Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2023 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
import Mathlib.RingTheory.Ideal.Basic
#align_import algebra.monoid_algebra.ideal from "leanprover-community/mathlib"@"72c366d047... | Mathlib/Algebra/MonoidAlgebra/Ideal.lean | 23 | 58 | theorem MonoidAlgebra.mem_ideal_span_of_image [Monoid G] [Semiring k] {s : Set G}
{x : MonoidAlgebra k G} :
x ∈ Ideal.span (MonoidAlgebra.of k G '' s) ↔ ∀ m ∈ x.support, ∃ m' ∈ s, ∃ d, m = d * m' := by |
let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' := fun {x y} hx hy m hm => by
classical exact (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
zero_mem' := fun m hm => by cases hm
smul_mem' := fun x... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 259 | 267 | theorem adjointify_η_ε (X : C) :
F.map ((adjointifyη η ε).hom.app X) ≫ ε.hom.app (F.obj X) = 𝟙 (F.obj X) := by |
dsimp [adjointifyη,Trans.trans]
simp only [comp_id, assoc, map_comp]
have := ε.hom.naturality (F.map (η.inv.app X)); dsimp at this; rw [this]; clear this
rw [← assoc _ _ (F.map _)]
have := ε.hom.naturality (ε.inv.app <| F.obj X); dsimp at this; rw [this]; clear this
have := (ε.app <| F.obj X).hom_inv_id; d... |
/-
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 | 2,111 | 2,112 | theorem mem_normalizer_iff'' {g : G} : g ∈ H.normalizer ↔ ∀ h : G, h ∈ H ↔ g⁻¹ * h * g ∈ H := by |
rw [← inv_mem_iff (x := g), mem_normalizer_iff, inv_inv]
|
/-
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.CompleteLattice
import Mathlib.Order.Synonym
import Mathlib.Order.Hom.Set
import Mathlib.Order.Bounds.Basic
#align_import order.galois_connectio... | Mathlib/Order/GaloisConnection.lean | 578 | 580 | theorem l_biInf_u [CompleteLattice α] [CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x}
{p : ι → Prop} (f : ∀ (i) (_ : p i), β) : l (⨅ (i) (hi), u (f i hi)) = ⨅ (i) (hi), f i hi := by |
simp only [iInf_subtype', gi.l_iInf_u]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.floor from "leanprover-community/mathlib"@"bf6a01357ff5684b... | Mathlib/MeasureTheory/Function/Floor.lean | 59 | 62 | theorem MeasurableSet.image_fract [BorelSpace R] {s : Set R} (hs : MeasurableSet s) :
MeasurableSet (Int.fract '' s) := by |
simp only [Int.image_fract, sub_eq_add_neg, image_add_right']
exact MeasurableSet.iUnion fun m => (measurable_add_const _ hs).inter measurableSet_Ico
|
/-
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.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.No... | Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean | 447 | 450 | theorem map_add_add (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x x' : E) (y y' : Fₗ) :
f (x + x') (y + y') = f x y + f.deriv₂ (x, y) (x', y') + f x' y' := by |
simp only [map_add, add_apply, coe_deriv₂, add_assoc]
abel
|
/-
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.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.units from "leanprover-community... | Mathlib/Data/Fintype/Units.lean | 36 | 40 | theorem Fintype.card_eq_card_units_add_one [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card α = Fintype.card αˣ + 1 := by |
rw [eq_comm, Fintype.card_congr unitsEquivNeZero]
have := Fintype.card_congr (Equiv.sumCompl (· = (0 : α)))
rwa [Fintype.card_sum, add_comm, Fintype.card_subtype_eq] at this
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Mo... | Mathlib/Data/Nat/Nth.lean | 265 | 266 | theorem nth_zero_of_exists [DecidablePred p] (h : ∃ n, p n) : nth p 0 = Nat.find h := by |
rw [nth_zero]; convert Nat.sInf_def h
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,025 | 1,025 | theorem tan_pi : tan π = 0 := by | rw [tan_periodic.eq, tan_zero]
|
/-
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 | 673 | 678 | theorem chain_dartAdj_darts {d : G.Dart} {v w : V} (h : d.snd = v) (p : G.Walk v w) :
List.Chain G.DartAdj d p.darts := by |
induction p generalizing d with
| nil => exact List.Chain.nil
-- Porting note: needed to defer `h` and `rfl` to help elaboration
| cons h' p ih => exact List.Chain.cons (by exact h) (ih (by rfl))
|
/-
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.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 288 | 290 | theorem changeForm.associated_neg_proof [Invertible (2 : R)] :
(QuadraticForm.associated (R := R) (M := M) (-Q)).toQuadraticForm = 0 - Q := by |
simp [QuadraticForm.toQuadraticForm_associated]
|
/-
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.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/... | Mathlib/Algebra/Lie/Solvable.lean | 164 | 166 | theorem derivedSeries_eq_bot_iff (k : ℕ) :
derivedSeries R I k = ⊥ ↔ derivedSeriesOfIdeal R L k I = ⊥ := by |
rw [← derivedSeries_eq_derivedSeriesOfIdeal_map, map_eq_bot_iff, ker_incl, eq_bot_iff]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
#align_import category_theory.natural... | Mathlib/CategoryTheory/NatIso.lean | 168 | 169 | theorem naturality_1 (α : F ≅ G) (f : X ⟶ Y) : α.inv.app X ≫ F.map f ≫ α.hom.app Y = G.map f := 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, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 597 | 598 | theorem mem_iSup {x : Set α} {f : ι → Filter α} : x ∈ iSup f ↔ ∀ i, x ∈ f i := by |
simp only [← Filter.mem_sets, iSup_sets_eq, iff_self_iff, mem_iInter]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
#align_import category_theory.abelian.diagram_lemmas.four fr... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 143 | 152 | theorem isIso_of_epi_of_isIso_of_isIso_of_mono (h₀ : Epi (app' φ 0)) (h₁ : IsIso (app' φ 1))
(h₂ : IsIso (app' φ 3)) (h₃ : Mono (app' φ 4)) : IsIso (app' φ 2) := by |
dsimp at h₀ h₁ h₂ h₃
have : Mono (app' φ 2) := by
apply mono_of_epi_of_mono_of_mono (δlastFunctor.map φ) (R₁.exact_iff_δlast.1 hR₁).1
(R₂.exact_iff_δlast.1 hR₂).1 <;> dsimp <;> infer_instance
have : Epi (app' φ 2) := by
apply epi_of_epi_of_epi_of_mono (δ₀Functor.map φ) (R₁.exact_iff_δ₀.1 hR₁).2
... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.NAry
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.SupClosed
#align_import data.set.sups from "leanprover-community/mathlib"@"20715f... | Mathlib/Data/Set/Sups.lean | 429 | 438 | theorem lowerClosure_infs [SemilatticeInf α] (s t : Set α) :
lowerClosure (s ⊼ t) = lowerClosure s ⊓ lowerClosure t := by |
ext a
simp only [SetLike.mem_coe, mem_lowerClosure, Set.mem_infs, exists_and_left, exists_prop,
LowerSet.coe_sup, Set.mem_inter_iff]
constructor
· rintro ⟨_, ⟨b, hb, c, hc, rfl⟩, ha⟩
exact ⟨⟨b, hb, ha.trans inf_le_left⟩, c, hc, ha.trans inf_le_right⟩
· rintro ⟨⟨b, hb, hab⟩, c, hc, hac⟩
exact ⟨_, ... |
/-
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
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib... | Mathlib/NumberTheory/Multiplicity.lean | 395 | 402 | theorem pow_sub_pow (hyx : y < x) (hxy : p ∣ x - y) (hx : ¬p ∣ x) {n : ℕ} (hn : n ≠ 0) :
padicValNat p (x ^ n - y ^ n) = padicValNat p (x - y) + padicValNat p n := by |
rw [← PartENat.natCast_inj, Nat.cast_add]
iterate 3 rw [padicValNat_def, PartENat.natCast_get]
· exact multiplicity.Nat.pow_sub_pow hp.out hp1 hxy hx n
· exact hn.bot_lt
· exact Nat.sub_pos_of_lt hyx
· exact Nat.sub_pos_of_lt (Nat.pow_lt_pow_left hyx hn)
|
/-
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.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 73 | 95 | theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b := by |
suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by
exact IntervalIntegrable.trans (this a).symm (this b)
have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by
intro c hc
rw [intervalIntegrable_iff, uIoc_of_le hc]
have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x... |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,220 | 1,222 | theorem singletonSubgraph_eq_induce {v : V} :
G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by |
ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 318 | 319 | theorem deriv_const_mul_field (u : 𝕜') : deriv (fun y => u * v y) x = u * deriv v x := by |
simp only [mul_comm u, deriv_mul_const_field]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 908 | 912 | theorem subset_eval_image_pi (ht : (s.pi t).Nonempty) (i : ι) : t i ⊆ eval i '' s.pi t := by |
classical
obtain ⟨f, hf⟩ := ht
refine fun y hy => ⟨update f i y, fun j hj => ?_, update_same _ _ _⟩
obtain rfl | hji := eq_or_ne j i <;> simp [*, hf _ hj]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe... | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 187 | 191 | theorem IsLocalMin.hasFDerivAt_eq_zero (h : IsLocalMin f a) (hf : HasFDerivAt f f' a) : f' = 0 := by |
ext y
apply (h.on univ).hasFDerivWithinAt_eq_zero hf.hasFDerivWithinAt <;>
rw [posTangentConeAt_univ] <;>
apply mem_univ
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 899 | 901 | theorem fract_add_nat (a : α) (m : ℕ) : fract (a + m) = fract a := by |
rw [fract]
simp
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 659 | 669 | theorem Basis.det_unitsSMul (e : Basis ι R M) (w : ι → Rˣ) :
(e.unitsSMul w).det = (↑(∏ i, w i)⁻¹ : R) • e.det := by |
ext f
change
(Matrix.det fun i j => (e.unitsSMul w).repr (f j) i) =
(↑(∏ i, w i)⁻¹ : R) • Matrix.det fun i j => e.repr (f j) i
simp only [e.repr_unitsSMul]
convert Matrix.det_mul_column (fun i => (↑(w i)⁻¹ : R)) fun i j => e.repr (f j) i
-- porting note (#10745): was `simp [← Finset.prod_inv_distri... |
/-
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 | 36 | 37 | theorem uniqueMDiffOn_iff_uniqueDiffOn : UniqueMDiffOn 𝓘(𝕜, E) s ↔ UniqueDiffOn 𝕜 s := by |
simp [UniqueMDiffOn, UniqueDiffOn, uniqueMDiffWithinAt_iff_uniqueDiffWithinAt]
|
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 380 | 388 | theorem unit_mul_pow_congr_unit {ϖ : R} (hirr : Irreducible ϖ) (u v : Rˣ) (m n : ℕ)
(h : ↑u * ϖ ^ m = v * ϖ ^ n) : u = v := by |
obtain rfl : m = n := unit_mul_pow_congr_pow hirr hirr u v m n h
rw [← sub_eq_zero] at h
rw [← sub_mul, mul_eq_zero] at h
cases' h with h h
· rw [sub_eq_zero] at h
exact mod_cast h
· apply (hirr.ne_zero (pow_eq_zero h)).elim
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 987 | 988 | theorem terminates_congr {c₁ c₂ : Computation α} (h : c₁ ~ c₂) : Terminates c₁ ↔ Terminates c₂ := by |
simp only [terminates_iff, exists_congr h]
|
/-
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.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 295 | 296 | theorem log_ne_zero {x : ℝ} : log x ≠ 0 ↔ x ≠ 0 ∧ x ≠ 1 ∧ x ≠ -1 := by |
simpa only [not_or] using log_eq_zero.not
|
/-
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.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 109 | 112 | theorem dvd_of_mem_divisors {m : ℕ} (h : n ∈ divisors m) : n ∣ m := by |
cases m
· apply dvd_zero
· simp [mem_divisors.1 h]
|
/-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... | Mathlib/Data/Nat/MaxPowDiv.lean | 87 | 99 | theorem pow_dvd (p n : ℕ) : p ^ (p.maxPowDiv n) ∣ n := by |
dsimp [maxPowDiv]
rw [go]
by_cases h : (1 < p ∧ 0 < n ∧ n % p = 0)
· have : n / p < n := by apply Nat.div_lt_self <;> aesop
rw [if_pos h]
have ⟨c,hc⟩ := pow_dvd p (n / p)
rw [go_succ, pow_succ]
nth_rw 2 [← mod_add_div' n p]
rw [h.right.right, zero_add]
exact ⟨c,by nth_rw 1 [hc]; ac_rfl⟩... |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 1,110 | 1,118 | theorem IsPathConnected.exists_path_through_family' {n : ℕ}
{s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ i, p i ∈ s) :
∃ (γ : Path (p 0) (p n)) (t : Fin (n + 1) → I), (∀ t, γ t ∈ s) ∧ ∀ i, γ (t i) = p i := by |
rcases h.exists_path_through_family p hp with ⟨γ, hγ⟩
rcases hγ with ⟨h₁, h₂⟩
simp only [range, mem_setOf_eq] at h₂
rw [range_subset_iff] at h₁
choose! t ht using h₂
exact ⟨γ, t, h₁, ht⟩
|
/-
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.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 276 | 292 | theorem degree_add_divByMonic (hq : Monic q) (h : degree q ≤ degree p) :
degree q + degree (p /ₘ q) = degree p := by |
nontriviality R
have hdiv0 : p /ₘ q ≠ 0 := by rwa [Ne, divByMonic_eq_zero_iff hq, not_lt]
have hlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0 := by
rwa [Monic.def.1 hq, one_mul, Ne, leadingCoeff_eq_zero]
have hmod : degree (p %ₘ q) < degree (q * (p /ₘ q)) :=
calc
degree (p %ₘ q) < degree q := d... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 459 | 463 | theorem count_maximal (w : HeightOneSpectrum R) :
count K v (w.asIdeal : FractionalIdeal R⁰ K) = if w = v then 1 else 0 := by |
split_ifs with h
· rw [h, count_self]
· exact count_maximal_coprime K v h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 482 | 486 | theorem totalDegree_multiset_prod (s : Multiset (MvPolynomial σ R)) :
s.prod.totalDegree ≤ (s.map MvPolynomial.totalDegree).sum := by |
refine Quotient.inductionOn s fun l => ?_
rw [Multiset.quot_mk_to_coe, Multiset.prod_coe, Multiset.map_coe, Multiset.sum_coe]
exact totalDegree_list_prod l
|
/-
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 | 73 | 73 | theorem rpow_natCast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n := by | simpa using rpow_intCast x n
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,765 | 1,773 | theorem sumCongr_refl_swap {α β : Sort _} [DecidableEq α] [DecidableEq β] (i j : β) :
Equiv.Perm.sumCongr (Equiv.refl α) (Equiv.swap i j) = Equiv.swap (Sum.inr i) (Sum.inr j) := by |
ext x
cases x
· simp [Sum.map, swap_apply_of_ne_of_ne]
· simp only [Equiv.sumCongr_apply, Sum.map, coe_refl, comp_id, Sum.elim_inr, comp_apply,
swap_apply_def, Sum.inr.injEq]
split_ifs <;> rfl
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 76 | 78 | theorem coe_inv_of_nonzero {J : FractionalIdeal R₁⁰ K} (h : J ≠ 0) :
(↑J⁻¹ : Submodule R₁ K) = IsLocalization.coeSubmodule K ⊤ / (J : Submodule R₁ K) := by |
simp_rw [inv_nonzero _ h, coe_one, coe_mk, IsLocalization.coeSubmodule_top]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 411 | 412 | theorem mul_left (m₁ m₂ n : ℕ) : qrSign (m₁ * m₂) n = qrSign m₁ n * qrSign m₂ n := by |
simp_rw [qrSign, Nat.cast_mul, map_mul, jacobiSym.mul_left]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
#align_import number_theory.p... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 465 | 468 | theorem norm_lt_one_mul {z1 z2 : ℤ_[p]} (hz2 : ‖z2‖ < 1) : ‖z1 * z2‖ < 1 :=
calc
‖z1 * z2‖ = ‖z1‖ * ‖z2‖ := by | simp
_ < 1 := mul_lt_one_of_nonneg_of_lt_one_right (norm_le_one _) (norm_nonneg _) hz2
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 239 | 240 | theorem HasFiniteIntegral.neg {f : α → β} (hfi : HasFiniteIntegral f μ) :
HasFiniteIntegral (-f) μ := by | simpa [HasFiniteIntegral] using hfi
|
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | Mathlib/Data/List/Sym.lean | 49 | 61 | theorem left_mem_of_mk_mem_sym2 {xs : List α} {a b : α}
(h : s(a, b) ∈ xs.sym2) : a ∈ xs := by |
induction xs with
| nil => exact (not_mem_nil _ h).elim
| cons x xs ih =>
rw [mem_cons]
rw [mem_sym2_cons_iff] at h
obtain (h | ⟨c, hc, h⟩ | h) := h
· rw [Sym2.eq_iff, ← and_or_left] at h
exact .inl h.1
· rw [Sym2.eq_iff] at h
obtain (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) := h <;> simp [hc]
... |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 110 | 121 | theorem intDegree_add_le {x y : RatFunc K} (hy : y ≠ 0) (hxy : x + y ≠ 0) :
intDegree (x + y) ≤ max (intDegree x) (intDegree y) := by |
by_cases hx : x = 0
· simp only [hx, zero_add, ne_eq] at hxy
simp [hx, hxy]
rw [intDegree_add hxy, ←
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hx y.denom_ne_zero,
mul_comm y.denom, ←
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hy x.denom_ne_zero,
... |
/-
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.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 171 | 174 | theorem map_coord_zero {m : ∀ i, M₁ i} (i : ι) (h : m i = 0) : f m = 0 := by |
classical
have : (0 : R) • (0 : M₁ i) = 0 := by simp
rw [← update_eq_self i m, h, ← this, f.map_smul, zero_smul R (M := M₂)]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-... | Mathlib/Data/TypeVec.lean | 491 | 494 | theorem const_iff_true {α : TypeVec n} {i x p} : ofRepeat (TypeVec.const p α i x) ↔ p := by |
induction i with
| fz => rfl
| fs _ ih => erw [TypeVec.const, @ih (drop α) x]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 454 | 457 | theorem continuous_sub_left {a : ℝ≥0∞} (a_ne_top : a ≠ ∞) : Continuous (a - ·) := by |
change Continuous (Function.uncurry Sub.sub ∘ (a, ·))
refine continuousOn_sub.comp_continuous (Continuous.Prod.mk a) fun x => ?_
simp only [a_ne_top, Ne, mem_setOf_eq, Prod.mk.inj_iff, false_and_iff, not_false_iff]
|
/-
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.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 92 | 94 | theorem Separable.of_mul_right {f g : R[X]} (h : (f * g).Separable) : g.Separable := by |
rw [mul_comm] at h
exact h.of_mul_left
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 219 | 224 | theorem divInt_add_divInt (n₁ n₂ : Int) {d₁ d₂} (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) :
n₁ /. d₁ + n₂ /. d₂ = (n₁ * d₂ + n₂ * d₁) /. (d₁ * d₂) := by |
rcases Int.eq_nat_or_neg d₁ with ⟨_, rfl | rfl⟩ <;>
rcases Int.eq_nat_or_neg d₂ with ⟨_, rfl | rfl⟩ <;>
simp_all [-Int.natCast_mul, Int.ofNat_eq_zero, Int.neg_eq_zero, divInt_neg', Int.mul_neg,
Int.ofNat_mul_ofNat, Int.neg_add, Int.neg_mul, mkRat_add_mkRat]
|
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 177 | 182 | theorem normUnit_content {p : R[X]} : normUnit (content p) = 1 := by |
by_cases hp0 : p.content = 0
· simp [hp0]
· ext
apply mul_left_cancel₀ hp0
erw [← normalize_apply, normalize_content, mul_one]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 438 | 439 | theorem testBit_lor (m n k) : testBit (lor m n) k = (testBit m k || testBit n k) := by |
rw [← bitwise_or, testBit_bitwise]
|
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 304 | 305 | theorem hasGradientAtFilter_const : HasGradientAtFilter (fun _ => c) 0 x L := by |
rw [HasGradientAtFilter, map_zero]; apply hasFDerivAtFilter_const c x L
|
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb... | Mathlib/Data/Real/EReal.lean | 1,031 | 1,035 | theorem toReal_sub {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :
toReal (x - y) = toReal x - toReal y := by |
lift x to ℝ using ⟨hx, h'x⟩
lift y to ℝ using ⟨hy, h'y⟩
rfl
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 111 | 120 | theorem ortho_smul_left {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₁} (ha : a ≠ 0) :
IsOrtho B x y ↔ IsOrtho B (a • x) y := by |
dsimp only [IsOrtho]
constructor <;> intro H
· rw [map_smulₛₗ₂, H, smul_zero]
· rw [map_smulₛₗ₂, smul_eq_zero] at H
cases' H with H H
· rw [map_eq_zero I₁] at H
trivial
· exact H
|
/-
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 | 272 | 279 | theorem map₂_distrib_le_right {m : δ → γ → ε} {n : α → β → δ} {m₁ : α → γ → α'} {m₂ : β → γ → β'}
{n' : α' → β' → ε} (h_distrib : ∀ a b c, m (n a b) c = n' (m₁ a c) (m₂ b c)) :
map₂ m (map₂ n f g) h ≤ map₂ n' (map₂ m₁ f h) (map₂ m₂ g h) := by |
rintro s ⟨t₁, ⟨u, hu, w₁, hw₁, ht₁⟩, t₂, ⟨v, hv, w₂, hw₂, ht₂⟩, hs⟩
refine ⟨_, image2_mem_map₂ hu hv, w₁ ∩ w₂, inter_mem hw₁ hw₂, ?_⟩
refine (image2_distrib_subset_right h_distrib).trans ((image2_subset ?_ ?_).trans hs)
· exact (image2_subset_left inter_subset_left).trans ht₁
· exact (image2_subset_left inte... |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 495 | 499 | theorem HasDerivAt.clm_apply (hc : HasDerivAt c c' x) (hu : HasDerivAt u u' x) :
HasDerivAt (fun y => (c y) (u y)) (c' (u x) + c x u') x := by |
have := (hc.hasFDerivAt.clm_apply hu.hasFDerivAt).hasDerivAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
|
/-
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.CategoryTheory.Limits.KanExtension
import Mathlib.CategoryTheory.Sites.CoverPreserving
import Mathlib.CategoryTheory.Sites.Sheafification
#align_import cate... | Mathlib/CategoryTheory/Sites/CoverLifting.lean | 246 | 258 | theorem gluedSection_isAmalgamation : x.IsAmalgamation (gluedSection ℱ hS hx) := by |
intro V fV hV
-- Porting note: next line was `ext W`
-- Now `ext` can't see that `ran` is defined as a limit.
-- See https://github.com/leanprover-community/mathlib4/issues/5229
refine limit.hom_ext (fun (W : StructuredArrow (op V) G.op) ↦ ?_)
simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 488 | 490 | theorem integrable_const_iff {c : β} : Integrable (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ := by |
have : AEStronglyMeasurable (fun _ : α => c) μ := aestronglyMeasurable_const
rw [Integrable, and_iff_right this, hasFiniteIntegral_const_iff]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | Mathlib/Analysis/Calculus/MeanValue.lean | 833 | 846 | theorem Convex.mul_sub_le_image_sub_of_le_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
(hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C}
(hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x) :
∀ᵉ (x ∈ D) (y ∈ D), x ≤ y → C * (y - x) ≤ f y - f x := by |
intro x hx y hy hxy
cases' eq_or_lt_of_le hxy with hxy' hxy'
· rw [hxy', sub_self, sub_self, mul_zero]
have hxyD : Icc x y ⊆ D := hD.ordConnected.out hx hy
have hxyD' : Ioo x y ⊆ interior D :=
subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩
obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 153 | 156 | theorem mem_degrees {p : MvPolynomial σ R} {i : σ} :
i ∈ p.degrees ↔ ∃ d, p.coeff d ≠ 0 ∧ i ∈ d.support := by |
classical
simp only [degrees_def, Multiset.mem_sup, ← mem_support_iff, Finsupp.mem_toMultiset, exists_prop]
|
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 870 | 874 | theorem hausdorffMeasure_image (hf : Isometry f) (hd : 0 ≤ d ∨ Surjective f) (s : Set X) :
μH[d] (f '' s) = μH[d] s := by |
simp only [hausdorffMeasure, ← OuterMeasure.coe_mkMetric, ← OuterMeasure.comap_apply]
rw [OuterMeasure.isometry_comap_mkMetric _ hf (hd.imp_left _)]
exact ENNReal.monotone_rpow_of_nonneg
|
/-
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
#align_import analysis.box_integr... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 240 | 241 | theorem iUnion_eq_empty : π₁.iUnion = ∅ ↔ π₁ = ⊥ := by |
simp [← injective_boxes.eq_iff, Finset.ext_iff, Prepartition.iUnion, imp_false]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 676 | 682 | theorem _root_.Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
[SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
StronglyMeasurable f := by |
letI := pseudoMetrizableSpacePseudoMetric β
nontriviality β; inhabit β
exact ⟨SimpleFunc.approxOn f hf Set.univ default (Set.mem_univ _), fun x ↦
SimpleFunc.tendsto_approxOn hf (Set.mem_univ _) (by rw [closure_univ]; simp)⟩
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 382 | 384 | theorem coeff_index_single_X [DecidableEq σ] (s t : σ) :
coeff R (single t 1) (X s : MvPowerSeries σ R) = if t = s then 1 else 0 := by |
simp only [coeff_X, single_left_inj (one_ne_zero : (1 : ℕ) ≠ 0)]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#... | Mathlib/Order/Interval/Set/Pi.lean | 164 | 165 | theorem image_update_Ico_left (f : ∀ i, α i) (i : ι) (a : α i) :
update f i '' Ico a (f i) = Ico (update f i a) f := by | simpa using image_update_Ico f i a (f i)
|
/-
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.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tact... | Mathlib/RingTheory/Artinian.lean | 210 | 213 | theorem monotone_stabilizes_iff_artinian :
(∀ f : ℕ →o (Submodule R M)ᵒᵈ, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsArtinian R M := by |
rw [isArtinian_iff_wellFounded]
exact WellFounded.monotone_chain_condition.symm
|
/-
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.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 113 | 115 | theorem iteratedDerivWithin_zero : iteratedDerivWithin 0 f s = f := by |
ext x
simp [iteratedDerivWithin]
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 333 | 337 | theorem _root_.Real.toNNReal_sum_of_nonneg {α} {s : Finset α} {f : α → ℝ}
(hf : ∀ a, a ∈ s → 0 ≤ f a) :
Real.toNNReal (∑ a ∈ s, f a) = ∑ a ∈ s, Real.toNNReal (f a) := by |
rw [← coe_inj, NNReal.coe_sum, Real.coe_toNNReal _ (Finset.sum_nonneg hf)]
exact Finset.sum_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 477 | 479 | theorem HasLineDerivAt.of_comp {v : E'} (hf : HasLineDerivAt 𝕜 (f ∘ L) f' x v) :
HasLineDerivAt 𝕜 f f' (L x) (L v) := by |
simpa [HasLineDerivAt] using hf
|
/-
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.Order.Filter.Cofinite
#align_import data.analysis.filter from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Computa... | Mathlib/Data/Analysis/Filter.lean | 347 | 352 | theorem ne_bot_iff {f : Filter α} (F : f.Realizer) : f ≠ ⊥ ↔ ∀ a : F.σ, (F.F a).Nonempty := by |
rw [not_iff_comm, ← le_bot_iff, F.le_iff Realizer.bot, not_forall]
simp only [Set.not_nonempty_iff_eq_empty]
exact ⟨fun ⟨x, e⟩ _ ↦ ⟨x, le_of_eq e⟩, fun h ↦
let ⟨x, h⟩ := h ()
⟨x, le_bot_iff.1 h⟩⟩
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,236 | 2,238 | theorem conjTranspose_eq_zero [AddMonoid α] [StarAddMonoid α] {M : Matrix m n α} :
Mᴴ = 0 ↔ M = 0 := by |
rw [← conjTranspose_inj (A := M), conjTranspose_zero]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Ring.Pi
import Ma... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 794 | 795 | theorem const_le {x y : α} : const x ≤ const y ↔ x ≤ y := by |
rw [le_iff_lt_or_eq]; exact or_congr const_lt const_equiv
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 174 | 177 | theorem encard_eq_encard_iff_encard_diff_eq_encard_diff (h : (s ∩ t).Finite) :
s.encard = t.encard ↔ (s \ t).encard = (t \ s).encard := by |
rw [← encard_diff_add_encard_inter s t, ← encard_diff_add_encard_inter t s, inter_comm t s,
WithTop.add_right_cancel_iff h.encard_lt_top.ne]
|
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.D... | Mathlib/CategoryTheory/CofilteredSystem.lean | 82 | 101 | theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
[IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
[∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by |
-- Step 1: lift everything to the `max u v w` universe.
let J' : Type max w v u := AsSmall.{max w v} J
let down : J' ⥤ J := AsSmall.down
let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
haveI : ∀ i... |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,483 | 1,485 | theorem LowerSet.iSup_Iic (s : Set α) : ⨆ a ∈ s, LowerSet.Iic a = lowerClosure s := by |
ext
simp
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 457 | 458 | theorem hasLineDerivAt_zero : HasLineDerivAt 𝕜 f 0 x 0 := by |
simp [HasLineDerivAt, hasDerivAt_const]
|
/-
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.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 487 | 493 | theorem orthocenter_mem_altitude (t : Triangle ℝ P) {i₁ : Fin 3} :
t.orthocenter ∈ t.altitude i₁ := by |
obtain ⟨i₂, i₃, h₁₂, h₂₃, h₁₃⟩ : ∃ i₂ i₃, i₁ ≠ i₂ ∧ i₂ ≠ i₃ ∧ i₁ ≠ i₃ := by
-- Porting note (#11043): was `decide!`
fin_cases i₁ <;> decide
rw [orthocenter_eq_mongePoint, t.altitude_eq_mongePlane h₁₂ h₁₃ h₂₃]
exact t.mongePoint_mem_mongePlane
|
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Probability/ConditionalProbability.lean | 110 | 115 | theorem cond_toMeasurable_eq :
μ[|(toMeasurable μ s)] = μ[|s] := by |
unfold cond
by_cases hnt : μ s = ∞
· simp [hnt]
· simp [Measure.restrict_toMeasurable hnt]
|
/-
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, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 105 | 105 | theorem compl_apply_diag [Zero α] [One α] (i : V) : A.compl i i = 0 := by | simp [compl]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 789 | 807 | theorem span_image_eq_map_total (s : Set α) :
span R (v '' s) = Submodule.map (Finsupp.total α M R v) (supported R R s) := by |
apply span_eq_of_le
· intro x hx
rw [Set.mem_image] at hx
apply Exists.elim hx
intro i hi
exact ⟨_, Finsupp.single_mem_supported R 1 hi.1, by simp [hi.2]⟩
· refine map_le_iff_le_comap.2 fun z hz => ?_
have : ∀ i, z i • v i ∈ span R (v '' s) := by
intro c
haveI := Classical.decPred... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 636 | 644 | theorem finset_sum_iSup_nat {α} {ι} [SemilatticeSup ι] {s : Finset α} {f : α → ι → ℝ≥0∞}
(hf : ∀ a, Monotone (f a)) : (∑ a ∈ s, iSup (f a)) = ⨆ n, ∑ a ∈ s, f a n := by |
refine Finset.induction_on s ?_ ?_
· simp
· intro a s has ih
simp only [Finset.sum_insert has]
rw [ih, iSup_add_iSup_of_monotone (hf a)]
intro i j h
exact Finset.sum_le_sum fun a _ => hf a h
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
#align_import ring_theory.fractional_ideal from "lean... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 583 | 584 | theorem coe_mul (I J : FractionalIdeal S P) : (↑(I * J) : Submodule R P) = I * J := by |
simp only [mul_def, coe_mk]
|
/-
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.MeasureTheory.Measure.AEMeasurable
#align_import measure_theory.group.arithmetic from "leanprover-community/mathlib"@"a75898643b2d774cced9ae7c0b28c2... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 915 | 920 | theorem List.aemeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
AEMeasurable l.prod μ := by |
induction' l with f l ihl; · exact aemeasurable_one
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset... | Mathlib/GroupTheory/Exponent.lean | 252 | 256 | theorem lcm_orderOf_dvd_exponent [Fintype G] :
(Finset.univ : Finset G).lcm orderOf ∣ exponent G := by |
apply Finset.lcm_dvd
intro g _
exact order_dvd_exponent g
|
/-
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.MeasureTheory.Decomposition.SignedLebesgue
import Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure
#align_import measure_theory.decompos... | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | 466 | 471 | theorem integrable_rnDeriv_smul_iff [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
[SigmaFinite μ] {f : α → E} :
Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ := by |
nth_rw 2 [← withDensity_rnDeriv_eq μ ν hμν]
rw [← integrable_withDensity_iff_integrable_smul' (E := E)
(measurable_rnDeriv μ ν) (rnDeriv_lt_top μ ν)]
|
/-
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.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 372 | 382 | theorem toList_formPerm_isRotated_self (l : List α) (hl : 2 ≤ l.length) (hn : Nodup l) (x : α)
(hx : x ∈ l) : toList (formPerm l) x ~r l := by |
obtain ⟨k, hk, rfl⟩ := get_of_mem hx
have hr : l ~r l.rotate k := ⟨k, rfl⟩
rw [formPerm_eq_of_isRotated hn hr]
rw [get_eq_get_rotate l k k]
simp only [Nat.mod_eq_of_lt k.2, tsub_add_cancel_of_le (le_of_lt k.2), Nat.mod_self]
erw [toList_formPerm_nontrivial]
· simp
· simpa using hl
· simpa using hn
|
/-
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 | 703 | 704 | theorem isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by |
simp [isOpen_iff_nhds, mem_nhds_iff]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-c... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 752 | 783 | theorem right_half_plane_of_bounded_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
(hexp : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * abs z ^ c))
(hre : IsBoundedUnder (· ≤ ·) atTop fun x : ℝ => ‖f x‖) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
(hz : 0 ≤ z.re) : ‖f z‖ ≤ C := by |
-- For each `ε < 0`, the function `fun z ↦ exp (ε * z) • f z` satisfies assumptions of
-- `right_half_plane_of_tendsto_zero_on_real`, hence `‖exp (ε * z) • f z‖ ≤ C` for all `ε < 0`.
-- Taking the limit as `ε → 0`, we obtain the required inequality.
suffices ∀ᶠ ε : ℝ in 𝓝[<] 0, ‖exp (ε * z) • f z‖ ≤ C by
... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 228 | 238 | theorem mem_support_cycleOf_iff : y ∈ support (f.cycleOf x) ↔ SameCycle f x y ∧ x ∈ support f := by |
by_cases hx : f x = x
· rw [(cycleOf_eq_one_iff _).mpr hx]
simp [hx]
· rw [mem_support, cycleOf_apply]
split_ifs with hy
· simp only [hx, hy, iff_true_iff, Ne, not_false_iff, and_self_iff, mem_support]
rcases hy with ⟨k, rfl⟩
rw [← not_mem_support]
simpa using hx
· simpa [hx] us... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 171 | 175 | theorem cast_card_erase_of_mem {R} [AddGroupWithOne R] {s : Finset α} (hs : a ∈ s) :
((s.erase a).card : R) = s.card - 1 := by |
rw [card_erase_of_mem hs, Nat.cast_sub, Nat.cast_one]
rw [Nat.add_one_le_iff, Finset.card_pos]
exact ⟨a, hs⟩
|
/-
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.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 307 | 309 | theorem coeff_monomial_mul (p : R[X]) (n d : ℕ) (r : R) :
coeff (monomial n r * p) (d + n) = r * coeff p d := by |
rw [← C_mul_X_pow_eq_monomial, mul_assoc, coeff_C_mul, X_pow_mul, coeff_mul_X_pow]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 469 | 474 | theorem tmul_sum (m : M) {α : Type*} (s : Finset α) (n : α → N) :
(m ⊗ₜ[R] ∑ a ∈ s, n a) = ∑ a ∈ s, m ⊗ₜ[R] n a := by |
classical
induction' s using Finset.induction with a s has ih h
· simp
· simp [Finset.sum_insert has, tmul_add, ih]
|
/-
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.GroupWithZero.Units.Basic
import Mathli... | Mathlib/Algebra/GroupWithZero/Divisibility.lean | 71 | 78 | theorem dvdNotUnit_of_dvd_of_not_dvd {a b : α} (hd : a ∣ b) (hnd : ¬b ∣ a) : DvdNotUnit a b := by |
constructor
· rintro rfl
exact hnd (dvd_zero _)
· rcases hd with ⟨c, rfl⟩
refine ⟨c, ?_, rfl⟩
rintro ⟨u, rfl⟩
simp at hnd
|
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.EffectiveEpi.Basic
/-!
# Effective epimorphic sieves
We define the notion of effective epimorphic (... | Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean | 132 | 142 | theorem Sieve.effectiveEpimorphic_singleton {X Y : C} (f : Y ⟶ X) :
(Presieve.singleton f).EffectiveEpimorphic ↔ (EffectiveEpi f) := by |
constructor
· intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h
· rintro ⟨h⟩
show Nonempty _
rw [Sieve.generateSingleton_eq]
apply Nonempty.map (isColimitOfEffectiveEpiStruct _) h
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 46 | 48 | theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y := by |
simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_inj]
|
/-
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 | 250 | 252 | theorem _root_.Polynomial.toLaurent_C_mul_eq (r : R) (f : R[X]) :
toLaurent (Polynomial.C r * f) = C r * toLaurent f := by |
simp only [_root_.map_mul, Polynomial.toLaurent_C]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 293 | 295 | theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a := by |
simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left,
insert_def]
|
/-
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.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 69 | 70 | theorem cast_to_int [AddGroupWithOne α] (n : PosNum) : ((n : ℤ) : α) = n := by |
rw [← to_nat_to_int, Int.cast_natCast, cast_to_nat]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 346 | 347 | theorem degree_of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly.degree ≤ 0 := by |
simpa only [of_c_eq_zero ha hb hc] using degree_C_le
|
/-
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.Subgroup.Basic
import Mathlib.GroupTheory.Submonoid.Center
#align_import group_theory.subgroup.basic from "leanprover-community/mathlib"@"4be5... | Mathlib/GroupTheory/Subgroup/Center.lean | 130 | 131 | theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h := by |
rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 251 | 251 | theorem preimage_neg_Ioo : -Ioo a b = Ioo (-b) (-a) := by | simp [← Ioi_inter_Iio, inter_comm]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 398 | 398 | theorem arccos_eq_zero {x} : arccos x = 0 ↔ 1 ≤ x := by | simp [arccos, sub_eq_zero]
|
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