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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 459 | 465 | theorem integral_sub_average (μ : Measure α) [IsFiniteMeasure μ] (f : α → E) :
∫ x, f x - ⨍ a, f a ∂μ ∂μ = 0 := by |
by_cases hf : Integrable f μ
· rw [integral_sub hf (integrable_const _), integral_average, sub_self]
refine integral_undef fun h => hf ?_
convert h.add (integrable_const (⨍ a, f a ∂μ))
exact (sub_add_cancel _ _).symm
|
/-
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
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 71 | 73 | theorem Homeomorph.mulLeft_symm (a : G) : (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹ := by |
ext
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.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,318 | 1,334 | theorem Valid'.balance'_lemma {α l l' r r'} (H1 : BalancedSz l' r')
(H2 : Nat.dist (@size α l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l') :
2 * @size α r ≤ 9 * size l + 5 ∨ size r ≤ 3 := by |
suffices @size α r ≤ 3 * (size l + 1) by
rcases Nat.eq_zero_or_pos (size l) with l0 | l0
· apply Or.inr; rwa [l0] at this
change 1 ≤ _ at l0; apply Or.inl; omega
rcases H2 with (⟨hl, rfl⟩ | ⟨hr, rfl⟩) <;> rcases H1 with (h | ⟨_, h₂⟩)
· exact le_trans (Nat.le_add_left _ _) (le_trans h (Nat.le_add_left... |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 111 | 113 | theorem length_mul_ge_length_sub_length' (w₁ w₂ : W) :
ℓ w₂ - ℓ w₁ ≤ ℓ (w₁ * w₂) := by |
simpa [Nat.sub_le_of_le_add, add_comm] using cs.length_mul_le w₁⁻¹ (w₁ * w₂)
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 301 | 307 | theorem map_nhds_eq (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse) {x : E}
(hs : s ∈ 𝓝 x) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : map f (𝓝 x) = 𝓝 (f x) := by |
refine
le_antisymm ((hf.continuousOn x (mem_of_mem_nhds hs)).continuousAt hs) (le_map fun t ht => ?_)
have : f '' (s ∩ t) ∈ 𝓝 (f x) :=
(hf.mono_set inter_subset_left).image_mem_nhds f'symm (inter_mem hs ht) hc
exact mem_of_superset this (image_subset _ inter_subset_right)
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Contraction.lean | 277 | 279 | theorem lTensorHomEquivHomLTensor_apply (x : P ⊗[R] (M →ₗ[R] Q)) :
lTensorHomEquivHomLTensor R M P Q x = lTensorHomToHomLTensor R M P Q x := by |
rw [← LinearEquiv.coe_toLinearMap, lTensorHomEquivHomLTensor_toLinearMap]
|
/-
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.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 964 | 966 | theorem diam_triple : diam ({x, y, z} : Set α) = max (max (edist x y) (edist x z)) (edist y z) := by |
simp only [diam_insert, iSup_insert, iSup_singleton, diam_singleton, ENNReal.max_zero_right,
ENNReal.sup_eq_max]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 2,386 | 2,387 | theorem toSubgraph_cons_nil_eq_subgraphOfAdj (h : G.Adj u v) :
(cons h nil).toSubgraph = G.subgraphOfAdj h := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set α` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 196 | 210 | theorem inducedOuterMeasure_eq_iInf (s : Set α) :
inducedOuterMeasure m P0 m0 s = ⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht := by |
apply le_antisymm
· simp only [le_iInf_iff]
intro t ht hs
refine le_trans (measure_mono hs) ?_
exact le_of_eq (inducedOuterMeasure_eq' _ msU m_mono _)
· refine le_iInf ?_
intro f
refine le_iInf ?_
intro hf
refine le_trans ?_ (extend_iUnion_le_tsum_nat' _ msU _)
refine le_iInf ?_
... |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.Restrict
#align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bcea... | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 48 | 52 | theorem mk {s t : Set α} (hs : μ s = 0) (ht : ν t = 0) (hst : univ ⊆ s ∪ t) :
MutuallySingular μ ν := by |
use toMeasurable μ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs
refine measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx ?_) ht
exact subset_toMeasurable _ _ hxs
|
/-
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 | 369 | 372 | theorem finAddFlip_apply_mk_left {k : ℕ} (h : k < m) (hk : k < m + n := Nat.lt_add_right n h)
(hnk : n + k < n + m := Nat.add_lt_add_left h n) :
finAddFlip (⟨k, hk⟩ : Fin (m + n)) = ⟨n + k, hnk⟩ := by |
convert finAddFlip_apply_castAdd ⟨k, h⟩ n
|
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variable {α : Type*}
namespace List
... | Mathlib/Data/List/Iterate.lean | 48 | 52 | theorem iterate_add (f : α → α) (a : α) (m n : ℕ) :
iterate f a (m + n) = iterate f a m ++ iterate f (f^[m] a) n := by |
induction m generalizing a with
| zero => simp
| succ n ih => rw [iterate, add_right_comm, iterate, ih, Nat.iterate, cons_append]
|
/-
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.Algebra.CharP.Invertible
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomia... | Mathlib/RingTheory/Polynomial/Dickson.lean | 196 | 262 | theorem dickson_one_one_zmod_p (p : ℕ) [Fact p.Prime] : dickson 1 (1 : ZMod p) p = X ^ p := by |
-- Recall that `dickson_one_one_eval_add_inv` characterises `dickson 1 1 p`
-- as a polynomial that maps `x + x⁻¹` to `x ^ p + (x⁻¹) ^ p`.
-- Since `X ^ p` also satisfies this property in characteristic `p`,
-- we can use a variant on `Polynomial.funext` to conclude that these polynomials are equal.
-- For t... |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Bitraversable.Basic
#align_import control.bitraversable.lemmas from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
/-!
#... | Mathlib/Control/Bitraversable/Lemmas.lean | 110 | 112 | theorem tfst_eq_fst_id {α α' β} (f : α → α') (x : t α β) :
tfst (F := Id) (pure ∘ f) x = pure (fst f x) := by |
apply bitraverse_eq_bimap_id
|
/-
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.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 325 | 331 | theorem support_finset_sum [DecidableEq β] [AddCommMonoid M] {s : Finset α} {f : α → β →₀ M} :
(Finset.sum s f).support ⊆ s.biUnion fun x => (f x).support := by |
rw [← Finset.sup_eq_biUnion]
induction' s using Finset.cons_induction_on with a s ha ih
· rfl
· rw [Finset.sum_cons, Finset.sup_cons]
exact support_add.trans (Finset.union_subset_union (Finset.Subset.refl _) ih)
|
/-
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 | 96 | 97 | theorem T_sub_one (n : ℤ) : T R (n - 1) = 2 * X * T R n - T R (n + 1) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 1)
|
/-
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.Topology.Algebra.Ring.Ideal
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed_space.units from "leanprover-community/mathl... | Mathlib/Analysis/NormedSpace/Units.lean | 159 | 174 | theorem inverse_one_sub_norm : (fun t : R => inverse (1 - t)) =O[𝓝 0] (fun _t => 1 : R → ℝ) := by |
simp only [IsBigO, IsBigOWith, Metric.eventually_nhds_iff]
refine ⟨‖(1 : R)‖ + 1, (2 : ℝ)⁻¹, by norm_num, fun t ht ↦ ?_⟩
rw [dist_zero_right] at ht
have ht' : ‖t‖ < 1 := by
have : (2 : ℝ)⁻¹ < 1 := by cancel_denoms
linarith
simp only [inverse_one_sub t ht', norm_one, mul_one, Set.mem_setOf_eq]
chang... |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,470 | 1,470 | theorem range_incl : N.incl.range = N := by | simp [← LieSubmodule.coe_toSubmodule_eq_iff]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"... | Mathlib/Data/Sum/Basic.lean | 213 | 213 | theorem isLeft_left (h : LiftRel r s x (inl c)) : x.isLeft := by | cases h; rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 481 | 483 | theorem ne_zero_of_mem_factors {p a : α} (h : p ∈ factors a) : a ≠ 0 := by |
rintro rfl
simp at h
|
/-
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.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 53 | 56 | theorem MellinConvergent.cpow_smul {f : ℝ → E} {s a : ℂ} :
MellinConvergent (fun t => (t : ℂ) ^ a • f t) s ↔ MellinConvergent f (s + a) := by |
refine integrableOn_congr_fun (fun t ht => ?_) measurableSet_Ioi
simp_rw [← sub_add_eq_add_sub, cpow_add _ _ (ofReal_ne_zero.2 <| ne_of_gt ht), mul_smul]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLat... | Mathlib/Algebra/Tropical/BigOperators.lean | 70 | 74 | theorem untrop_prod [AddCommMonoid R] (s : Finset S) (f : S → Tropical R) :
untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i) := by |
convert Multiset.untrop_prod (s.val.map f)
simp only [Multiset.map_map, Function.comp_apply]
rfl
|
/-
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 | 66 | 71 | theorem preimage_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R ⁻¹' sphere y (dist y c) =
insert c (perpBisector c (inversion c R y) : Set P) := by |
ext x
rcases eq_or_ne x c with rfl | hx; · simp [dist_comm]
rw [mem_preimage, mem_sphere, ← inversion_mem_perpBisector_inversion_iff hR] <;> simp [*]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 276 | 283 | theorem Memℒp.locallyIntegrable [IsLocallyFiniteMeasure μ] {f : X → E} {p : ℝ≥0∞}
(hf : Memℒp f p μ) (hp : 1 ≤ p) : LocallyIntegrable f μ := by |
intro x
rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
have : Fact (μ U < ⊤) := ⟨h'U⟩
refine ⟨U, hU, ?_⟩
rw [IntegrableOn, ← memℒp_one_iff_integrable]
apply (hf.restrict U).memℒp_of_exponent_le hp
|
/-
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.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 783 | 791 | theorem raised_iff {n m} : Raised n m ↔ n ≤ m ∧ m ≤ n + 1 := by |
constructor
· rintro (rfl | rfl)
· exact ⟨le_rfl, Nat.le_succ _⟩
· exact ⟨Nat.le_succ _, le_rfl⟩
· rintro ⟨h₁, h₂⟩
rcases eq_or_lt_of_le h₁ with (rfl | h₁)
· exact Or.inl rfl
· exact Or.inr (le_antisymm h₂ h₁)
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 238 | 240 | theorem fixedBy_eq_univ_iff_eq_one {m : M} : fixedBy α m = Set.univ ↔ m = 1 := by |
rw [← (smul_left_injective' (M := M) (α := α)).eq_iff, Set.eq_univ_iff_forall]
simp_rw [Function.funext_iff, one_smul, mem_fixedBy]
|
/-
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, Scott Morrison
-/
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathli... | Mathlib/Data/List/Range.lean | 87 | 90 | theorem take_range (m n : ℕ) : take m (range n) = range (min m n) := by |
apply List.ext_get
· simp
· simp (config := { contextual := true }) [← get_take, Nat.lt_min]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 84 | 89 | theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) :
ContDiffAt ℝ n (fun x : ℝ => (e x).re) z := by |
have A : ContDiffAt ℝ n ((↑) : ℝ → ℂ) z := ofRealCLM.contDiff.contDiffAt
have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ
have C : ContDiffAt ℝ n re (e z) := reCLM.contDiff.contDiffAt
exact C.comp z (B.comp z A)
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba5... | Mathlib/Order/Bounded.lean | 322 | 325 | theorem unbounded_le_inter_not_le [SemilatticeSup α] (a : α) :
Unbounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Unbounded (· ≤ ·) s := by |
rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]
exact bounded_le_inter_not_le a
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 490 | 493 | theorem det_updateColumn_add_smul_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) (c : R) :
det (updateColumn A i fun k => A k i + c • A k j) = det A := by |
rw [← det_transpose, ← updateRow_transpose, ← det_transpose A]
exact det_updateRow_add_smul_self Aᵀ hij c
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,453 | 1,462 | theorem integral_smul_const {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [CompleteSpace E]
(f : X → 𝕜) (c : E) :
∫ x, f x • c ∂μ = (∫ x, f x ∂μ) • c := by |
by_cases hf : Integrable f μ
· exact ((1 : 𝕜 →L[𝕜] 𝕜).smulRight c).integral_comp_comm hf
· by_cases hc : c = 0
· simp [hc, integral_zero, smul_zero]
rw [integral_undef hf, integral_undef, zero_smul]
rw [integrable_smul_const hc]
simp_rw [hf, not_false_eq_true]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Uni Marx
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
i... | Mathlib/CategoryTheory/Category/RelCat.lean | 93 | 122 | theorem rel_iso_iff {X Y : RelCat} (r : X ⟶ Y) :
IsIso (C := RelCat) r ↔ ∃ f : (Iso (C := Type) X Y), graphFunctor.map f.hom = r := by |
constructor
· intro h
have h1 := congr_fun₂ h.hom_inv_id
have h2 := congr_fun₂ h.inv_hom_id
simp only [RelCat.Hom.rel_comp_apply₂, RelCat.Hom.rel_id_apply₂, eq_iff_iff] at h1 h2
obtain ⟨f, hf⟩ := Classical.axiomOfChoice (fun a => (h1 a a).mpr rfl)
obtain ⟨g, hg⟩ := Classical.axiomOfChoice (fun ... |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 112 | 117 | theorem mul_of'_divOf (x : k[G]) (a : G) : x * of' k G a /ᵒᶠ a = x := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
rw [AddMonoidAlgebra.divOf_apply, of'_apply, mul_single_apply_aux, mul_one]
intro c
rw [add_comm]
exact add_right_inj _
|
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,503 | 1,504 | theorem lintegral_dirac' (a : α) {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂dirac a = f a := by |
simp [lintegral_congr_ae (ae_eq_dirac' hf)]
|
/-
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.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.No... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 845 | 854 | theorem AffineMap.continuous_linear_iff {f : P →ᵃ[𝕜] P₂} : Continuous f.linear ↔ Continuous f := by |
inhabit P
have :
(f.linear : V → V₂) =
(AffineIsometryEquiv.vaddConst 𝕜 <| f default).toHomeomorph.symm ∘
f ∘ (AffineIsometryEquiv.vaddConst 𝕜 default).toHomeomorph := by
ext v
simp
rw [this]
simp only [Homeomorph.comp_continuous_iff, Homeomorph.comp_continuous_iff']
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
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 | 1,731 | 1,733 | theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 := by |
have := Real.sin_sq_add_cos_sq x
simp_all [add_comm, abs, normSq, sq, sin_ofReal_re, cos_ofReal_re, mul_re]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 955 | 961 | theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t := by |
induction' n with n ihn generalizing x
· rfl
· refine (eventually_nhds_nhdsWithin.2 h).mono fun y hy => ?_
simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]
rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 102 | 105 | theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by |
unfold partialSum -- Porting note: added
continuity
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 140 | 164 | theorem Basis.le_span {J : Set M} (v : Basis ι R M) (hJ : span R J = ⊤) : #(range v) ≤ #J := by |
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite J
· rw [← Cardinal.lift_le, Cardinal.mk_range_eq_of_injective v.injective, Cardinal.mk_fintype J]
convert Cardinal.lift_le.{v}.2 (basis_le_span' v hJ)
simp
· let S : J → Set ι := fun j => ↑(v.repr j).support
let S' : J → Set M :=... |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 54 | 57 | theorem norm_image_of_norm_zero [SemilinearMapClass 𝓕 σ₁₂ E F] (f : 𝓕) (hf : Continuous f) {x : E}
(hx : ‖x‖ = 0) : ‖f x‖ = 0 := by |
rw [← mem_closure_zero_iff_norm, ← specializes_iff_mem_closure, ← map_zero f] at *
exact hx.map hf
|
/-
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, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_... | Mathlib/Topology/FiberBundle/Basic.lean | 801 | 808 | theorem isOpen_target_of_mem_pretrivializationAtlas_inter (e e' : Pretrivialization F (π F E))
(he' : e' ∈ a.pretrivializationAtlas) :
IsOpen (e'.toPartialEquiv.target ∩ e'.toPartialEquiv.symm ⁻¹' e.source) := by |
letI := a.totalSpaceTopology
obtain ⟨u, hu1, hu2⟩ := continuousOn_iff'.mp (a.continuous_symm_of_mem_pretrivializationAtlas he')
e.source (a.isOpen_source e)
rw [inter_comm, hu2]
exact hu1.inter e'.open_target
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.r... | Mathlib/Data/ENNReal/Operations.lean | 535 | 541 | theorem sum_lt_sum_of_nonempty {s : Finset α} (hs : s.Nonempty) {f g : α → ℝ≥0∞}
(Hlt : ∀ i ∈ s, f i < g i) : ∑ i ∈ s, f i < ∑ i ∈ s, g i := by |
induction hs using Finset.Nonempty.cons_induction with
| singleton => simp [Hlt _ (Finset.mem_singleton_self _)]
| cons _ _ _ _ ih =>
simp only [Finset.sum_cons, forall_mem_cons] at Hlt ⊢
exact ENNReal.add_lt_add Hlt.1 (ih Hlt.2)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,403 | 1,406 | theorem _root_.Function.invFunOn_injOn_image [Nonempty α] (f : α → β) (s : Set α) :
Set.InjOn (invFunOn f s) (f '' s) := by |
rintro _ ⟨x, hx, rfl⟩ _ ⟨x', hx', rfl⟩ he
rw [← invFunOn_apply_eq (f := f) hx, he, invFunOn_apply_eq (f := f) hx']
|
/-
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
-/
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c73... | Mathlib/Topology/Maps.lean | 627 | 629 | theorem isOpenMap_iff (hg : OpenEmbedding g) :
IsOpenMap f ↔ IsOpenMap (g ∘ f) := by |
simp_rw [isOpenMap_iff_nhds_le, ← map_map, comp, ← hg.map_nhds_eq, Filter.map_le_map_iff hg.inj]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,928 | 2,935 | theorem Rel.countP_eq (r : α → α → Prop) [IsTrans α r] [IsSymm α r] {s t : Multiset α} (x : α)
[DecidablePred (r x)] (h : Rel r s t) : countP (r x) s = countP (r x) t := by |
induction' s using Multiset.induction_on with y s ih generalizing t
· rw [rel_zero_left.mp h]
· obtain ⟨b, bs, hb1, hb2, rfl⟩ := rel_cons_left.mp h
rw [countP_cons, countP_cons, ih hb2]
simp only [decide_eq_true_eq, Nat.add_right_inj]
exact (if_congr ⟨fun h => _root_.trans h hb1, fun h => _root_.tran... |
/-
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
-/
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Isomorphisms.lean | 88 | 93 | theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
Function.Surjective (quotientInfToSupQuotient p p') := by |
rw [← range_eq_top, quotientInfToSupQuotient, range_liftQ, eq_top_iff']
rintro ⟨x, hx⟩; rcases mem_sup.1 hx with ⟨y, hy, z, hz, rfl⟩
use ⟨y, hy⟩; apply (Submodule.Quotient.eq _).2
simp only [mem_comap, map_sub, coeSubtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Contraction.lean | 113 | 118 | theorem map_dualTensorHom (f : Module.Dual R M) (p : P) (g : Module.Dual R N) (q : Q) :
TensorProduct.map (dualTensorHom R M P (f ⊗ₜ[R] p)) (dualTensorHom R N Q (g ⊗ₜ[R] q)) =
dualTensorHom R (M ⊗[R] N) (P ⊗[R] Q) (dualDistrib R M N (f ⊗ₜ g) ⊗ₜ[R] p ⊗ₜ[R] q) := by |
ext m n
simp only [compr₂_apply, mk_apply, map_tmul, dualTensorHom_apply, dualDistrib_apply, ←
smul_tmul_smul]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Directed
import Mathlib.Logic.Equiv.Set
#align_import order.complete_boolean_algebra from "le... | Mathlib/Order/CompleteBooleanAlgebra.lean | 229 | 230 | theorem disjoint_iSup_iff {f : ι → α} : Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) := by |
simpa only [disjoint_comm] using @iSup_disjoint_iff
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 155 | 156 | theorem map₂_left [NeBot g] : map₂ (fun x _ => x) f g = f := by |
rw [← map_prod_eq_map₂, map_fst_prod]
|
/-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
impo... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 133 | 141 | theorem cardinal_le_max_transcendence_basis (hv : IsTranscendenceBasis R v) :
#K ≤ max (max #R #ι) ℵ₀ :=
calc
#K ≤ max #(Algebra.adjoin R (Set.range v)) ℵ₀ :=
letI := isAlgClosure_of_transcendence_basis v hv
Algebra.IsAlgebraic.cardinal_mk_le_max _ _
_ = max #(MvPolynomial ι R) ℵ₀ := by | rw [Cardinal.eq.2 ⟨hv.1.aevalEquiv.toEquiv⟩]
_ ≤ max (max (max #R #ι) ℵ₀) ℵ₀ := max_le_max MvPolynomial.cardinal_mk_le_max le_rfl
_ = _ := by simp [max_assoc]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 557 | 559 | theorem Antiperiodic.const_inv_smul₀ [AddCommMonoid α] [Neg β] [DivisionSemiring γ] [Module γ α]
(h : Antiperiodic f c) {a : γ} (ha : a ≠ 0) : Antiperiodic (fun x => f (a⁻¹ • x)) (a • c) := by |
simpa only [inv_inv] using h.const_smul₀ (inv_ne_zero ha)
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594b... | Mathlib/Algebra/Polynomial/BigOperators.lean | 269 | 283 | theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < Multiset.card t) :
(t.map fun x => X - C x).prod.coeff ((Multiset.card t) - 1) = -t.sum := by |
nontriviality R
convert multiset_prod_X_sub_C_nextCoeff (by assumption)
rw [nextCoeff, if_neg]
swap
· rw [natDegree_multiset_prod_of_monic]
swap
· simp only [Multiset.mem_map]
rintro _ ⟨_, _, rfl⟩
apply monic_X_sub_C
simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map]
obtain ⟨x, ... |
/-
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
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,073 | 1,077 | theorem moebius_ne_zero_iff_squarefree {n : ℕ} : μ n ≠ 0 ↔ Squarefree n := by |
constructor <;> intro h
· contrapose! h
simp [h]
· simp [h, pow_ne_zero]
|
/-
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
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 317 | 322 | theorem mul_smul' (f g : ArithmeticFunction R) (h : ArithmeticFunction M) :
(f * g) • h = f • g • h := by |
ext n
simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, Finset.sum_sigma']
apply Finset.sum_nbij' (fun ⟨⟨_i, j⟩, ⟨k, l⟩⟩ ↦ ⟨(k, l * j), (l, j)⟩)
(fun ⟨⟨i, _j⟩, ⟨k, l⟩⟩ ↦ ⟨(i * k, l), (i, k)⟩) <;> aesop (add simp mul_assoc)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 259 | 260 | theorem card_fintypeIio : Fintype.card (Set.Iio b) = b := by |
rw [Fintype.card_ofFinset, card_Iio]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 907 | 910 | theorem sum_centroidWeightsIndicator_eq_one_of_card_ne_zero [CharZero k] [Fintype ι]
(h : card s ≠ 0) : ∑ i, s.centroidWeightsIndicator k i = 1 := by |
rw [sum_centroidWeightsIndicator]
exact s.sum_centroidWeights_eq_one_of_card_ne_zero k h
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 602 | 614 | theorem ContinuousOn.aestronglyMeasurable [TopologicalSpace α] [TopologicalSpace β]
[h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [PseudoMetrizableSpace β]
{f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) :
AEStronglyMeasurable f (μ.restrict s) := by |
borelize β
refine
aestronglyMeasurable_iff_aemeasurable_separable.2
⟨hf.aemeasurable hs, f '' s, ?_,
mem_of_superset (self_mem_ae_restrict hs) (subset_preimage_image _ _)⟩
cases h.out
· rw [image_eq_range]
exact isSeparable_range <| continuousOn_iff_continuous_restrict.1 hf
· exact .of_... |
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 3,148 | 3,156 | theorem length_eraseIdx_add_one {l : List ι} {i : ℕ} (h : i < l.length) :
(l.eraseIdx i).length + 1 = l.length := calc
(l.eraseIdx i).length + 1
_ = (l.take i ++ l.drop (i + 1)).length + 1 := by | rw [eraseIdx_eq_take_drop_succ]
_ = (l.take i).length + (l.drop (i + 1)).length + 1 := by rw [length_append]
_ = i + (l.drop (i + 1)).length + 1 := by rw [length_take_of_le (le_of_lt h)]
_ = i + (l.length - (i + 1)) + 1 := by rw [length_drop]
_ = (i + 1) + (l.length - (i + 1)... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_t... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 160 | 168 | theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) := by |
refine
le_antisymm (iInf_le_of_le a <| iInf₂_le b Subset.rfl)
(le_iInf fun a' => le_iInf fun b' => le_iInf fun h => ENNReal.coe_le_coe.2 ?_)
rcases le_or_lt b a with ab | ab
· rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))]
apply zero_le
cases' (Ioc_subset_Ioc_iff ab).1 h with h₁ h₂
exa... |
/-
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.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Co... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 172 | 176 | theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
AnalyticOn 𝕜 (_root_.deriv^[n] f) s := by |
induction' n with n IH
· exact h
· simpa only [Function.iterate_succ', Function.comp_apply] using IH.deriv
|
/-
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.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathli... | Mathlib/Data/Nat/PSub.lean | 85 | 93 | theorem psub_eq_none {m n : ℕ} : psub m n = none ↔ m < n := by |
cases s : psub m n <;> simp [eq_comm]
· show m < n
refine lt_of_not_ge fun h => ?_
cases' le.dest h with k e
injection s.symm.trans (psub_eq_some.2 <| (add_comm _ _).trans e)
· show n ≤ m
rw [← psub_eq_some.1 s]
apply Nat.le_add_left
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 601 | 603 | theorem le_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z := by |
rw [← rpow_le_rpow_iff_of_neg _ hx hz, rpow_inv_rpow _ hz.ne] <;> positivity
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
/-!
# D... | Mathlib/SetTheory/Game/Domineering.lean | 101 | 106 | theorem card_of_mem_right {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) : 2 ≤ Finset.card b := by |
have w₁ : m ∈ b := (Finset.mem_inter.1 h).1
have w₂ := fst_pred_mem_erase_of_mem_right h
have i₁ := Finset.card_erase_lt_of_mem w₁
have i₂ := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w₂)
exact Nat.lt_of_le_of_lt i₂ i₁
|
/-
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.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "l... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 131 | 136 | theorem ContDiff.euclidean_dist (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) (h : ∀ x, f x ≠ g x) :
ContDiff ℝ n fun x => Euclidean.dist (f x) (g x) := by |
simp only [Euclidean.dist]
apply ContDiff.dist ℝ
exacts [(toEuclidean (E := G)).contDiff.comp hf,
(toEuclidean (E := G)).contDiff.comp hg, fun x => toEuclidean.injective.ne (h x)]
|
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.SingleObj
import Mathlib.GroupTheory.GroupAction.Quotient
impor... | Mathlib/CategoryTheory/Action.lean | 89 | 89 | theorem back_coe (x : ActionCategory M X) : ↑x.back = x := by | cases x; rfl
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.q... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 131 | 133 | theorem mk_ker {I : Ideal R} : ker (Quotient.mk I) = I := by |
ext
rw [ker, mem_comap, Submodule.mem_bot, Quotient.eq_zero_iff_mem]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 334 | 344 | theorem _root_.Wbtw.angle₂₁₃_eq_zero_of_ne {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) (hp₂p₁ : p₂ ≠ p₁) :
∠ p₂ p₁ p₃ = 0 := by |
rw [angle, angle_eq_zero_iff]
rcases h with ⟨r, ⟨hr0, hr1⟩, rfl⟩
have hr0' : r ≠ 0 := by
rintro rfl
simp at hp₂p₁
replace hr0 := hr0.lt_of_ne hr0'.symm
refine ⟨vsub_ne_zero.2 hp₂p₁, r⁻¹, inv_pos.2 hr0, ?_⟩
rw [AffineMap.lineMap_apply, vadd_vsub_assoc, vsub_self, add_zero, smul_smul, inv_mul_cancel ... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 239 | 254 | theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S : Ideal R}
[hpi : IsPrime S] (hS : S ≠ ⊥) : IsMaximal S :=
isMaximal_iff.2
⟨(ne_top_iff_one S).1 hpi.1, by
intro T x hST hxS hxT
cases' (mem_iff_generator_dvd _).1 (hST <| generator_mem S) with z hz
cases hpi.mem_... |
rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
exact (hxS <| hTS hxT).elim
| inr h =>
cases' (mem_iff_generator_dvd _).1 h with y hy
have : generator S ≠ 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS
rw [← mul_one (generator S), hy, mul_left_comm... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra... | Mathlib/LinearAlgebra/Pi.lean | 317 | 320 | theorem iInf_comap_proj :
⨅ i, comap (proj i : ((i : ι) → φ i) →ₗ[R] φ i) (p i) = pi Set.univ p := by |
ext x
simp
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib... | Mathlib/Analysis/Calculus/LHopital.lean | 434 | 445 | theorem lhopital_zero_atTop (hdf : ∀ᶠ x : ℝ in atTop, DifferentiableAt ℝ f x)
(hg' : ∀ᶠ x : ℝ in atTop, deriv g x ≠ 0) (hftop : Tendsto f atTop (𝓝 0))
(hgtop : Tendsto g atTop (𝓝 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :
Tendsto (fun x => f x / g x) atTop l := by |
have hdg : ∀ᶠ x in atTop, DifferentiableAt ℝ g x := hg'.mp
(eventually_of_forall fun _ hg' =>
by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))
have hdf' : ∀ᶠ x in atTop, HasDerivAt f (deriv f x) x :=
hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)
have hdg' :... |
/-
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 | 1,615 | 1,618 | theorem integral_comp_mul_deriv' {f f' g : ℝ → ℝ} (h : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
(h' : ContinuousOn f' (uIcc a b)) (hg : ContinuousOn g (f '' [[a, b]])) :
(∫ x in a..b, (g ∘ f) x * f' x) = ∫ x in f a..f b, g x := by |
simpa [mul_comm] using integral_comp_smul_deriv' h h' hg
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "lea... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 150 | 157 | theorem snorm_add_lt_top {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
snorm (f + g) p μ < ∞ :=
calc
snorm (f + g) p μ ≤ LpAddConst p * (snorm f p μ + snorm g p μ) :=
snorm_add_le' hf.aestronglyMeasurable hg.aestronglyMeasurable p
_ < ∞ := by |
apply ENNReal.mul_lt_top (LpAddConst_lt_top p).ne
exact (ENNReal.add_lt_top.2 ⟨hf.2, hg.2⟩).ne
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 277 | 282 | theorem analyticAt_exp_of_mem_ball (x : 𝔸) (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
AnalyticAt 𝕂 (exp 𝕂) x := by |
by_cases h : (expSeries 𝕂 𝔸).radius = 0
· rw [h] at hx; exact (ENNReal.not_lt_zero hx).elim
· have h := pos_iff_ne_zero.mpr h
exact (hasFPowerSeriesOnBall_exp_of_radius_pos h).analyticAt_of_mem hx
|
/-
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.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathli... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 103 | 105 | theorem completedRiemannZeta₀_one_sub (s : ℂ) :
completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s := by |
rw [← completedHurwitzZetaEven₀_zero, ← completedCosZeta₀_zero, completedHurwitzZetaEven₀_one_sub]
|
/-
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, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc... | Mathlib/Data/Real/Pi/Bounds.lean | 179 | 180 | theorem pi_lt_315 : π < 3.15 := by |
pi_upper_bound [140 / 99, 279 / 151, 51 / 26, 412 / 207]
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 489 | 491 | theorem equiv_comp_equiv_apply_eq_self (x : L) :
equiv j i p (equiv i j p x) = x := by |
rw [equiv_comp_equiv_apply, equiv_self_apply]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 60 | 66 | theorem ext {P Q : Karoubi C} (h_X : P.X = Q.X) (h_p : P.p ≫ eqToHom h_X = eqToHom h_X ≫ Q.p) :
P = Q := by |
cases P
cases Q
dsimp at h_X h_p
subst h_X
simpa only [mk.injEq, heq_eq_eq, true_and, eqToHom_refl, comp_id, id_comp] using h_p
|
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | Mathlib/Topology/Compactification/OnePoint.lean | 152 | 153 | theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range some ↔ x = ∞ := by |
rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff]
|
/-
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.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathl... | Mathlib/Order/JordanHolder.lean | 102 | 106 | theorem isMaximal_inf_right_of_isMaximal_sup {x y : X} (hxz : IsMaximal x (x ⊔ y))
(hyz : IsMaximal y (x ⊔ y)) : IsMaximal (x ⊓ y) y := by |
rw [inf_comm]
rw [sup_comm] at hxz hyz
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,695 | 1,696 | theorem prod_range_div {M : Type*} [CommGroup M] (f : ℕ → M) (n : ℕ) :
(∏ i ∈ range n, f (i + 1) / f i) = f n / f 0 := by | apply prod_range_induction <;> simp
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 95 | 98 | theorem imageToKernel_zero_left [HasKernels V] [HasZeroObject V] {w} :
imageToKernel (0 : A ⟶ B) g w = 0 := by |
ext
simp
|
/-
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
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 309 | 322 | theorem Nodup.erase_get [DecidableEq α] {l : List α} (hl : l.Nodup) :
∀ i : Fin l.length, l.erase (l.get i) = l.eraseIdx ↑i := by |
induction l with
| nil => simp
| cons a l IH =>
intro i
cases i using Fin.cases with
| zero => simp
| succ i =>
rw [nodup_cons] at hl
rw [erase_cons_tail]
· simp [IH hl.2]
· rw [beq_iff_eq, get_cons_succ']
exact mt (· ▸ l.get_mem i i.isLt) hl.1
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monoton... | Mathlib/Data/Nat/Totient.lean | 129 | 135 | theorem totient_even {n : ℕ} (hn : 2 < n) : Even n.totient := by |
haveI : Fact (1 < n) := ⟨one_lt_two.trans hn⟩
haveI : NeZero n := NeZero.of_gt hn
suffices 2 = orderOf (-1 : (ZMod n)ˣ) by
rw [← ZMod.card_units_eq_totient, even_iff_two_dvd, this]
exact orderOf_dvd_card
rw [← orderOf_units, Units.coe_neg_one, orderOf_neg_one, ringChar.eq (ZMod n) n, if_neg hn.ne']
|
/-
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.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 568 | 570 | theorem continuousWithinAt_Ioo_iff_Ioi (h : a < b) :
ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by |
simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsWithin_Ioi h]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Geometry.Manifold.VectorBundle.Basic
import Mathlib.Topology.VectorBundle.Hom
#align_import geometry.manifold.vector_bundle.hom from "leanprover-c... | Mathlib/Geometry/Manifold/VectorBundle/Hom.lean | 44 | 52 | theorem smoothOn_continuousLinearMapCoordChange
[SmoothVectorBundle F₁ E₁ IB] [SmoothVectorBundle F₂ E₂ IB] [MemTrivializationAtlas e₁]
[MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂] [MemTrivializationAtlas e₂'] :
SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂)
(continuousLinearMapCoo... |
have h₁ := smoothOn_coordChangeL IB e₁' e₁
have h₂ := smoothOn_coordChangeL IB e₂ e₂'
refine (h₁.mono ?_).cle_arrowCongr (h₂.mono ?_) <;> mfld_set_tac
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 389 | 419 | theorem natSepDegree_mul_eq_iff (g : F[X]) :
(f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ (f = 0 ∧ g = 0) ∨ IsCoprime f g := by |
by_cases h : f * g = 0
· rw [mul_eq_zero] at h
wlog hf : f = 0 generalizing f g
· simpa only [mul_comm, add_comm, and_comm,
isCoprime_comm] using this g f h.symm (h.resolve_left hf)
rw [hf, zero_mul, natSepDegree_zero, zero_add, isCoprime_zero_left, isUnit_iff, eq_comm,
natSepDegree_eq_ze... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 509 | 509 | theorem xor_iff_iff_not : Xor' a b ↔ (a ↔ ¬b) := by | simp only [← @xor_not_right a, not_not]
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 821 | 828 | theorem degree_sum_le (s : Finset ι) (f : ι → R[X]) :
degree (∑ i ∈ s, f i) ≤ s.sup fun b => degree (f b) :=
Finset.cons_induction_on s (by simp only [sum_empty, sup_empty, degree_zero, le_refl])
fun a s has ih =>
calc
degree (∑ i ∈ cons a s has, f i) ≤ max (degree (f a)) (degree (∑ i ∈ s, f i)) := ... |
rw [Finset.sum_cons]; exact degree_add_le _ _
_ ≤ _ := by rw [sup_cons, sup_eq_max]; exact max_le_max le_rfl ih
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 569 | 579 | theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
LocallyIntegrable f μ := by |
intro x
rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U :=
exists_Icc_mem_subset_of_mem_nhds hU
have ab : a ≤ b := xab.1.trans xab.2
refine ⟨Icc a b, hab, ?_⟩
exact
(hmono.monotoneOn _).integrableOn_of_measure_ne_top (i... |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 134 | 154 | theorem isTree_iff_existsUnique_path :
G.IsTree ↔ Nonempty V ∧ ∀ v w : V, ∃! p : G.Walk v w, p.IsPath := by |
classical
rw [isTree_iff, isAcyclic_iff_path_unique]
constructor
· rintro ⟨hc, hu⟩
refine ⟨hc.nonempty, ?_⟩
intro v w
let q := (hc v w).some.toPath
use q
simp only [true_and_iff, Path.isPath]
intro p hp
specialize hu ⟨p, hp⟩ q
exact Subtype.ext_iff.mp hu
· rintro ⟨hV, h⟩
r... |
/-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.GroupTheory.OrderOfElement
/-!
# Fixed-point-free automorphi... | Mathlib/GroupTheory/FixedPointFree.lean | 42 | 44 | theorem commutatorMap_injective : Function.Injective (commutatorMap φ) := by |
refine fun x y h ↦ inv_mul_eq_one.mp <| hφ _ ?_
rwa [map_mul, map_inv, eq_inv_mul_iff_mul_eq, ← mul_assoc, ← eq_div_iff_mul_eq', ← division_def]
|
/-
Copyright (c) 2022 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.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 274 | 291 | theorem exists_large_sphere (n d : ℕ) :
∃ k, ((d ^ n :) / (n * d ^ 2 :) : ℝ) ≤ (sphere n d k).card := by |
obtain ⟨k, -, hk⟩ := exists_large_sphere_aux n d
refine ⟨k, ?_⟩
obtain rfl | hn := n.eq_zero_or_pos
· simp
obtain rfl | hd := d.eq_zero_or_pos
· simp
refine (div_le_div_of_nonneg_left ?_ ?_ ?_).trans hk
· exact cast_nonneg _
· exact cast_add_one_pos _
simp only [← le_sub_iff_add_le', cast_mul, ← mu... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,332 | 1,334 | theorem preimage_mul_left_singleton :
preimage {b} (a * ·) (mul_right_injective _).injOn = {a⁻¹ * b} := by |
classical rw [← image_mul_left', image_singleton]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,150 | 1,160 | theorem colimit.map_post {D : Type u'} [Category.{v'} D] [HasColimitsOfShape J D]
(H : C ⥤ D) :/- H (colimit F) ⟶ H (colimit G) ⟶ colimit (G ⋙ H) vs
H (colimit F) ⟶ colimit (F ⋙ H) ⟶ colimit (G ⋙ H) -/
colimit.post
F H ≫
H.map (colim.map α) =
colim.map (whiskerRight α ... |
ext
rw [← assoc, colimit.ι_post, ← H.map_comp, colimit.ι_map, H.map_comp]
rw [← assoc, colimit.ι_map, assoc, colimit.ι_post]
rfl
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 141 | 143 | theorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by |
subst_vars
rfl
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 626 | 696 | theorem cont_eval_fix {f k v} (fok : Code.Ok f) :
Turing.eval step (stepNormal f (Cont.fix f k) v) =
f.fix.eval v >>= fun v => Turing.eval step (Cfg.ret k v) := by |
refine Part.ext fun x => ?_
simp only [Part.bind_eq_bind, Part.mem_bind_iff]
constructor
· suffices ∀ c, x ∈ eval step c → ∀ v c', c = Cfg.then c' (Cont.fix f k) →
Reaches step (stepNormal f Cont.halt v) c' →
∃ v₁ ∈ f.eval v, ∃ v₂ ∈ if List.headI v₁ = 0 then pure v₁.tail else f.fix.eval v₁.tail,
... |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,379 | 1,381 | theorem comap_id (K : Subgroup N) : K.comap (MonoidHom.id _) = K := by |
ext
rfl
|
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.RingTheory.OreLocalization.OreSet
import Mathlib.Algebra.Group.Submonoid.Operations
#align_import ring_theory.ore_local... | Mathlib/RingTheory/OreLocalization/Basic.lean | 689 | 690 | theorem add_oreDiv {r r' : X} {s : S} : r /ₒ s + r' /ₒ s = (r + r') /ₒ s := by |
simp [oreDiv_add_char s s 1 1 (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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import data.polynomial.induction fro... | Mathlib/Algebra/Polynomial/Induction.lean | 75 | 78 | theorem span_le_of_C_coeff_mem (cf : ∀ i : ℕ, C (f.coeff i) ∈ I) :
Ideal.span { g | ∃ i, g = C (f.coeff i) } ≤ I := by |
simp only [@eq_comm _ _ (C _)]
exact (Ideal.span_le.trans range_subset_iff).mpr cf
|
/-
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.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.f... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,193 | 1,197 | theorem reduce.min (H : Red (reduce L₁) L₂) : reduce L₁ = L₂ := by |
induction' H with L1 L' L2 H1 H2 ih
· rfl
· cases' H1 with L4 L5 x b
exact reduce.not H2
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.