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) 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 | 1,230 | 1,233 | theorem tendsto_sum_nat_add (f : ℕ → ℝ≥0) : Tendsto (fun i => ∑' k, f (k + i)) atTop (𝓝 0) := by |
rw [← tendsto_coe]
convert _root_.tendsto_sum_nat_add fun i => (f i : ℝ)
norm_cast
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 260 | 264 | theorem Convex.translate_preimage_right (hs : Convex 𝕜 s) (z : E) :
Convex 𝕜 ((fun x => z + x) ⁻¹' s) := by |
intro x hx y hy a b ha hb hab
have h := hs hx hy ha hb hab
rwa [smul_add, smul_add, add_add_add_comm, ← add_smul, hab, one_smul] at 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.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of options
... | Mathlib/Data/Option/NAry.lean | 130 | 131 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a := by |
cases a <;> cases b <;> simp [h_comm]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 539 | 544 | theorem tendsto_zero_of_tendsto_zero_mass {γ : Type*} {F : Filter γ} {μs : γ → FiniteMeasure Ω}
(mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)) : Tendsto μs F (𝓝 0) := by |
rw [tendsto_iff_forall_testAgainstNN_tendsto]
intro f
convert tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f
rw [zero_testAgainstNN_apply]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 167 | 167 | theorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by | rw [add_comm, map_add_one, add_comm 1]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 584 | 608 | theorem hasSum_two_pi_I_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
(hf : CircleIntegrable f c R) (hw : abs w < R) :
HasSum (fun n : ℕ => ∮ z in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z)
(∮ z in C(c, R), (z - (c + w))⁻¹ • f z) := by |
have hR : 0 < R := (Complex.abs.nonneg w).trans_lt hw
have hwR : abs w / R ∈ Ico (0 : ℝ) 1 :=
⟨div_nonneg (Complex.abs.nonneg w) hR.le, (div_lt_one hR).2 hw⟩
refine intervalIntegral.hasSum_integral_of_dominated_convergence
(fun n θ => ‖f (circleMap c R θ)‖ * (abs w / R) ^ n) (fun n => ?_) (fun n => ?_)... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,212 | 1,214 | theorem lTensor_bij_iff_rTensor_bij :
Function.Bijective (lTensor M f) ↔ Function.Bijective (rTensor M f) := by |
simp [← comm_comp_rTensor_comp_comm_eq]
|
/-
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 | 1,099 | 1,099 | theorem cosh_add_sinh : cosh x + sinh x = exp x := by | rw [← ofReal_inj]; simp
|
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
/-!
# `L^p` distance on products of two metric space... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 161 | 164 | theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
edist f g =
(if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) := by |
convert if_pos rfl
|
/-
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.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 505 | 509 | theorem intervalIntegral_eq_integral_uIoc (f : ℝ → E) (a b : ℝ) (μ : Measure ℝ) :
∫ x in a..b, f x ∂μ = (if a ≤ b then 1 else -1 : ℝ) • ∫ x in Ι a b, f x ∂μ := by |
split_ifs with h
· simp only [integral_of_le h, uIoc_of_le h, one_smul]
· simp only [integral_of_ge (not_le.1 h).le, uIoc_of_lt (not_le.1 h), neg_one_smul]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 523 | 531 | theorem WSameSide.trans {s : AffineSubspace R P} {x y z : P} (hxy : s.WSameSide x y)
(hyz : s.WSameSide y z) (hy : y ∉ s) : s.WSameSide x z := by |
rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩
rw [wSameSide_iff_exists_left hp₂, or_iff_right hy] at hyz
rcases hyz with ⟨p₃, hp₃, hyz⟩
refine ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz ?_⟩
refine fun h => False.elim ?_
rw [vsub_eq_zero_iff_eq] at h
exact hy (h.symm ▸ hp₂)
|
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Init.Data.List.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.Con... | Mathlib/Data/Nat/Bits.lean | 609 | 610 | theorem one_bits : Nat.bits 1 = [true] := by |
convert bit1_bits 0
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | Mathlib/Topology/Sheaves/Stalks.lean | 113 | 114 | theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
(s) : germ F x (F.map i.op s) = germ F (i x) s := by | rw [← comp_apply, germ_res]
|
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheo... | Mathlib/Algebra/BigOperators/Fin.lean | 457 | 482 | theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
((ofFn f).take i).prod = ∏ j ∈ Finset.univ.filter fun j : Fin n => j.val < i, f j := by |
induction i with
| zero =>
simp
| succ i IH =>
by_cases h : i < n
· have : i < length (ofFn f) := by rwa [length_ofFn f]
rw [prod_take_succ _ _ this]
have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
((Finset.univ : Finset (Fin n)).filter fun j => j.val <... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7... | Mathlib/Algebra/RingQuot.lean | 478 | 481 | theorem lift_mkRingHom_apply (f : R →+* T) {r : R → R → Prop} (w : ∀ ⦃x y⦄, r x y → f x = f y) (x) :
lift ⟨f, w⟩ (mkRingHom r x) = f x := by |
simp_rw [lift_def, preLift_def, mkRingHom_def]
rfl
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,599 | 1,601 | theorem ContDiffAt.smul {f : E → 𝕜} {g : E → F} (hf : ContDiffAt 𝕜 n f x)
(hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x => f x • g x) x := by |
rw [← contDiffWithinAt_univ] at *; exact hf.smul hg
|
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 396 | 400 | theorem agree'_refl {n : ℕ} (x : M F) : Agree' n x x := by |
induction' n with _ n_ih generalizing x <;>
induction x using PFunctor.M.casesOn' <;> constructor <;> try rfl
intros
apply n_ih
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 540 | 540 | theorem not_isRoot_C (r a : R) (hr : r ≠ 0) : ¬IsRoot (C r) a := by | simpa using hr
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 439 | 440 | theorem cospanExt_inv_app_one : (cospanExt iX iY iZ wf wg).inv.app WalkingCospan.one = iZ.inv := by |
dsimp [cospanExt]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulA... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,207 | 1,210 | theorem coprod_inl_inr [ContinuousAdd M₁] [ContinuousAdd M'₁] :
(ContinuousLinearMap.inl R₁ M₁ M'₁).coprod (ContinuousLinearMap.inr R₁ M₁ M'₁) =
ContinuousLinearMap.id R₁ (M₁ × M'₁) := by |
apply coe_injective; apply LinearMap.coprod_inl_inr
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 73 | 74 | theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := by |
rw [← coe_nonempty, coe_Ioo, Set.nonempty_Ioo]
|
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d8... | Mathlib/Data/Bool/Count.lean | 60 | 71 | theorem count_not_eq_count (hl : Chain' (· ≠ ·) l) (h2 : Even (length l)) (b : Bool) :
count (!b) l = count b l := by |
cases' l with x l
· rfl
rw [length_cons, Nat.even_add_one, Nat.not_even_iff] at h2
suffices count (!x) (x :: l) = count x (x :: l) by
-- Porting note: old proof is
-- cases b <;> cases x <;> try exact this;
cases b <;> cases x <;>
revert this <;> simp only [Bool.not_false, Bool.not_true] <;> in... |
/-
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 | 1,139 | 1,139 | theorem sin_pi : sin π = 0 := by | rw [← ofReal_sin, Real.sin_pi]; simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 1,110 | 1,123 | theorem IsCompact.binary_compact_cover {K U V : Set X}
(hK : IsCompact K) (hU : IsOpen U) (hV : IsOpen V) (h2K : K ⊆ U ∪ V) :
∃ K₁ K₂ : Set X, IsCompact K₁ ∧ IsCompact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂ := by |
have hK' : IsCompact (closure K) := hK.closure
have : SeparatedNhds (closure K \ U) (closure K \ V) := by
apply SeparatedNhds.of_isCompact_isCompact_isClosed (hK'.diff hU) (hK'.diff hV)
(isClosed_closure.sdiff hV)
rw [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]
exact hK.closure_s... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 268 | 270 | theorem truncateFun_sub (x y : 𝕎 R) :
truncateFun n (x - y) = truncateFun n x - truncateFun n y := by |
witt_truncateFun_tac
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.Normed.Fi... | Mathlib/Analysis/NormedSpace/Basic.lean | 69 | 70 | theorem norm_zsmul [NormedSpace 𝕜 E] (n : ℤ) (x : E) : ‖n • x‖ = ‖(n : 𝕜)‖ * ‖x‖ := by |
rw [← norm_smul, ← Int.smul_one_eq_cast, smul_assoc, one_smul]
|
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 338 | 340 | theorem hasStrictDerivAt_const_rpow {a : ℝ} (ha : 0 < a) (x : ℝ) :
HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a) x := by |
simpa using (hasStrictDerivAt_const _ _).rpow (hasStrictDerivAt_id x) ha
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 542 | 545 | theorem image_to_eq_image [HasImages V] [HasEqualizers V] {i j : ι} (r : c.Rel i j) :
imageSubobject (C.dTo j) = imageSubobject (C.d i j) := by |
rw [C.dTo_eq r]
apply imageSubobject_iso_comp
|
/-
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.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2... | Mathlib/Order/Filter/Lift.lean | 252 | 254 | theorem tendsto_lift' {m : γ → β} {l : Filter γ} :
Tendsto m l (f.lift' h) ↔ ∀ s ∈ f, ∀ᶠ a in l, m a ∈ h s := by |
simp only [Filter.lift', tendsto_lift, tendsto_principal, comp]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 179 | 180 | theorem natCast_get {x : PartENat} (h : x.Dom) : (x.get h : PartENat) = x := by |
exact Part.ext' (iff_of_true trivial h) fun _ _ => rfl
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 116 | 116 | theorem factorization_zero : factorization 0 = 0 := by | ext; simp [factorization]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 120 | 130 | theorem mapAccumr₂_mapAccumr₂_left_right
(f₁ : γ → β → σ₁ → σ₁ × φ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr₂ f₂ xs ys s₂).snd ys s₁)
= let m := mapAccumr₂ (fun x y (s₁, s₂) =>
let r₂ := f₂ x y s₂
let r₁ := f₁ r₂.snd y s₁
((r₁.fst, r₂.fst), r₁.sn... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Logic.Function.Basic
#align_import logic.is_empty from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
/-!
# Types that... | Mathlib/Logic/IsEmpty.lean | 201 | 202 | theorem isEmpty_psum {α β} : IsEmpty (PSum α β) ↔ IsEmpty α ∧ IsEmpty β := by |
simp only [← not_nonempty_iff, nonempty_psum, not_or]
|
/-
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 | 1,332 | 1,334 | theorem prod_ite_irrel (p : Prop) [Decidable p] (s : Finset α) (f g : α → β) :
∏ x ∈ s, (if p then f x else g x) = if p then ∏ x ∈ s, f x else ∏ x ∈ s, g x := by |
split_ifs with h <;> rfl
|
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
#align_import algebra.grou... | Mathlib/Algebra/Order/Ring/Basic.lean | 161 | 162 | theorem lt_self_pow (h : 1 < a) (hm : 1 < m) : a < a ^ m := by |
simpa only [pow_one] using pow_lt_pow_right h hm
|
/-
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 | 1,030 | 1,033 | theorem isPathConnected_singleton (x : X) : IsPathConnected ({x} : Set X) := by |
refine ⟨x, rfl, ?_⟩
rintro y rfl
exact JoinedIn.refl rfl
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 277 | 293 | theorem isLittleO_iff_nat_mul_le_aux (h₀ : (∀ x, 0 ≤ ‖f x‖) ∨ ∀ x, 0 ≤ ‖g x‖) :
f =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g x‖ := by |
constructor
· rintro H (_ | n)
· refine (H.def one_pos).mono fun x h₀' => ?_
rw [Nat.cast_zero, zero_mul]
refine h₀.elim (fun hf => (hf x).trans ?_) fun hg => hg x
rwa [one_mul] at h₀'
· have : (0 : ℝ) < n.succ := Nat.cast_pos.2 n.succ_pos
exact (isBigOWith_inv this).1 (H.def' <| in... |
/-
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 | 2,887 | 2,899 | theorem rel_of_forall {m1 m2 : Multiset α} {r : α → α → Prop} (h : ∀ a b, a ∈ m1 → b ∈ m2 → r a b)
(hc : card m1 = card m2) : m1.Rel r m2 := by |
revert m1
refine @(m2.induction_on ?_ ?_)
· intro m _h hc
rw [rel_zero_right, ← card_eq_zero, hc, card_zero]
· intro a t ih m h hc
rw [card_cons] at hc
obtain ⟨b, hb⟩ := card_pos_iff_exists_mem.1 (show 0 < card m from hc.symm ▸ Nat.succ_pos _)
obtain ⟨m', rfl⟩ := exists_cons_of_mem hb
refin... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin, Andrew Yang
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoidal.End
... | Mathlib/CategoryTheory/Shift/Basic.lean | 532 | 535 | theorem shift_shiftFunctorCompIsoId_neg_add_self_inv_app (n : A) (X : C) :
((shiftFunctorCompIsoId C (-n) n (neg_add_self n)).inv.app X)⟦-n⟧' =
(shiftFunctorCompIsoId C n (-n) (add_neg_self n)).inv.app (X⟦-n⟧) := by |
apply shift_shiftFunctorCompIsoId_inv_app
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
im... | Mathlib/CategoryTheory/Category/Basic.lean | 252 | 254 | theorem id_of_comp_left_id (f : X ⟶ X) (w : ∀ {Y : C} (g : X ⟶ Y), f ≫ g = g) : f = 𝟙 X := by |
convert w (𝟙 X)
simp
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 945 | 946 | theorem num_div_dvd' (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q) ∣ p := by | simpa using num_div_dvd p hq
|
/-
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.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62... | Mathlib/Algebra/Group/Support.lean | 309 | 311 | theorem image_inter_mulSupport_eq {s : Set β} {g : β → α} :
g '' s ∩ mulSupport f = g '' (s ∩ mulSupport (f ∘ g)) := by |
rw [mulSupport_comp_eq_preimage f g, image_inter_preimage]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import d... | Mathlib/Data/PNat/Factors.lean | 292 | 299 | theorem factorMultiset_mul (n m : ℕ+) :
factorMultiset (n * m) = factorMultiset n + factorMultiset m := by |
let u := factorMultiset n
let v := factorMultiset m
have : n = u.prod := (prod_factorMultiset n).symm; rw [this]
have : m = v.prod := (prod_factorMultiset m).symm; rw [this]
rw [← PrimeMultiset.prod_add]
repeat' rw [PrimeMultiset.factorMultiset_prod]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 271 | 275 | theorem whisker_assoc (X : C) {Y Y' : C} (f : Y ⟶ Y') (Z : C) :
(X ◁ f) ▷ Z = (α_ X Y Z).hom ≫ X ◁ f ▷ Z ≫ (α_ X Y' Z).inv := by |
simp only [← id_tensorHom, ← tensorHom_id]
rw [← assoc, ← associator_naturality]
simp
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,308 | 1,309 | theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by |
rw [← sInter_image, image_id']
|
/-
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.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 328 | 338 | theorem ClassGroup.mk0_integralRep [IsDedekindDomain R]
(I : (FractionalIdeal R⁰ (FractionRing R))ˣ) :
ClassGroup.mk0 ⟨ClassGroup.integralRep I, ClassGroup.integralRep_mem_nonZeroDivisors I.ne_zero⟩
= ClassGroup.mk I := by |
rw [← ClassGroup.mk_mk0 (FractionRing R), eq_comm, ClassGroup.mk_eq_mk]
have fd_ne_zero : (algebraMap R (FractionRing R)) I.1.den ≠ 0 := by
exact IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (SetLike.coe_mem _)
refine ⟨Units.mk0 (algebraMap R _ I.1.den) fd_ne_zero, ?_⟩
apply Units.ext
rw [mul_com... |
/-
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.Computability.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 507 | 510 | theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a := by |
rw [eval, Nat.unpaired, Nat.unpair_pair]
simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only []
rw [Nat.rec_zero]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | Mathlib/Analysis/MeanInequalitiesPow.lean | 126 | 129 | theorem pow_sum_div_card_le_sum_pow (f : ι → ℝ≥0) (n : ℕ) :
(∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by |
simpa only [← NNReal.coe_le_coe, NNReal.coe_sum, Nonneg.coe_div, NNReal.coe_pow] using
@Real.pow_sum_div_card_le_sum_pow ι s (((↑) : ℝ≥0 → ℝ) ∘ f) n fun _ _ => NNReal.coe_nonneg _
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 591 | 636 | theorem continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ univ)) :
ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) := by |
/- First get rid of the case where the space is not locally compact. Then `g` vanishes everywhere
and the conclusion is trivial. -/
by_cases H : ∀ p ∈ s, ∀ x, g p x = 0
· apply (continuousOn_const (c := 0)).congr
rintro ⟨p, x⟩ ⟨hp, -⟩
apply integral_eq_zero_of_ae (eventually_of_forall (fun y ↦ ?_))
... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Yaël Dillies
-/
import Mathlib.Data.Real.NNReal
import Mathlib.Tactic.GCongr.Core
#align_import analysis.normed.group.sem... | Mathlib/Analysis/Normed/Group/Seminorm.lean | 526 | 527 | theorem toZeroHom_eq_coe : ⇑p.toZeroHom = p := by |
rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 706 | 709 | theorem zpow_pow_orderOf : (x ^ i) ^ orderOf x = 1 := by |
by_cases h : IsOfFinOrder x
· rw [← zpow_natCast, ← zpow_mul, mul_comm, zpow_mul, zpow_natCast, pow_orderOf_eq_one, one_zpow]
· rw [orderOf_eq_zero h, _root_.pow_zero]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 350 | 378 | theorem prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t := by |
apply le_antisymm
· set S := toMeasurable μ s
set T := toMeasurable ν t
have hSTm : MeasurableSet (S ×ˢ T) :=
(measurableSet_toMeasurable _ _).prod (measurableSet_toMeasurable _ _)
calc
μ.prod ν (s ×ˢ t) ≤ μ.prod ν (S ×ˢ T) := by gcongr <;> apply subset_toMeasurable
_ = μ S * ν T := b... |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 286 | 290 | theorem realize_foldr_inf (l : List (L.BoundedFormula α n)) (v : α → M) (xs : Fin n → M) :
(l.foldr (· ⊓ ·) ⊤).Realize v xs ↔ ∀ φ ∈ l, BoundedFormula.Realize φ v xs := by |
induction' l with φ l ih
· simp
· simp [ih]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "lean... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 215 | 217 | theorem toOuterMeasure_apply_inter_support :
p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s := by |
simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support]
|
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 181 | 184 | theorem _root_.Algebra.adjoin.powerBasis'_gen (hx : IsIntegral R x) :
(adjoin.powerBasis' hx).gen = ⟨x, SetLike.mem_coe.1 <| subset_adjoin <| mem_singleton x⟩ := by |
rw [Algebra.adjoin.powerBasis', PowerBasis.map_gen, AdjoinRoot.powerBasis'_gen, equivAdjoin,
AlgEquiv.ofBijective_apply, Minpoly.toAdjoin, liftHom_root]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 848 | 849 | theorem coprod.diag_comp {X Y : C} [HasBinaryCoproduct X X] (f : X ⟶ Y) :
codiag X ≫ f = coprod.desc f f := by | simp
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 515 | 520 | theorem sOppSide_iff_exists_right {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y) := by |
rw [SOppSide, and_comm, wOppSide_iff_exists_right h, and_assoc, and_congr_right_iff,
and_congr_right_iff]
rintro _ hy
rw [or_iff_right hy]
|
/-
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.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift... | Mathlib/Order/WithBot.lean | 336 | 340 | theorem unbot'_lt_iff {a : WithBot α} {b c : α} (h : a = ⊥ → b < c) :
a.unbot' b < c ↔ a < c := by |
induction a
· simpa [bot_lt_coe] using h rfl
· simp
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 712 | 712 | theorem mul_self_norm (z : K) : ‖z‖ * ‖z‖ = normSq z := by | rw [normSq_eq_def', sq]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,716 | 1,721 | theorem prime_pow_le_iff_le_bcount [DecidableEq (Associates α)] {m p : Associates α}
(h₁ : m ≠ 0) (h₂ : Irreducible p) {k : ℕ} : p ^ k ≤ m ↔ k ≤ bcount ⟨p, h₂⟩ m.factors := by |
rcases Associates.exists_non_zero_rep h₁ with ⟨m, hm, rfl⟩
have := nontrivial_of_ne _ _ hm
rw [bcount, factors_mk, Multiset.le_count_iff_replicate_le, ← factors_le, factors_prime_pow,
factors_mk, WithTop.coe_le_coe] <;> assumption
|
/-
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, Mario Carneiro
-/
/-!
# Definitions and properties of `coprime`
-/
namespace Nat
/-!
### `coprime`
See also `nat.coprime_of_dvd` and `nat.coprime_o... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 57 | 59 | theorem coprime_div_gcd_div_gcd
(H : 0 < gcd m n) : Coprime (m / gcd m n) (n / gcd m n) := by |
rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintyp... | Mathlib/Data/Finset/Sort.lean | 197 | 199 | theorem orderEmbOfFin_zero {s : Finset α} {k : ℕ} (h : s.card = k) (hz : 0 < k) :
orderEmbOfFin s h ⟨0, hz⟩ = s.min' (card_pos.mp (h.symm ▸ hz)) := by |
simp only [orderEmbOfFin_apply, Fin.val_mk, sorted_zero_eq_min']
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 467 | 469 | theorem map_adjoin_singleton (e : A →ₐ[R] B) (x : A) :
(adjoin R {x}).map e = adjoin R {e x} := by |
rw [map_adjoin, Set.image_singleton]
|
/-
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.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprove... | Mathlib/Analysis/Complex/ReImTopology.lean | 99 | 100 | theorem interior_setOf_im_le (a : ℝ) : interior { z : ℂ | z.im ≤ a } = { z | z.im < a } := by |
simpa only [interior_Iic] using interior_preimage_im (Iic a)
|
/-
Copyright (c) 2020 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.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,118 | 1,124 | theorem _root_.Continuous.memℒp_top_of_hasCompactSupport
{X : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
{f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f) (μ : Measure X) : Memℒp f ⊤ μ := by |
borelize E
rcases hf.bounded_above_of_compact_support h'f with ⟨C, hC⟩
apply memℒp_top_of_bound ?_ C (Filter.eventually_of_forall hC)
exact (hf.stronglyMeasurable_of_hasCompactSupport h'f).aestronglyMeasurable
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 1,014 | 1,016 | theorem not_anisotropic_iff_exists (Q : QuadraticForm R M) :
¬Anisotropic Q ↔ ∃ x, x ≠ 0 ∧ Q x = 0 := by |
simp only [Anisotropic, not_forall, exists_prop, and_comm]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 122 | 127 | theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
μ.withDensity (r • f) = r • μ.withDensity f := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf]
simp only [Pi.smul_apply, smul_eq_mul]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Noetheri... | Mathlib/Topology/NoetherianSpace.lean | 87 | 101 | theorem noetherianSpace_TFAE :
TFAE [NoetherianSpace α,
WellFounded fun s t : Closeds α => s < t,
∀ s : Set α, IsCompact s,
∀ s : Opens α, IsCompact (s : Set α)] := by |
tfae_have 1 ↔ 2
· refine (noetherianSpace_iff α).trans (Opens.compl_bijective.2.wellFounded_iff ?_)
exact (@OrderIso.compl (Set α)).lt_iff_lt.symm
tfae_have 1 ↔ 4
· exact noetherianSpace_iff_opens α
tfae_have 1 → 3
· exact @NoetherianSpace.isCompact α _
tfae_have 3 → 4
· exact fun h s => h s
tfae... |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 477 | 481 | theorem continuous_testAgainstNN_eval (f : Ω →ᵇ ℝ≥0) :
Continuous fun μ : FiniteMeasure Ω => μ.testAgainstNN f := by |
show Continuous ((fun φ : WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0) => φ f) ∘ toWeakDualBCNN)
refine Continuous.comp ?_ (toWeakDualBCNN_continuous (Ω := Ω))
exact WeakBilin.eval_continuous (𝕜 := ℝ≥0) (E := (Ω →ᵇ ℝ≥0) →L[ℝ≥0] ℝ≥0) _ _
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
/-... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 109 | 115 | theorem ergodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
Ergodic (e ∘ f ∘ e.symm) μ' ↔ Ergodic f μ := by |
have : MeasurePreserving (e ∘ f ∘ e.symm) μ' μ' ↔ MeasurePreserving f μ μ := by
rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]
replace h : PreErgodic (e ∘ f ∘ e.symm) μ' ↔ PreErgodic f μ := h.preErgodic_conjugate_iff
exact ⟨fun hf => { this.mp hf.toMeasurePreserving, h.mp hf.toPreErgodic w... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Mathport.Rename
/-!
# Basic facts about `Thunk`.
-/
set_option autoImplicit true
namespace Thunk
#align thunk.mk Thunk.mk
-- Porting note: Added `Thunk.... | Mathlib/Lean/Thunk.lean | 20 | 24 | theorem ext {α : Type u} {a b : Thunk α} (eq : a.get = b.get) : a = b := by |
have ⟨_⟩ := a
have ⟨_⟩ := b
congr
exact funext fun _ ↦ eq
|
/-
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.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520... | Mathlib/Data/ENNReal/Inv.lean | 403 | 406 | theorem le_of_forall_nnreal_lt {x y : ℝ≥0∞} (h : ∀ r : ℝ≥0, ↑r < x → ↑r ≤ y) : x ≤ y := by |
refine le_of_forall_ge_of_dense fun r hr => ?_
lift r to ℝ≥0 using ne_top_of_lt hr
exact h r hr
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,105 | 2,108 | theorem exists_of_lt_bmex {o : Ordinal} {f : ∀ a < o, Ordinal} {a} (ha : a < bmex o f) :
∃ i hi, f i hi = a := by |
cases' exists_of_lt_mex ha with i hi
exact ⟨_, typein_lt_self i, hi⟩
|
/-
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.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 606 | 613 | theorem comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ⊆ s := by |
obtain ⟨w, w_in, w_sub⟩ : ∃ w ∈ 𝓤 α, w ○ w ⊆ s := comp_mem_uniformity_sets hs
use symmetrizeRel w, symmetrize_mem_uniformity w_in, symmetric_symmetrizeRel w
have : symmetrizeRel w ⊆ w := symmetrizeRel_subset_self w
calc symmetrizeRel w ○ symmetrizeRel w
_ ⊆ w ○ w := by mono
_ ⊆ s := w_sub
|
/-
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 | 431 | 432 | theorem finrank_eq_card_finset_basis {ι : Type w} {b : Finset ι} (h : Basis b R M) :
finrank R M = Finset.card b := by | rw [finrank_eq_card_basis h, Fintype.card_coe]
|
/-
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.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Prefixes, suffixes... | Mathlib/Data/List/Infix.lean | 430 | 432 | theorem inits_reverse (l : List α) : inits (reverse l) = reverse (map reverse l.tails) := by |
rw [tails_eq_inits l]
simp [reverse_involutive.comp_self, reverse_map]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 742 | 771 | theorem eq_comp_δ_of_not_surjective' {n : ℕ} {Δ : SimplexCategory} (θ : Δ ⟶ mk (n + 1))
(i : Fin (n + 2)) (hi : ∀ x, θ.toOrderHom x ≠ i) : ∃ θ' : Δ ⟶ mk n, θ = θ' ≫ δ i := by |
by_cases h : i < Fin.last (n + 1)
· use θ ≫ σ (Fin.castPred i h.ne)
ext1
ext1
ext1 x
simp only [len_mk, Category.assoc, comp_toOrderHom, OrderHom.comp_coe, Function.comp_apply]
by_cases h' : θ.toOrderHom x ≤ i
· simp only [σ, mkHom, Hom.toOrderHom_mk, OrderHom.coe_mk]
-- This used to ... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 429 | 437 | theorem All.upperBound?_ub {t : RBNode α} (hp : t.All p) (H : ∀ {x}, ub = some x → p x) :
t.upperBound? cut ub = some x → p x := by |
induction t generalizing ub with
| nil => exact H
| node _ _ _ _ ihl ihr =>
simp [upperBound?]; split
· exact ihl hp.2.1 fun | rfl => hp.1
· exact ihr hp.2.2 H
· exact fun | rfl => hp.1
|
/-
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 | 344 | 347 | theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀ x, Decidable (x ∈ t)]
{a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ t] a) l)
(h₁ : Tendsto g (𝓝[s ∩ tᶜ] a) l) : Tendsto (piecewise t f g) (𝓝[s] a) l := by |
apply Tendsto.piecewise <;> rwa [← nhdsWithin_inter']
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,031 | 3,033 | theorem map_le_map_iff' {f : G →* N} {H K : Subgroup G} :
H.map f ≤ K.map f ↔ H ⊔ f.ker ≤ K ⊔ f.ker := by |
simp only [map_le_map_iff, sup_le_iff, le_sup_right, and_true_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 821 | 822 | theorem nat_add_eq {a : Cardinal} (n : ℕ) (ha : ℵ₀ ≤ a) : n + a = a := by |
rw [add_comm, add_nat_eq n ha]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 301 | 301 | theorem orElse_none' (x : Option α) : x.orElse (fun _ ↦ none) = x := by | cases x <;> rfl
|
/-
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.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Sets.Ope... | Mathlib/Topology/Category/TopCat/Opens.lean | 347 | 350 | theorem openEmbedding_obj_top {X : TopCat} (U : Opens X) :
U.openEmbedding.isOpenMap.functor.obj ⊤ = U := by |
ext1
exact Set.image_univ.trans Subtype.range_coe
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.LinearM... | Mathlib/Algebra/Module/Equiv.lean | 433 | 437 | theorem eq_comp_toLinearMap_symm (f : M₂ →ₛₗ[σ₂₃] M₃) (g : M₁ →ₛₗ[σ₁₃] M₃) :
f = g.comp e₁₂.symm.toLinearMap ↔ f.comp e₁₂.toLinearMap = g := by |
constructor <;> intro H <;> ext
· simp [H, e₁₂.toEquiv.eq_comp_symm f g]
· simp [← H, ← e₁₂.toEquiv.eq_comp_symm f g]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 875 | 877 | theorem find_le (n : ℕ) (h : P n) : find P ≤ n := by |
rw [le_coe_iff]
exact ⟨⟨_, h⟩, @Nat.find_min' P _ _ _ h⟩
|
/-
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 | 1,046 | 1,051 | theorem IsPathConnected.preimage_coe {U W : Set X} (hW : IsPathConnected W) (hWU : W ⊆ U) :
IsPathConnected (((↑) : U → X) ⁻¹' W) := by |
rcases hW with ⟨x, x_in, hx⟩
use ⟨x, hWU x_in⟩, by simp [x_in]
rintro ⟨y, hyU⟩ hyW
exact ⟨(hx hyW).joined_subtype.somePath.map (continuous_inclusion hWU), by simp⟩
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 171 | 172 | theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by | simp [weightedSMul]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 245 | 245 | theorem monomial_zero_eq_C_apply (a : R) : monomial R 0 a = C R a := by | simp
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import... | Mathlib/NumberTheory/DiophantineApproximation.lean | 93 | 132 | theorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by |
let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋
have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _
have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)
conv in |_| ≤ _ => rw [mul_comm, le_div_iff hn, ← abs_of_pos hn, ← abs_mul]
let D := Icc (0 : ℤ) n
by_cases H : ∃ m ∈ D, f m = ... |
/-
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 | 1,302 | 1,305 | theorem degree_cubic_le : degree (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d) ≤ 3 := by |
simpa only [add_assoc] using
degree_add_le_of_degree_le (degree_C_mul_X_pow_le 3 a)
(le_trans degree_quadratic_le <| WithBot.coe_le_coe.mpr <| Nat.le_succ 2)
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 1,026 | 1,032 | theorem ball_zero_absorbs_ball_zero (p : Seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
Absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) := by |
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩
refine .of_norm ⟨r, fun a ha x hx => ?_⟩
rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero]
rw [p.mem_ball_zero] at hx
exact hx.trans (hr.trans_le <| by gcongr)
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 164 | 176 | theorem ofArrows_bind {ι : Type*} (Z : ι → C) (g : ∀ i : ι, Z i ⟶ X)
(j : ∀ ⦃Y⦄ (f : Y ⟶ X), ofArrows Z g f → Type*) (W : ∀ ⦃Y⦄ (f : Y ⟶ X) (H), j f H → C)
(k : ∀ ⦃Y⦄ (f : Y ⟶ X) (H i), W f H i ⟶ Y) :
((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) =
ofArrows (fun i : Σi, j _ (ofArrows.mk ... |
funext Y
ext f
constructor
· rintro ⟨_, _, _, ⟨i⟩, ⟨i'⟩, rfl⟩
exact ofArrows.mk (Sigma.mk _ _)
· rintro ⟨i⟩
exact bind_comp _ (ofArrows.mk _) (ofArrows.mk _)
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 671 | 674 | theorem realize_relabel {φ : L.Formula α} {g : α → β} {v : β → M} :
(φ.relabel g).Realize v ↔ φ.Realize (v ∘ g) := by |
rw [Realize, Realize, relabel, BoundedFormula.realize_relabel, iff_eq_eq, Fin.castAdd_zero]
exact congr rfl (funext finZeroElim)
|
/-
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 | 806 | 809 | theorem degree_erase_lt (hp : p ≠ 0) : degree (p.erase (natDegree p)) < degree p := by |
apply lt_of_le_of_ne (degree_erase_le _ _)
rw [degree_eq_natDegree hp, degree, support_erase]
exact fun h => not_mem_erase _ _ (mem_of_max h)
|
/-
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.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@... | Mathlib/Data/List/Intervals.lean | 143 | 148 | theorem chain'_succ (n m : ℕ) : Chain' (fun a b => b = succ a) (Ico n m) := by |
by_cases h : n < m
· rw [eq_cons h]
exact chain_succ_range' _ _ 1
· rw [eq_nil_of_le (le_of_not_gt h)]
trivial
|
/-
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 | 57 | 62 | theorem LocallyIntegrableOn.mono (hf : LocallyIntegrableOn f s μ) {g : X → F}
(hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
LocallyIntegrableOn g s μ := by |
intro x hx
rcases hf x hx with ⟨t, t_mem, ht⟩
exact ⟨t, t_mem, Integrable.mono ht hg.restrict (ae_restrict_of_ae h)⟩
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 547 | 548 | theorem bind_map {γ} (f : α → β) (x) (g : β → Part γ) :
(map f x).bind g = x.bind fun y => g (f y) := by | rw [← bind_some_eq_map, bind_assoc]; simp
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 545 | 553 | theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω := by |
intro N M hNM ω
simp only [upcrossingsBefore]
by_cases hemp : {n : ℕ | upperCrossingTime a b f N n ω < N}.Nonempty
· refine csSup_le_csSup (upperCrossingTime_lt_bddAbove hab) hemp fun n hn => ?_
rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn]
exact lt_of_lt_of_le hn hNM
· r... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 132 | 135 | theorem generateFrom_eq_prod {C : Set (Set α)} {D : Set (Set β)} (hC : generateFrom C = ‹_›)
(hD : generateFrom D = ‹_›) (h2C : IsCountablySpanning C) (h2D : IsCountablySpanning D) :
generateFrom (image2 (· ×ˢ ·) C D) = Prod.instMeasurableSpace := by |
rw [← hC, ← hD, generateFrom_prod_eq h2C h2D]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 344 | 360 | theorem sheaf_condition : (Presieve.ofArrows X π).IsSheafFor P ↔
Nonempty (IsLimit (Fork.ofι (forkMap P X π) (w P X π))) := by |
rw [Types.type_equalizer_iff_unique, isSheafFor_arrows_iff]
erw [← Equiv.forall_congr_left (Types.productIso _).toEquiv.symm]
simp_rw [← compatible_iff, ← Iso.toEquiv_fun, Equiv.apply_symm_apply]
apply forall₂_congr
intro x _
apply exists_unique_congr
intro t
erw [Equiv.eq_symm_apply]
constructor
·... |
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