprime (f i) (f j))
(hprod : ∏ i ∈ s, f i = c ^ n) : ∀ i ∈ s, ∃ d : R, f i = d ^ n := by |
classical
intro i hi
rw [← insert_erase hi, prod_insert (not_mem_erase i s)] at hprod
refine
exists_eq_pow_of_mul_eq_pow_of_coprime
(IsCoprime.prod_right fun j hj => h i hi j (erase_subset i s hj) fun hij => ?_) hprod
rw [hij] at hj
exact (s.not_mem_erase _) hj
|
/-
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.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 1,040 | 1,042 | theorem pi_trans (ei : ∀ i, PartialEquiv (αi i) (βi i)) (ei' : ∀ i, PartialEquiv (βi i) (γi i)) :
(PartialEquiv.pi ei).trans (PartialEquiv.pi ei') = .pi fun i ↦ (ei i).trans (ei' i) := by |
ext <;> simp [forall_and]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "lea... | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 223 | 228 | theorem homeomorphCircle_apply (hT : T ≠ 0) (x : AddCircle T) :
homeomorphCircle hT x = toCircle x := by |
induction' x using QuotientAddGroup.induction_on' with x
rw [homeomorphCircle, Homeomorph.trans_apply,
homeomorphAddCircle_apply_mk, homeomorphCircle'_apply_mk, toCircle_apply_mk]
ring_nf
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 83 | 87 | theorem withDensity_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) :
μ.withDensity f = μ.withDensity g := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, withDensity_apply _ hs]
exact lintegral_congr_ae (ae_restrict_of_ae h)
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 134 | 134 | theorem T_neg_two : T R (-2) = 2 * X ^ 2 - 1 := by | simp [T_two]
|
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