Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
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/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 151 | 153 | theorem Valid'.node4L_lemma₁ {a b c d : ℕ} (lr₂ : 3 * (b + c + 1 + d) ≤ 16 * a + 9)
(mr₂ : b + c + 1 ≤ 3 * d) (mm₁ : b ≤ 3 * c) : b < 3 * a + 1 := by | omega |
/-
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.Order.Disjoint
/-!
# The order on `Prop`
Instances on `Prop` such as `DistribLattice`, `BoundedOrder`, `LinearOrder`.
-/
/-- Propositions form a d... | Mathlib/Order/PropInstances.lean | 72 | 80 | theorem codisjoint_iff [∀ i, OrderTop (α' i)] {f g : ∀ i, α' i} :
Codisjoint f g ↔ ∀ i, Codisjoint (f i) (g i) :=
@disjoint_iff _ (fun i => (α' i)ᵒᵈ) _ _ _ _
theorem isCompl_iff [∀ i, BoundedOrder (α' i)] {f g : ∀ i, α' i} :
IsCompl f g ↔ ∀ i, IsCompl (f i) (g i) := by | simp_rw [_root_.isCompl_iff, disjoint_iff, codisjoint_iff, forall_and]
end Pi |
/-
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
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | Mathlib/Data/List/Nodup.lean | 165 | 168 | theorem count_eq_of_nodup [DecidableEq α] {a : α} {l : List α} (d : Nodup l) :
count a l = if a ∈ l then 1 else 0 := by | split_ifs with h
· exact count_eq_one_of_mem d h |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 552 | 555 | theorem eval₂_T_neg_n (n : ℕ) : eval₂ f x (T (-n)) = x⁻¹ ^ n := by | rw [←mk'_one_X_pow]
unfold eval₂
rw [IsLocalization.lift_mk'_spec, map_one, coe_eval₂RingHom, eval₂_X_pow, ←mul_pow, |
/-
Copyright (c) 2022 Daniel Roca González. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Roca González
-/
import Mathlib.Analysis.InnerProductSpace.Dual
/-!
# The Lax-Milgram Theorem
We consider a Hilbert space `V` over `ℝ`
equipped with a bounded bilinear f... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 80 | 82 | theorem range_eq_top (coercive : IsCoercive B) : range B♯ = ⊤ := by | haveI := coercive.isClosed_range.completeSpace_coe
rw [← (range B♯).orthogonal_orthogonal] |
/-
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.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | Mathlib/Topology/MetricSpace/Bounded.lean | 123 | 125 | theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by | simpa only [dist_comm _ c] using comap_dist_right_atTop c |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Set.UnorderedInterval
/-!
# Locally... | Mathlib/Order/Interval/Finset/Defs.lean | 1,156 | 1,160 | theorem map_subtype_embedding_Ici (hp : ∀ ⦃a x⦄, a ≤ x → p a → p x) :
(Ici a).map (Embedding.subtype p) = (Ici a : Finset α) := by | rw [subtype_Ici_eq]
exact Finset.subtype_map_of_mem fun x hx => hp (mem_Ici.1 hx) a.prop |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 242 | 244 | theorem nhdsNE_neBot (x : α) : NeBot (𝓝[≠] x) := by | rw [← mem_closure_iff_nhdsWithin_neBot, Metric.mem_closure_iff]
rintro ε ε0 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.Connected.T... | Mathlib/Topology/DiscreteQuotient.lean | 377 | 388 | theorem finsetClopens_inj [CompactSpace X] :
(finsetClopens X).Injective := by | apply Function.Injective.of_comp (f := Set.image (fun (t : Clopens X) ↦ t.carrier) ∘ Finset.toSet)
rw [comp_finsetClopens]
intro ⟨_, _⟩ ⟨_, _⟩ h
congr
rw [Setoid.classes_inj]
exact h
/--
The discrete quotients of a compact space are in bijection with a subtype of the type of
`Finset (Clopens X)`. |
/-
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
/-!
# Compositions
A compositio... | Mathlib/Combinatorics/Enumerative/Composition.lean | 207 | 210 | theorem length_pos_iff : 0 < c.length ↔ 0 < n := by | simp [pos_iff_ne_zero]
alias ⟨_, length_pos_of_pos⟩ := length_pos_iff |
/-
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.Order.Filter.CountableInter
/-!
# Filters with countable intersections and countable separating families
In this file we prove some facts about a f... | Mathlib/Order/Filter/CountableSeparatingOn.lean | 156 | 170 | theorem exists_subset_subsingleton_mem_of_forall_separating (p : Set α → Prop)
{s : Set α} [h : HasCountableSeparatingOn α p s] (hs : s ∈ l)
(hl : ∀ U, p U → U ∈ l ∨ Uᶜ ∈ l) : ∃ t, t ⊆ s ∧ t.Subsingleton ∧ t ∈ l := by | rcases h.1 with ⟨S, hSc, hSp, hS⟩
refine ⟨s ∩ ⋂₀ (S ∩ l.sets) ∩ ⋂ (U ∈ S) (_ : Uᶜ ∈ l), Uᶜ, ?_, ?_, ?_⟩
· exact fun _ h ↦ h.1.1
· intro x hx y hy
simp only [mem_sInter, mem_inter_iff, mem_iInter, mem_compl_iff] at hx hy
refine hS x hx.1.1 y hy.1.1 (fun s hsS ↦ ?_)
cases hl s (hSp s hsS) with
| inl... |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 72 | 74 | theorem pow_two : t ^ 2 = 1 := by | rcases ht with ⟨w, i, rfl⟩
simp |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Order.Antichain
import Mathlib.Topology.ContinuousOn
/-!
# Left and right continuity
In this file we prove a few lemmas about left and right cont... | Mathlib/Topology/Order/LeftRight.lean | 111 | 112 | theorem nhdsLT_sup_nhdsGE (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a := by | rw [← nhdsWithin_union, Iio_union_Ici, nhdsWithin_univ] |
/-
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.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,258 | 1,259 | theorem norm_div_eq_zero_iff : ‖a / b‖ = 0 ↔ a = b := by | rw [norm_eq_zero', div_eq_one] |
/-
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.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | Mathlib/FieldTheory/Separable.lean | 66 | 67 | theorem separable_of_subsingleton [Subsingleton R] (f : R[X]) : f.Separable := by | simp [Separable, IsCoprime, eq_iff_true_of_subsingleton] |
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 739 | 740 | theorem infDist_le_infDist_add_hausdorffDist (fin : hausdorffEdist s t ≠ ⊤) :
infDist x t ≤ infDist x s + hausdorffDist s t := by | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... | Mathlib/FieldTheory/Minpoly/Basic.lean | 96 | 97 | theorem ne_one [Nontrivial B] : minpoly A x ≠ 1 := by | intro h |
/-
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.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 138 | 138 | theorem card_bind : card (s.bind f) = (s.map (card ∘ f)).sum := by | simp [bind] |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | Mathlib/Probability/Process/Stopping.lean | 865 | 878 | theorem memLp_stoppedValue [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
(hu : ∀ n, MemLp (u n) p μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) : MemLp (stoppedValue u τ) p μ :=
memLp_stoppedValue_of_mem_finset hτ hu fun ω => Finset.mem_Iic.mpr (hbdd ω)
theorem integrable_stoppedValue_of_mem_finset (hτ : IsStoppingTime... | simp_rw [← memLp_one_iff_integrable] at hu ⊢
exact memLp_stoppedValue_of_mem_finset hτ hu hbdd
variable (ι)
theorem integrable_stoppedValue [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
(hu : ∀ n, Integrable (u n) μ) {N : ι} (hbdd : ∀ ω, τ ω ≤ N) : |
/-
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.Interval.Set.Basic
import Mathlib.Order.Hom.Set
/-!
# Lemmas about images of inte... | Mathlib/Order/Interval/Set/OrderIso.lean | 58 | 59 | theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) := by | rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm] |
/-
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.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | Mathlib/GroupTheory/CoprodI.lean | 584 | 586 | theorem prod_smul (m) : ∀ w : Word M, prod (m • w) = m * prod w := by | induction m using CoprodI.induction_on with
| one => |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 497 | 499 | theorem affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one {w : ι → k} {p : ι → P}
(hw : s.weightedVSub p w = (0 : V)) {i : ι} [DecidablePred (· ≠ i)] (his : i ∈ s)
(hwi : w i = -1) : {x ∈ s | x ≠ i}.affineCombination k p w = p i := by | |
/-
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.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invariants (f : α → α)... | Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 58 | 60 | theorem measurable_invariants_dom {f : α → α} {g : α → β} :
Measurable[invariants f] g ↔ Measurable g ∧ ∀ s, MeasurableSet s → (g ∘ f) ⁻¹' s = g ⁻¹' s := by | simp only [Measurable, ← forall_and]; rfl |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 336 | 339 | theorem rightUnitor_inv_braiding (X : C) : (ρ_ X).inv ≫ (β_ X (𝟙_ C)).hom = (λ_ X).inv := by | apply (cancel_mono (λ_ X).hom).1
simp only [assoc, braiding_leftUnitor, Iso.inv_hom_id] |
/-
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.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... | Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 37 | 42 | theorem inversion_mem_perpBisector_inversion_iff (hR : R ≠ 0) (hx : x ≠ c) (hy : y ≠ c) :
inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c := by | rw [mem_perpBisector_iff_dist_eq, dist_inversion_inversion hx hy, dist_inversion_center]
have hx' := dist_ne_zero.2 hx
have hy' := dist_ne_zero.2 hy
-- takes 300ms, but the "equivalent" simp call fails -> hard to speed up |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 539 | 540 | theorem left_vsub_lineMap (p₀ p₁ : P1) (c : k) : p₀ -ᵥ lineMap p₀ p₁ c = c • (p₀ -ᵥ p₁) := by | rw [← neg_vsub_eq_vsub_rev, lineMap_vsub_left, ← smul_neg, neg_vsub_eq_vsub_rev] |
/-
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 | 180 | 191 | theorem IsLindelof.indexed_countable_subcover {ι : Type v} [Nonempty ι]
(hs : IsLindelof s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) :
∃ f : ℕ → ι, s ⊆ ⋃ n, U (f n) := by | obtain ⟨c, ⟨c_count, c_cov⟩⟩ := hs.elim_countable_subcover U hUo hsU
rcases c.eq_empty_or_nonempty with rfl | c_nonempty
· simp only [mem_empty_iff_false, iUnion_of_empty, iUnion_empty] at c_cov
simp only [subset_eq_empty c_cov rfl, empty_subset, exists_const]
obtain ⟨f, f_surj⟩ := (Set.countable_iff_exists_s... |
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Kernel.Composition.Comp
/-!
# Invariance of measures along a kernel
We say that a measure `μ` is invariant with respect to a kernel `κ` if its ... | Mathlib/Probability/Kernel/Invariance.lean | 57 | 60 | theorem comp_const_apply_eq_bind (κ : Kernel α β) (μ : Measure α) (a : α) :
(κ ∘ₖ const α μ) a = μ.bind κ := by | rw [← const_apply (μ.bind κ) a, const_bind_eq_comp_const κ μ] |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 86 | 90 | theorem eval_at_1 (n ν : ℕ) : (bernsteinPolynomial R n ν).eval 1 = if ν = n then 1 else 0 := by | rw [bernsteinPolynomial]
split_ifs with h
· subst h; simp
· obtain hνn | hnν := Ne.lt_or_lt h |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteStability
import Mathlib.RingTheory.Ideal.Quotient.Nilpotent
import Mathlib.RingTheory.Kaehler.Basic
import Mathlib.RingTheory.Localization.... | Mathlib/RingTheory/Unramified/Basic.lean | 138 | 151 | theorem ext_of_iInf [FormallyUnramified R A] (hI : ⨅ i, I ^ i = ⊥) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ := by | have (i : ℕ) :
(Ideal.Quotient.mkₐ R (I ^ i)).comp g₁ = (Ideal.Quotient.mkₐ R (I ^ i)).comp g₂ := by
by_cases hi : i = 0
· ext x
have : Subsingleton (B ⧸ I ^ i) := by
rw [hi, pow_zero, Ideal.one_eq_top]
infer_instance
exact Subsingleton.elim _ _
apply ext (I.map (algebraMap... |
/-
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.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 82 | 86 | theorem rel_join {r : α → β → Prop} {s t} (h : Rel (Rel r) s t) : Rel r s.join t.join := by | induction h with
| zero => simp
| cons hab hst ih => simpa using hab.add ih |
/-
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.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... | Mathlib/RingTheory/PowerSeries/Order.lean | 121 | 129 | theorem order_eq_nat {φ : R⟦X⟧} {n : ℕ} :
order φ = n ↔ coeff R n φ ≠ 0 ∧ ∀ i, i < n → coeff R i φ = 0 := by | classical
rcases eq_or_ne φ 0 with (rfl | hφ)
· simp
simp [order, dif_neg hφ, Nat.find_eq_iff]
/-- The order of a formal power series is exactly `n` if the `n`th coefficient is nonzero,
and the `i`th coefficient is `0` for all `i < n`. -/ |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | Mathlib/GroupTheory/Coxeter/Basic.lean | 364 | 364 | theorem wordProd_cons (i : B) (ω : List B) : π (i :: ω) = s i * π ω := by | simp [wordProd] |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, Patrick Massot
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.RCLike.Basic
/-!
# A collection of specific limit computations for `RCLike`
-... | Mathlib/Analysis/SpecificLimits/RCLike.lean | 19 | 22 | theorem RCLike.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (𝓝 0) := by | convert tendsto_algebraMap_inverse_atTop_nhds_zero_nat 𝕜
simp |
/-
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.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 561 | 566 | theorem parts_partition {σ : Perm α} :
σ.partition.parts = σ.cycleType + Multiset.replicate (Fintype.card α - #σ.support) 1 :=
rfl
theorem filter_parts_partition_eq_cycleType {σ : Perm α} :
((partition σ).parts.filter fun n => 2 ≤ n) = σ.cycleType := by | |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | Mathlib/NumberTheory/Multiplicity.lean | 46 | 48 | theorem dvd_geom_sum₂_iff_of_dvd_sub' {x y p : R} (h : p ∣ x - y) :
(p ∣ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) ↔ p ∣ n * x ^ (n - 1) := by | rw [geom_sum₂_comm, dvd_geom_sum₂_iff_of_dvd_sub]; simpa using h.neg_right |
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | Mathlib/LinearAlgebra/Matrix/Block.lean | 134 | 136 | theorem blockTriangular_blockDiagonal [DecidableEq α] (d : α → Matrix m m R) :
BlockTriangular (blockDiagonal d) Prod.snd := by | rintro ⟨i, i'⟩ ⟨j, j'⟩ h |
/-
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.Normal.... | Mathlib/FieldTheory/SeparableDegree.lean | 315 | 319 | theorem natSepDegree_eq_zero (h : f.natDegree = 0) : f.natSepDegree = 0 := by | linarith only [natSepDegree_le_natDegree f, h]
@[simp]
theorem natSepDegree_C (x : F) : (C x).natSepDegree = 0 := natSepDegree_eq_zero _ (natDegree_C _) |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | Mathlib/GroupTheory/Perm/Sign.lean | 375 | 385 | theorem sign_mul (f g : Perm α) : sign (f * g) = sign f * sign g :=
MonoidHom.map_mul sign f g
@[simp]
theorem sign_trans (f g : Perm α) : sign (f.trans g) = sign g * sign f := by | rw [← mul_def, sign_mul]
@[simp]
theorem sign_one : sign (1 : Perm α) = 1 :=
MonoidHom.map_one sign |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 196 | 200 | theorem norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle (x + y) y) = ‖x + y‖ := by | rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two h |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 169 | 170 | theorem left_lt_add_div_two : a < (a + b) / 2 ↔ a < b := by | simp [lt_div_iff₀, mul_two] |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | Mathlib/Algebra/RingQuot.lean | 69 | 70 | theorem Rel.sub_right {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) :
Rel r (a - b) (a - c) := by | simp only [sub_eq_add_neg, h.neg.add_right] |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.AlgebraicGeometry.StructureSheaf
import Mathlib.RingTheory.Localization.Localizatio... | Mathlib/AlgebraicGeometry/Spec.lean | 338 | 342 | theorem toPushforwardStalk_comp :
f ≫ StructureSheaf.toPushforwardStalk f p =
StructureSheaf.toStalk R p ≫
(TopCat.Presheaf.stalkFunctor _ _).map (Spec.sheafedSpaceMap f).c := by | rw [StructureSheaf.toStalk, Category.assoc, TopCat.Presheaf.stalkFunctor_map_germ] |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | Mathlib/Analysis/Convex/Strict.lean | 231 | 233 | theorem StrictConvex.add_right (hs : StrictConvex 𝕜 s) (z : E) :
StrictConvex 𝕜 ((fun x => x + z) '' s) := by | simpa only [add_comm] using hs.add_left z |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | Mathlib/Topology/MetricSpace/Infsep.lean | 215 | 224 | theorem einfsep_eq_top_iff : s.einfsep = ∞ ↔ s.Subsingleton :=
⟨subsingleton_of_einfsep_eq_top, Subsingleton.einfsep⟩
theorem Nontrivial.einfsep_ne_top (hs : s.Nontrivial) : s.einfsep ≠ ∞ := by | contrapose! hs
rw [not_nontrivial_iff]
exact subsingleton_of_einfsep_eq_top hs
theorem Nontrivial.einfsep_lt_top (hs : s.Nontrivial) : s.einfsep < ∞ := by
rw [lt_top_iff_ne_top] |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | Mathlib/Data/Fin/Tuple/Basic.lean | 823 | 834 | theorem insertNth_comp_succAbove (i : Fin (n + 1)) (x : β) (p : Fin n → β) :
insertNth i x p ∘ i.succAbove = p :=
funext (insertNth_apply_succAbove i _ _)
theorem insertNth_eq_iff {p : Fin (n + 1)} {a : α p} {f : ∀ i, α (p.succAbove i)} {g : ∀ j, α j} :
insertNth p a f = g ↔ a = g p ∧ f = removeNth p g := by | simp [funext_iff, forall_iff_succAbove p, removeNth]
theorem eq_insertNth_iff {p : Fin (n + 1)} {a : α p} {f : ∀ i, α (p.succAbove i)} {g : ∀ j, α j} :
g = insertNth p a f ↔ g p = a ∧ removeNth p g = f := by
simpa [eq_comm] using insertNth_eq_iff |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | Mathlib/Data/Rel.lean | 265 | 265 | theorem inter_dom_subset_preimage_image (s : Set α) : s ∩ r.dom ⊆ r.preimage (r.image s) := by | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | Mathlib/Analysis/Analytic/Composition.lean | 272 | 280 | theorem comp_removeZero (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) :
q.comp p.removeZero = q.comp p := by | ext n; simp [FormalMultilinearSeries.comp]
end FormalMultilinearSeries
end Topological
variable [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup H] |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | Mathlib/Analysis/RCLike/Basic.lean | 494 | 496 | theorem div_re_ofReal {z : K} {r : ℝ} : re (z / r) = re z / r := by | rw [div_eq_inv_mul, div_eq_inv_mul, ← ofReal_inv, re_ofReal_mul] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | Mathlib/FieldTheory/Finite/Basic.lean | 241 | 251 | theorem card (p : ℕ) [CharP K p] : ∃ n : ℕ+, Nat.Prime p ∧ q = p ^ (n : ℕ) := by | haveI hp : Fact p.Prime := ⟨CharP.char_is_prime K p⟩
letI : Module (ZMod p) K := { (ZMod.castHom dvd_rfl K : ZMod p →+* _).toModule with }
obtain ⟨n, h⟩ := VectorSpace.card_fintype (ZMod p) K
rw [ZMod.card] at h
refine ⟨⟨n, ?_⟩, hp.1, h⟩
apply Or.resolve_left (Nat.eq_zero_or_pos n)
rintro rfl
rw [pow_zero... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... | Mathlib/RingTheory/WittVector/Defs.lean | 328 | 329 | theorem zsmul_coeff (m : ℤ) (x : 𝕎 R) (n : ℕ) :
(m • x).coeff n = peval (wittZSMul p m n) ![x.coeff] := by | |
/-
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.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | Mathlib/Algebra/Order/ToIntervalMod.lean | 372 | 374 | theorem toIocMod_zsmul_add' (a b : α) (m : ℤ) :
toIocMod hp (m • p + a) b = m • p + toIocMod hp a b := by | rw [add_comm, toIocMod_add_zsmul', add_comm] |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Defs
/-!
# Theorems about invertible elements in rings
-/
universe u
variable {R : Type u}
/... | Mathlib/Algebra/Ring/Invertible.lean | 44 | 46 | theorem invOf_sub_invOf [Ring R] (a b : R) [Invertible a] [Invertible b] :
⅟a - ⅟b = ⅟a * (b - a) * ⅟b := by | rw [mul_sub, invOf_mul_self, sub_mul, one_mul, mul_assoc, mul_invOf_self, mul_one] |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 326 | 330 | theorem mem_iff_lt_colLen {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ ↔ i < μ.colLen j := by | rw [← rowLen_transpose, ← mem_iff_lt_rowLen]
simp
theorem col_eq_prod {μ : YoungDiagram} {j : ℕ} : μ.col j = Finset.range (μ.colLen j) ×ˢ {j} := by |
/-
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.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 335 | 338 | theorem condExp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → E}
(hf : ∀ i ∈ s, Integrable (f i) μ) (m : MeasurableSpace α) :
μ[∑ i ∈ s, f i|m] =ᵐ[μ] ∑ i ∈ s, μ[f i|m] := by | classical |
/-
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.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... | Mathlib/RingTheory/HahnSeries/Basic.lean | 452 | 463 | theorem embDomain_single {f : Γ ↪o Γ'} {g : Γ} {r : R} :
embDomain f (single g r) = single (f g) r := by | ext g'
by_cases h : g' = f g
· simp [h]
rw [embDomain_notin_image_support, coeff_single_of_ne h]
by_cases hr : r = 0
· simp [hr]
rwa [support_single_of_ne hr, Set.image_singleton, Set.mem_singleton_iff]
theorem embDomain_injective {f : Γ ↪o Γ'} :
Function.Injective (embDomain f : HahnSeries Γ R → HahnS... |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 111 | 112 | theorem card_opow_le (a b : Ordinal) : (a ^ b).card ≤ max ℵ₀ (max a.card b.card) := by | obtain ⟨n, rfl⟩ | ha := eq_nat_or_omega0_le a |
/-
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.Order.Filter.CountableInter
/-!
# Filters with countable intersections and countable separating families
In this file we prove some facts about a f... | Mathlib/Order/Filter/CountableSeparatingOn.lean | 180 | 186 | theorem exists_singleton_mem_of_mem_of_forall_separating [Nonempty α] (p : Set α → Prop)
{s : Set α} [HasCountableSeparatingOn α p s] (hs : s ∈ l) (hl : ∀ U, p U → U ∈ l ∨ Uᶜ ∈ l) :
∃ a, {a} ∈ l := by | rcases s.eq_empty_or_nonempty with rfl | hne
· exact ‹Nonempty α›.elim fun a ↦ ⟨a, mem_of_superset hs (empty_subset _)⟩
· exact (exists_mem_singleton_mem_of_mem_of_nonempty_of_forall_separating p hs hne hl).imp fun _ ↦
And.right |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 136 | 138 | theorem mod_zero (a : R) : a % 0 = a := by | simpa only [zero_mul, zero_add] using div_add_mod a 0
theorem lt_one (a : R) : a ≺ (1 : R) → a = 0 := |
/-
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.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
/-!
# More basic logic properties
A few more logic le... | Mathlib/Logic/Lemmas.lean | 22 | 22 | theorem iff_left_comm {a b c : Prop} : (a ↔ (b ↔ c)) ↔ (b ↔ (a ↔ c)) := by | tauto |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 386 | 387 | theorem natCast_eq_last (n) : (n : Fin (n + 1)) = Fin.last n := by | ext; simp |
/-
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.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 326 | 332 | theorem ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim (hm : m ≤ m0) {f : α → E}
(hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf : FinStronglyMeasurable f (μ.trim hm)) : f =ᵐ[μ] 0 := ... | obtain ⟨t, ht_meas, htf_zero, htμ⟩ := hf.exists_set_sigmaFinite
haveI : SigmaFinite ((μ.restrict t).trim hm) := by rwa [restrict_trim hm μ ht_meas] at htμ
have htf_zero : f =ᵐ[μ.restrict tᶜ] 0 := by |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 251 | 266 | theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
(f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
(r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
Pairwise (Disjoint on t) ∧
... | rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with
⟨t, A, t_closed, st, t_approx, ht⟩
refine
⟨disjointed t, A, disjoint_disjointed _,
MeasurableSet.disjointed fun n => (t_closed n).measurableSet, ?_, ?_, ht⟩
· rw [iUnion_disjointed]; exact st
· intro n; exact... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 815 | 815 | theorem castNum_shiftLeft (m : Num) (n : Nat) : ↑(m <<< n) = (m : ℕ) <<< (n : ℕ) := by | |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.Multilinear.Basic
/-!
# Images of (von Neumann) bounded sets under continuous ... | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | 44 | 83 | theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by
classical
if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then
exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩
else
let _ : NontriviallyNormed... | have hfV : f ⁻¹' V ∈ 𝓝 0 := (map_continuous f).tendsto' _ _ f.map_zero hV
rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV
have : ∀ i, ∃ c : 𝕜, c ≠ 0 ∧ ∀ c' : 𝕜, ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i := fun i ↦ by
rw [isVonNBounded_pi_iff] at hs
have := (hs i).tendsto_smallSets_nhds... |
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
import Mathlib.CategoryTheory.Limits.Types.Colimits
/-!
# Concrete description of (co)limits in functor cate... | Mathlib/CategoryTheory/Limits/FunctorToTypes.lean | 25 | 29 | theorem jointly_surjective (k : K) {t : Cocone F} (h : IsColimit t) (x : t.pt.obj k)
[∀ k, HasColimit (F.flip.obj k)] : ∃ j y, x = (t.ι.app j).app k y := by | let hev := isColimitOfPreserves ((evaluation _ _).obj k) h
obtain ⟨j, y, rfl⟩ := Types.jointly_surjective _ hev x
exact ⟨j, y, by simp⟩ |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | Mathlib/Algebra/Polynomial/Monic.lean | 58 | 62 | theorem Monic.map [Semiring S] (f : R →+* S) (hp : Monic p) : Monic (p.map f) := by | unfold Monic
nontriviality
have : f p.leadingCoeff ≠ 0 := by
rw [show _ = _ from hp, f.map_one] |
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
/-!
# Minimal... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 114 | 118 | theorem _root_.IsIntegrallyClosed.minpoly.unique {s : S} {P : R[X]} (hmo : P.Monic)
(hP : Polynomial.aeval s P = 0)
(Pmin : ∀ Q : R[X], Q.Monic → Polynomial.aeval s Q = 0 → degree P ≤ degree Q) :
P = minpoly R s := by | have hs : IsIntegral R s := ⟨P, hmo, hP⟩ |
/-
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.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,218 | 1,220 | theorem deleteVerts_adj {u v : V} :
(G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by | simp [and_assoc] |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# A collection of specific asymptotic results
This fil... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 56 | 60 | theorem tendsto_pow_div_pow_atTop_zero [TopologicalSpace 𝕜] [OrderTopology 𝕜] {p q : ℕ}
(hpq : p < q) : Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop (𝓝 0) := by | rw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]
apply tendsto_zpow_atTop_zero
omega |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 904 | 912 | theorem IsBasis.isBasis_subset (hI : M.IsBasis I X) (hIY : I ⊆ Y) (hYX : Y ⊆ X) :
M.IsBasis I Y := by | rw [isBasis_iff (hYX.trans hI.subset_ground), and_iff_right hI.indep, and_iff_right hIY]
exact fun J hJ hIJ hJY ↦ hI.eq_of_subset_indep hJ hIJ (hJY.trans hYX)
@[simp] theorem isBasis_self_iff_indep : M.IsBasis I I ↔ M.Indep I := by
rw [isBasis_iff', and_iff_right rfl.subset, and_assoc, and_iff_left_iff_imp]
exac... |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 69 | 71 | theorem thickenedIndicatorAux_closure_eq (δ : ℝ) (E : Set α) :
thickenedIndicatorAux δ (closure E) = thickenedIndicatorAux δ E := by | simp +unfoldPartialApp only [thickenedIndicatorAux, infEdist_closure] |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | Mathlib/Probability/Kernel/CondDistrib.lean | 88 | 93 | theorem _root_.MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff
(hY : AEMeasurable Y μ) (hf : AEStronglyMeasurable f (μ.map fun a => (X a, Y a))) :
(∀ᵐ a ∂μ.map X, Integrable (fun ω => f (a, ω)) (condDistrib Y X μ a)) ∧
Integrable (fun a => ∫ ω, ‖f (a, ω)‖ ∂condDistrib Y X μ a) (μ.map ... | rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prodMk₀ hY] |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
/-!
# Bounded monotone sequences converge
In this file we prove a few theorems of the form “if the range of a... | Mathlib/Topology/Order/MonotoneConvergence.lean | 117 | 118 | theorem tendsto_atBot_ciSup (h_anti : Antitone f) (hbdd : BddAbove <| range f) :
Tendsto f atBot (𝓝 (⨆ i, f i)) := by | convert tendsto_atTop_ciSup h_anti.dual hbdd.dual using 1 |
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 419 | 422 | theorem isTheta_exp_comp_one {f : α → ℝ} :
(fun x => exp (f x)) =Θ[l] (fun _ => 1 : α → ℝ) ↔ IsBoundedUnder (· ≤ ·) l fun x => |f x| := by | simp only [← exp_zero, isTheta_exp_comp_exp_comp, sub_zero] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | Mathlib/Data/ZMod/Basic.lean | 536 | 539 | theorem cast_sub_one {R : Type*} [Ring R] {n : ℕ} (k : ZMod n) :
(cast (k - 1 : ZMod n) : R) = (if k = 0 then (n : R) else cast k) - 1 := by | split_ifs with hk
· rw [hk, zero_sub, ZMod.cast_neg_one] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | Mathlib/NumberTheory/ArithmeticFunction.lean | 861 | 865 | theorem isMultiplicative_pow {k : ℕ} : IsMultiplicative (pow k) :=
isMultiplicative_id.ppow
@[arith_mult]
theorem isMultiplicative_sigma {k : ℕ} : IsMultiplicative (σ k) := by | |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Polynomials supported by a set of variables
This file contains the definition and lemmas about `MvPolynomial.support... | Mathlib/Algebra/MvPolynomial/Supported.lean | 102 | 103 | theorem supported_le_supported_iff [Nontrivial R] : supported R s ≤ supported R t ↔ s ⊆ t := by | constructor |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.SelfAdjoint
/-... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 806 | 807 | theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} :
(↑(⨅ i, S i) : Set A) = ⋂ i, S i := by | simp [iInf] |
/-
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
/-!
# Projection from cardinal numbers to natural numbers
In this file we define `Cardinal.toNat` to be the natural projectio... | Mathlib/SetTheory/Cardinal/ToNat.lean | 155 | 156 | 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 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 239 | 265 | theorem HD_below_aux1 {f : Cb X Y} (C : ℝ) {x : X} :
BddBelow (range fun y : Y => f (inl x, inr y) + C) :=
let ⟨cf, hcf⟩ := f.isBounded_range.bddBelow
⟨cf + C, forall_mem_range.2 fun _ => add_le_add_right ((fun x => hcf (mem_range_self x)) _) _⟩
private theorem HD_bound_aux1 [Nonempty Y] (f : Cb X Y) (C : ℝ) :... | obtain ⟨Cf, hCf⟩ := f.isBounded_range.bddAbove
refine ⟨Cf + C, forall_mem_range.2 fun x => ?_⟩
calc
⨅ y, f (inl x, inr y) + C ≤ f (inl x, inr default) + C := ciInf_le (HD_below_aux1 C) default
_ ≤ Cf + C := add_le_add ((fun x => hCf (mem_range_self x)) _) le_rfl
theorem HD_below_aux2 {f : Cb X Y} (C : ℝ) {... |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 375 | 378 | theorem iInf_apply {ι} [Nonempty ι] (m : ι → OuterMeasure α) (s : Set α) :
(⨅ i, m i) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ i, m i (t n) := by | rw [iInf, sInf_apply (range_nonempty m)]
simp only [iInf_range] |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | Mathlib/Algebra/Polynomial/Coeff.lean | 222 | 227 | theorem coeff_mul_X_pow (p : R[X]) (n d : ℕ) :
coeff (p * Polynomial.X ^ n) (d + n) = coeff p d := by | rw [coeff_mul, Finset.sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one]
· rintro ⟨i, j⟩ h1 h2
rw [coeff_X_pow, if_neg, mul_zero]
rintro rfl |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 251 | 254 | theorem integrable_cexp_neg_mul_sum_add {ι : Type*} [Fintype ι] (hb : 0 < b.re) (c : ι → ℂ) :
Integrable (fun (v : ι → ℝ) ↦ cexp (- b * ∑ i, (v i : ℂ) ^ 2 + ∑ i, c i * v i)) := by | simp_rw [neg_mul, Finset.mul_sum]
exact integrable_cexp_neg_sum_mul_add (fun _ ↦ hb) c |
/-
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.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,101 | 1,104 | theorem deleteEdges_inter_edgeSet_right_eq :
G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by | ext <;> simp +contextual [imp_false] |
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Tactic.FieldSimp
import Mathli... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 190 | 203 | theorem ruzsa_triangle_inequality_mul_div_div (A B C : Finset G) :
#(A * C) * #B ≤ #(A / B) * #(B / C) := by | rw [div_eq_mul_inv, ← card_inv B, ← card_inv (B / C), inv_div', div_inv_eq_mul]
exact ruzsa_triangle_inequality_mul_mul_mul _ _ _
/-- **Ruzsa's triangle inequality**. Div-mul-div version. -/
@[to_additive "**Ruzsa's triangle inequality**. Sub-add-sub version."]
theorem ruzsa_triangle_inequality_div_mul_div (A B C : ... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 26 | 28 | theorem uniqueMDiffWithinAt_iff_uniqueDiffWithinAt :
UniqueMDiffWithinAt 𝓘(𝕜, E) s x ↔ UniqueDiffWithinAt 𝕜 s x := by | simp only [UniqueMDiffWithinAt, mfld_simps] |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 658 | 660 | theorem tensor_inv_hom_id {V W X Y Z : C} (f : V ≅ W) (g : X ⟶ Y) (h : Y ⟶ Z) :
(g ⊗ f.inv) ≫ (h ⊗ f.hom) = (g ⊗ 𝟙 W) ≫ (h ⊗ 𝟙 W) := by | rw [← tensor_comp, f.inv_hom_id]; simp [tensorHom_id] |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.Pointwise
/-!
# Submonoid of inverses
Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid ... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 202 | 203 | theorem leftInvEquiv_symm_eq_inv (x : S) : ((S.leftInvEquiv hS).symm x : M) = (x : M)⁻¹ := by | rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm] |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | Mathlib/Algebra/Polynomial/EraseLead.lean | 147 | 152 | theorem eraseLead_C_mul_X_pow (r : R) (n : ℕ) : eraseLead (C r * X ^ n) = 0 := by | rw [C_mul_X_pow_eq_monomial, eraseLead_monomial]
@[simp] lemma eraseLead_C_mul_X (r : R) : eraseLead (C r * X) = 0 := by
simpa using eraseLead_C_mul_X_pow _ 1 |
/-
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.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 633 | 636 | theorem isPartition_iff_iUnion_eq {π : Prepartition I} : π.IsPartition ↔ π.iUnion = I := by | simp_rw [IsPartition, Set.Subset.antisymm_iff, π.iUnion_subset, true_and, Set.subset_def,
mem_iUnion, Box.mem_coe] |
/-
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.Data.Set.Image
import Mathlib.Data.List.Defs
/-!
# Lemmas about `List`s and `Set.range`
In this file we prove lemmas about range of some operations... | Mathlib/Data/Set/List.lean | 52 | 57 | theorem range_list_getD (d : α) : (range fun n : Nat => l[n]?.getD d) = insert d { x | x ∈ l } :=
calc
(range fun n => l[n]?.getD d) = (fun o : Option α => o.getD d) '' range (l[·]?) := by | simp only [← range_comp, Function.comp_def]
rfl
_ = insert d { x | x ∈ l } := by |
/-
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.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 436 | 443 | theorem mem_restrict' : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : Set (ι → ℝ)) = ↑J ∩ ↑J' := by | simp only [mem_restrict, ← Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe]
@[mono]
theorem restrict_mono {π₁ π₂ : Prepartition I} (Hle : π₁ ≤ π₂) : π₁.restrict J ≤ π₂.restrict J := by
classical
refine ofWithBot_mono fun J₁ hJ₁ hne => ?_
rw [Finset.mem_image] at hJ₁; rcases hJ₁ with ⟨J₁, hJ₁, rfl⟩ |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | Mathlib/LinearAlgebra/Prod.lean | 147 | 154 | theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) := by | ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.snd, Prod.ext h.symm 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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 91 | 96 | theorem cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.cos (o.oangle (x + y) y) = ‖y‖ / ‖x + y‖ := by | rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).cos_oangle_add_right_of_oangle_eq_pi_div_two h |
/-
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 | 158 | 174 | theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
{t : L[[α]].Term β} {v : β → M} :
t.constantsToVars.realize (Sum.elim (fun a => ↑(L.con a)) v) = t.realize v := by | induction t with
| var => simp
| @func n f ts ih =>
cases n
· cases f
· simp only [realize, ih, constantsOn, constantsOnFunc, constantsToVars]
-- Porting note: below lemma does not work with simp for some reason
rw [withConstants_funMap_sumInl]
· simp only [realize, constantsToVa... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | Mathlib/Algebra/Divisibility/Units.lean | 219 | 222 | theorem IsRelPrime.mul_dvd_of_left_isPrimal (H : IsRelPrime x y) (H1 : x ∣ z) (H2 : y ∣ z)
(hx : IsPrimal x) : x * y ∣ z := by | rw [mul_comm]; exact H.symm.mul_dvd_of_right_isPrimal H2 H1 hx |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 292 | 296 | theorem tendsto_prod_filter_iff {c : β} :
Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun _ => c) p p' := by | simp_rw [nhds_eq_comap_uniformity, tendsto_comap_iff]
rfl |
/-
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 Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 865 | 867 | theorem injective_foldl_comp {l : List (α → α)} {f : α → α}
(hl : ∀ f ∈ l, Function.Injective f) (hf : Function.Injective f) :
Function.Injective (@List.foldl (α → α) (α → α) Function.comp f l) := by | |
/-
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.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 369 | 370 | theorem zero_testAgainstNN : (0 : FiniteMeasure Ω).testAgainstNN = 0 := by | funext |
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