Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, 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 | 1,476 | 1,486 | theorem exp_approx_succ {n} {x aβ bβ : β} (m : β) (eβ : n + 1 = m) (aβ bβ : β)
(e : |1 + x / m * aβ - aβ| β€ bβ - |x| / m * bβ)
(h : |exp x - expNear m x aβ| β€ |x| ^ m / m.factorial * bβ) :
|exp x - expNear n x aβ| β€ |x| ^ n / n.factorial * bβ := by |
refine (abs_sub_le _ _ _).trans ((add_le_add_right h _).trans ?_)
subst eβ; rw [expNear_succ, expNear_sub, abs_mul]
convert mul_le_mul_of_nonneg_left (a := |x| ^ n / β(Nat.factorial n))
(le_sub_iff_add_le'.1 e) ?_ using 1
Β· simp [mul_add, pow_succ', div_eq_mul_inv, abs_mul, abs_inv, β pow_abs, mul_inv, N... |
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ri... | Mathlib/RingTheory/Nilpotent/Defs.lean | 81 | 85 | theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
IsNilpotent (f r) := by |
use hr.choose
rw [β map_pow, hr.choose_spec, map_zero]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.Quot... | Mathlib/Analysis/Normed/Group/Quotient.lean | 380 | 389 | theorem IsQuotient.norm_lift {f : NormedAddGroupHom M N} (hquot : IsQuotient f) {Ξ΅ : β} (hΞ΅ : 0 < Ξ΅)
(n : N) : β m : M, f m = n β§ βmβ < βnβ + Ξ΅ := by |
obtain β¨m, rflβ© := hquot.surjective n
have nonemp : ((fun m' => βm + m'β) '' f.ker).Nonempty := by
rw [Set.image_nonempty]
exact β¨0, f.ker.zero_memβ©
rcases Real.lt_sInf_add_pos nonemp hΞ΅
with β¨_, β¨β¨x, hx, rflβ©, H : βm + xβ < sInf ((fun m' : M => βm + m'β) '' f.ker) + Ξ΅β©β©
exact β¨m + x, by rw [map_ad... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 347 | 348 | theorem equivMapDomain_zero {f : Ξ± β Ξ²} : equivMapDomain f (0 : Ξ± ββ M) = (0 : Ξ² ββ M) := by |
ext; simp only [equivMapDomain_apply, coe_zero, Pi.zero_apply]
|
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 620 | 628 | theorem SMulCommClass.of_mclosure_eq_top {N Ξ±} [Monoid M] [SMul N Ξ±] [MulAction M Ξ±] {s : Set M}
(htop : Submonoid.closure s = β€) (hs : β x β s, β (y : N) (z : Ξ±), x β’ y β’ z = y β’ x β’ z) :
SMulCommClass M N Ξ± := by |
refine β¨fun x => Submonoid.induction_of_closure_eq_top_left htop x ?_ ?_β©
Β· intro y z
rw [one_smul, one_smul]
Β· clear x
intro x hx x' hx' y z
rw [mul_smul, mul_smul, hx', hs x hx]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology... | Mathlib/Topology/TietzeExtension.lean | 269 | 272 | theorem exists_extension_norm_eq_of_closedEmbedding (f : X βα΅ β) {e : X β Y}
(he : ClosedEmbedding e) : β g : Y βα΅ β, βgβ = βfβ β§ g β e = f := by |
rcases exists_extension_norm_eq_of_closedEmbedding' f β¨e, he.continuousβ© he with β¨g, hg, rflβ©
exact β¨g, hg, rflβ©
|
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 133 | 136 | theorem mono (f : Ξ± β E) {s t : Set Ξ±} (hst : t β s) : eVariationOn f t β€ eVariationOn f s := by |
apply iSup_le _
rintro β¨n, β¨u, hu, utβ©β©
exact sum_le f n hu fun i => hst (ut i)
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d449... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 83 | 85 | theorem Equiv.pointReflection_midpoint_right (x y : P) :
(Equiv.pointReflection (midpoint R x y)) y = x := by |
rw [midpoint_comm, Equiv.pointReflection_midpoint_left]
|
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"... | Mathlib/Data/Rat/Cast/CharZero.lean | 119 | 120 | theorem cast_mk (a b : β€) : (a /. b : Ξ±) = a / b := by |
simp only [divInt_eq_div, cast_div, cast_intCast]
|
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 116 | 119 | theorem angle_add_le_pi_div_two_of_inner_eq_zero {x y : V} (h : βͺx, yβ« = 0) :
angle x (x + y) β€ Ο / 2 := by |
rw [angle_add_eq_arccos_of_inner_eq_zero h, Real.arccos_le_pi_div_two]
exact div_nonneg (norm_nonneg _) (norm_nonneg _)
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 746 | 749 | theorem Asymptotics.IsBigO.hasFDerivAt {xβ : E} {n : β} (h : f =O[π xβ] fun x => βx - xββ ^ n)
(hn : 1 < n) : HasFDerivAt f (0 : E βL[π] F) xβ := by |
rw [β nhdsWithin_univ] at h
exact (h.hasFDerivWithinAt (mem_univ _) hn).hasFDerivAt_of_univ
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel ... | Mathlib/Data/Rel.lean | 119 | 122 | theorem comp_left_id (r : Rel Ξ± Ξ²) : @Eq Ξ± β’ r = r := by |
unfold comp
ext x
simp
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymp... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 294 | 300 | theorem IsEquivalent.inv (huv : u ~[l] v) : (fun x β¦ (u x)β»ΒΉ) ~[l] fun x β¦ (v x)β»ΒΉ := by |
rw [isEquivalent_iff_exists_eq_mul] at *
rcases huv with β¨Ο, hΟ, hβ©
rw [β inv_one]
refine β¨fun x β¦ (Ο x)β»ΒΉ, Tendsto.invβ hΟ (by norm_num), ?_β©
convert h.inv
simp [mul_comm]
|
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathli... | Mathlib/Probability/StrongLaw.lean | 178 | 180 | theorem integral_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f ΞΌ) {A : β}
(hA : 0 β€ A) : β« x, truncation f A x βΞΌ = β« y in -A..A, y βMeasure.map f ΞΌ := by |
simpa using moment_truncation_eq_intervalIntegral hf hA one_ne_zero
|
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 466 | 472 | theorem IsClique.card_le_of_coloring {s : Finset V} (h : G.IsClique s) [Fintype Ξ±]
(C : G.Coloring Ξ±) : s.card β€ Fintype.card Ξ± := by |
rw [isClique_iff_induce_eq] at h
have f : G.induce βs βͺg G := Embedding.comap (Function.Embedding.subtype fun x => x β βs) G
rw [h] at f
convert Fintype.card_le_of_injective _ (C.comp f.toHom).injective_of_top_hom using 1
simp
|
/-
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 | 627 | 632 | theorem mono_snd_of_is_pullback_of_mono {t : PullbackCone f g} (ht : IsLimit t) [Mono f] :
Mono t.snd := by |
refine β¨fun {W} h k i => IsLimit.hom_ext ht ?_ iβ©
rw [β cancel_mono f, Category.assoc, Category.assoc, condition]
have := congrArg (Β· β« g) i; dsimp at this
rwa [Category.assoc, Category.assoc] at this
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.Separation
import Mathlib.Topology.LocallyConstant.Basic
#align_import topology.discrete_quotient f... | Mathlib/Topology/DiscreteQuotient.lean | 337 | 340 | theorem ofLE_map (cond : LEComap f A B) (h : B β€ B') (a : A) :
ofLE h (map f cond a) = map f (cond.mono le_rfl h) a := by |
rcases a with β¨β©
rfl
|
/-
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 | 1,022 | 1,026 | theorem _root_.Real.toNNReal_inv {x : β} : Real.toNNReal xβ»ΒΉ = (Real.toNNReal x)β»ΒΉ := by |
rcases le_total 0 x with hx | hx
Β· nth_rw 1 [β Real.coe_toNNReal x hx]
rw [β NNReal.coe_inv, Real.toNNReal_coe]
Β· rw [toNNReal_eq_zero.mpr hx, inv_zero, toNNReal_eq_zero.mpr (inv_nonpos.mpr hx)]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 407 | 410 | theorem mem_map {f : Ξ± β Ξ²} {b : Ξ²} {z : Sym2 Ξ±} : b β Sym2.map f z β β a, a β z β§ f a = b := by |
induction' z using Sym2.ind with x y
simp only [map_pair_eq, mem_iff, exists_eq_or_imp, exists_eq_left]
aesop
|
/-
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
-/
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathl... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 483 | 485 | theorem nondegenerate_toBilin'_iff_det_ne_zero {M : Matrix ΞΉ ΞΉ A} :
M.toBilin'.Nondegenerate β M.det β 0 := by |
rw [Matrix.nondegenerate_toBilin'_iff, Matrix.nondegenerate_iff_det_ne_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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.... | Mathlib/MeasureTheory/Group/Measure.lean | 193 | 200 | theorem forall_measure_preimage_mul_right_iff (ΞΌ : Measure G) :
(β (g : G) (A : Set G), MeasurableSet A β ΞΌ ((fun h => h * g) β»ΒΉ' A) = ΞΌ A) β
IsMulRightInvariant ΞΌ := by |
trans β g, map (Β· * g) ΞΌ = ΞΌ
Β· simp_rw [Measure.ext_iff]
refine forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => ?_
rw [map_apply (measurable_mul_const g) hA]
exact β¨fun h => β¨hβ©, fun h => h.1β©
|
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, YaΓ«l Dillies
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Top... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 207 | 214 | theorem iInter_halfspaces_eq (hsβ : Convex β s) (hsβ : IsClosed s) :
β l : E βL[β] β, { x | β y β s, l x β€ l y } = s := by |
rw [Set.iInter_setOf]
refine Set.Subset.antisymm (fun x hx => ?_) fun x hx l => β¨x, hx, le_rflβ©
by_contra h
obtain β¨l, s, hlA, hlβ© := geometric_hahn_banach_closed_point hsβ hsβ h
obtain β¨y, hy, hxyβ© := hx l
exact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl
|
/-
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.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Properties ... | Mathlib/Topology/LocalAtTarget.lean | 159 | 166 | theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
OpenEmbedding f β β i, OpenEmbedding ((U i).1.restrictPreimage f) := by |
simp_rw [openEmbedding_iff]
rw [forall_and]
apply and_congr
Β· apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
Β· simp_rw [Set.range_restrictPreimage]
apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f... | Mathlib/Data/Multiset/Fintype.lean | 248 | 251 | theorem Multiset.prod_eq_prod_coe [CommMonoid Ξ±] (m : Multiset Ξ±) : m.prod = β x : m, (x : Ξ±) := by |
congr
-- Porting note: `simp` fails with "maximum recursion depth has been reached"
erw [map_univ_coe]
|
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.h... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 55 | 56 | theorem hermite_succ (n : β) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by |
rw [hermite]
|
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 610 | 611 | theorem mem_closure_iff_infDist_zero (h : s.Nonempty) : x β closure s β infDist x s = 0 := by |
simp [mem_closure_iff_infEdist_zero, infDist, ENNReal.toReal_eq_zero_iff, infEdist_ne_top h]
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 408 | 418 | theorem floor_logb_natCast {b : β} {r : β} (hb : 1 < b) (hr : 0 β€ r) :
βlogb b rβ = Int.log b r := by |
obtain rfl | hr := hr.eq_or_lt
Β· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
have hb1' : 1 < (b : β) := Nat.one_lt_cast.mpr hb
apply le_antisymm
Β· rw [β Int.zpow_le_iff_le_log hb hr, β rpow_intCast b]
refine le_of_le_of_eq ?_ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr)
exact rpow_le_rpow_... |
/-
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.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 151 | 152 | theorem card_congr {Ξ± Ξ²} [Fintype Ξ±] [Fintype Ξ²] (f : Ξ± β Ξ²) : card Ξ± = card Ξ² := by |
rw [β ofEquiv_card f]; congr; apply Subsingleton.elim
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 93 | 100 | theorem rotate'_length_mul (l : List Ξ±) : β n : β, l.rotate' (l.length * n) = l
| 0 => by simp
| n + 1 =>
calc
l.rotate' (l.length * (n + 1)) =
(l.rotate' (l.length * n)).rotate' (l.rotate' (l.length * n)).length := by |
simp [-rotate'_length, Nat.mul_succ, rotate'_rotate']
_ = l := by rw [rotate'_length, rotate'_length_mul l n]
|
/-
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 SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 569 | 570 | theorem arg_coe_angle_eq_iff {x y : β} : (arg x : Real.Angle) = arg y β arg x = arg y := by |
simp_rw [β Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,039 | 1,039 | theorem zero_mod (b : Ordinal) : 0 % b = 0 := by | simp only [mod_def, zero_div, mul_zero, sub_self]
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,004 | 1,006 | theorem coe_eq_castSucc {a : Fin n} : (a : Fin (n + 1)) = castSucc a := by |
ext
exact val_cast_of_lt (Nat.lt.step a.is_lt)
|
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Completion
import Mathlib.Topology.Metri... | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 790 | 956 | theorem totallyBounded {t : Set GHSpace} {C : β} {u : β β β} {K : β β β}
(ulim : Tendsto u atTop (π 0)) (hdiam : β p β t, diam (univ : Set (GHSpace.Rep p)) β€ C)
(hcov : β p β t, β n : β, β s : Set (GHSpace.Rep p),
(#s) β€ K n β§ univ β β x β s, ball x (u n)) :
TotallyBounded t := by |
/- Let `Ξ΄>0`, and `Ξ΅ = Ξ΄/5`. For each `p`, we construct a finite subset `s p` of `p`, which
is `Ξ΅`-dense and has cardinality at most `K n`. Encoding the mutual distances of points
in `s p`, up to `Ξ΅`, we will get a map `F` associating to `p` finitely many data, and making
it possible to reconstruct `p` u... |
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 593 | 602 | theorem LinearMap.toMatrix_apply (f : Mβ ββ[R] Mβ) (i : m) (j : n) :
LinearMap.toMatrix vβ vβ f i j = vβ.repr (f (vβ j)) i := by |
rw [LinearMap.toMatrix, LinearEquiv.trans_apply, LinearMap.toMatrix'_apply,
LinearEquiv.arrowCongr_apply, Basis.equivFun_symm_apply, Finset.sum_eq_single j, if_pos rfl,
one_smul, Basis.equivFun_apply]
Β· intro j' _ hj'
rw [if_neg hj', zero_smul]
Β· intro hj
have := Finset.mem_univ j
contradicti... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Basic
/-!
# Matroid Independence and Basis axioms
Matroids in mathlib are defined axiomatically in terms of bases,
but can be described just... | Mathlib/Data/Matroid/IndepAxioms.lean | 265 | 281 | theorem _root_.Matroid.existsMaximalSubsetProperty_of_bdd {P : Set Ξ± β Prop}
(hP : β (n : β), β Y, P Y β Y.encard β€ n) (X : Set Ξ±) : ExistsMaximalSubsetProperty P X := by |
obtain β¨n, hPβ© := hP
rintro I hI hIX
have hfin : Set.Finite (ncard '' {Y | P Y β§ I β Y β§ Y β X}) := by
rw [finite_iff_bddAbove, bddAbove_def]
simp_rw [ENat.le_coe_iff] at hP
use n
rintro x β¨Y, β¨hY,-,-β©, rflβ©
obtain β¨nβ, heq, hleβ© := hP Y hY
rwa [ncard_def, heq, ENat.toNat_coe]
-- have... |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Co... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 469 | 474 | theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv π 0 f x = p 0 := by |
ext
convert (h.hasSum <| EMetric.mem_ball_self h.r_pos).tsum_eq.symm
Β· rw [iteratedFDeriv_zero_apply, add_zero]
Β· rw [tsum_eq_single 0 fun n hn β¦ by haveI := NeZero.mk hn; exact (p n).map_zero]
exact congr(p 0 $(Subsingleton.elim _ _))
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 797 | 798 | theorem coeFn_le (f g : Lp.simpleFunc G p ΞΌ) : (f : Ξ± β G) β€α΅[ΞΌ] g β f β€ g := by |
rw [β Subtype.coe_le_coe, β Lp.coeFn_le]
|
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 331 | 339 | theorem descPochhammer_succ_comp_X_sub_one (n : β) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) β’ (descPochhammer R n).comp (X - 1) := by |
suffices (descPochhammer β€ (n + 1)).comp (X - 1) =
descPochhammer β€ (n + 1) - (n + 1) * (descPochhammer β€ n).comp (X - 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this
nth_rw 2 [descPochhammer_succ_left]
rw [β sub_mul, descPochhammer_succ_right β€ n, mul_comp, mul_comm, s... |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,369 | 1,372 | theorem toMeasureOfLEZero_toSignedMeasure (hs : s β€[Set.univ] 0) :
(s.toMeasureOfLEZero Set.univ MeasurableSet.univ hs).toSignedMeasure = -s := by |
ext i hi
simp [hi, toMeasureOfLEZero_apply _ _ _ hi]
|
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 121 | 136 | theorem left_invariance {s : Set H} {x : H} {f : H β H'} {e' : PartialHomeomorph H' H'}
(he' : e' β G') (hfs : ContinuousWithinAt f s x) (hxe' : f x β e'.source) :
P (e' β f) s x β P f s x := by |
have h2f := hfs.preimage_mem_nhdsWithin (e'.open_source.mem_nhds hxe')
have h3f :=
((e'.continuousAt hxe').comp_continuousWithinAt hfs).preimage_mem_nhdsWithin <|
e'.symm.open_source.mem_nhds <| e'.mapsTo hxe'
constructor
Β· intro h
rw [hG.is_local_nhds h3f] at h
have h2 := hG.left_invariance'... |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.Norm
#align_import linear_algebra.free_module.norm from "leanprover-community/mathlib"@"90b0d53... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 74 | 84 | theorem finrank_quotient_span_eq_natDegree_norm [Algebra F S] [IsScalarTower F F[X] S]
(b : Basis ΞΉ F[X] S) {f : S} (hf : f β 0) :
FiniteDimensional.finrank F (S β§Έ span ({f} : Set S)) = (Algebra.norm F[X] f).natDegree := by |
haveI := Fintype.ofFinite ΞΉ
have h := span_singleton_eq_bot.not.2 hf
rw [natDegree_eq_of_degree_eq
(degree_eq_degree_of_associated <| associated_norm_prod_smith b hf)]
rw [natDegree_prod _ _ fun i _ => smithCoeffs_ne_zero b _ h i, finrank_quotient_eq_sum F h b]
-- finrank_quotient_eq_sum slow
congr w... |
/-
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 SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 156 | 165 | theorem exp_eq_one_iff {x : β} : exp x = 1 β β n : β€, x = n * (2 * Ο * I) := by |
constructor
Β· intro h
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos x.im (-Ο) with β¨n, hn, -β©
use -n
rw [Int.cast_neg, neg_mul, eq_neg_iff_add_eq_zero]
have : (x + n * (2 * Ο * I)).im β Set.Ioc (-Ο) Ο := by simpa [two_mul, mul_add] using hn
rw [β log_exp this.1 this.2, exp_periodic.int_... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Aesop
import Mathlib.Order.BoundedOrder
#align_import order.disjoint from "leanprover-community/mathlib"@"22c4d2ff43714b6ff724b2745ccfdc0f236a4a76"
/-!
# Dis... | Mathlib/Order/Disjoint.lean | 147 | 148 | theorem disjoint_assoc : Disjoint (a β b) c β Disjoint a (b β c) := by |
rw [disjoint_iff_inf_le, disjoint_iff_inf_le, inf_assoc]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 192 | 195 | theorem mk_eq_mk_iff {p q : Ξ± Γ Ξ±} : Sym2.mk p = Sym2.mk q β p = q β¨ p = q.swap := by |
cases p
cases q
simp only [eq_iff, Prod.mk.inj_iff, Prod.swap_prod_mk]
|
/-
Copyright (c) 2019 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 293 | 302 | theorem det_eq_one_of_finrank_eq_zero {π : Type*} [Field π] {M : Type*} [AddCommGroup M]
[Module π M] (h : FiniteDimensional.finrank π M = 0) (f : M ββ[π] M) :
LinearMap.det (f : M ββ[π] M) = 1 := by |
classical
refine @LinearMap.det_cases M _ π _ _ _ (fun t => t = 1) f ?_ rfl
intro s b
have : IsEmpty s := by
rw [β Fintype.card_eq_zero_iff]
exact (FiniteDimensional.finrank_eq_card_basis b).symm.trans h
exact Matrix.det_isEmpty
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76... | Mathlib/RingTheory/IsTensorProduct.lean | 60 | 65 | theorem TensorProduct.isTensorProduct : IsTensorProduct (TensorProduct.mk R M N) := by |
delta IsTensorProduct
convert_to Function.Bijective (LinearMap.id : M β[R] N ββ[R] M β[R] N) using 2
Β· apply TensorProduct.ext'
simp
Β· exact Function.bijective_id
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 553 | 556 | theorem le_rootMultiplicity_mul {p q : R[X]} (x : R) (hpq : p * q β 0) :
rootMultiplicity x p + rootMultiplicity x q β€ rootMultiplicity x (p * q) := by |
rw [le_rootMultiplicity_iff hpq, pow_add]
exact mul_dvd_mul (pow_rootMultiplicity_dvd p x) (pow_rootMultiplicity_dvd q x)
|
/-
Copyright (c) 2016 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathli... | Mathlib/Logic/Function/Basic.lean | 89 | 91 | theorem Injective.beq_eq {Ξ± Ξ² : Type*} [BEq Ξ±] [LawfulBEq Ξ±] [BEq Ξ²] [LawfulBEq Ξ²] {f : Ξ± β Ξ²}
(I : Injective f) {a b : Ξ±} : (f a == f b) = (a == b) := by |
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
|
/-
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 | 1,225 | 1,225 | theorem abs_zero : (0 : EReal).abs = 0 := by | rw [abs_eq_zero_iff]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,923 | 1,925 | theorem supports_biUnion {K : Option Ξ' β Finset Ξ'} {S} :
Supports (Finset.univ.biUnion K) S β β a, Supports (K a) S := by |
simp [Supports]; apply forall_swap
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,478 | 1,481 | theorem set_pi_mem_nhds {i : Set ΞΉ} {s : β a, Set (Ο a)} {x : β a, Ο a} (hi : i.Finite)
(hs : β a β i, s a β π (x a)) : pi i s β π x := by |
rw [pi_def, biInter_mem hi]
exact fun a ha => (continuous_apply a).continuousAt (hs a ha)
|
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 404 | 424 | theorem exists_norm_le_le_norm_sub_of_finset {c : π} (hc : 1 < βcβ) {R : β} (hR : βcβ < R)
(h : Β¬FiniteDimensional π E) (s : Finset E) : β x : E, βxβ β€ R β§ β y β s, 1 β€ βy - xβ := by |
let F := Submodule.span π (s : Set E)
haveI : FiniteDimensional π F :=
Module.finite_def.2
((Submodule.fg_top _).2 (Submodule.fg_def.2 β¨s, Finset.finite_toSet _, rflβ©))
have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
have : β x, x β F := by
contrapose! h
have ... |
/-
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 | 513 | 515 | theorem lt_add_one {x : PartENat} (hx : x β β€) : x < x + 1 := by |
rw [PartENat.lt_add_iff_pos_right hx]
norm_cast
|
/-
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, SΓ©bastien GouΓ«zel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 303 | 305 | theorem div_const {π : Type*} {f : β β π} [NormedField π] (h : IntervalIntegrable f ΞΌ a b)
(c : π) : IntervalIntegrable (fun x => f x / c) ΞΌ a b := by |
simpa only [div_eq_mul_inv] using mul_const h cβ»ΒΉ
|
/-
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.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.M... | Mathlib/Analysis/Convex/Integral.lean | 326 | 339 | theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace β E]
(h_le : βα΅ x βΞΌ, βf xβ β€ C) : f =α΅[ΞΌ] const Ξ± (β¨ x, f x βΞΌ) β¨ ββ¨ x, f x βΞΌβ < C := by |
rcases le_or_lt C 0 with hC0 | hC0
Β· have : f =α΅[ΞΌ] 0 := h_le.mono fun x hx => norm_le_zero_iff.1 (hx.trans hC0)
simp only [average_congr this, Pi.zero_apply, average_zero]
exact Or.inl this
by_cases hfi : Integrable f ΞΌ; swap
Β· simp [average_eq, integral_undef hfi, hC0, ENNReal.toReal_pos_iff]
rcase... |
/-
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.Polynomial.Expand
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
import Mathlib.RingTheory.Adjoin.... | Mathlib/RingTheory/IntegralClosure.lean | 623 | 637 | theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
p.leadingCoeff β’ normalizeScaleRoots p = scaleRoots p p.leadingCoeff := by |
ext
simp only [coeff_scaleRoots, normalizeScaleRoots, coeff_monomial, coeff_smul, Finset.smul_sum,
Ne, Finset.sum_ite_eq', finset_sum_coeff, smul_ite, smul_zero, mem_support_iff]
-- Porting note: added the following `simp only`
simp only [ge_iff_le, tsub_le_iff_right, smul_eq_mul, mul_ite, mul_one, mul_zer... |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 345 | 351 | theorem involute_eq_of_mem_odd {x : CliffordAlgebra Q} (h : x β evenOdd Q 1) : involute x = -x := by |
induction x, h using odd_induction with
| ΞΉ m => exact involute_ΞΉ _
| add x y _hx _hy ihx ihy =>
rw [map_add, ihx, ihy, neg_add]
| ΞΉ_mul_ΞΉ_mul mβ mβ x _hx ihx =>
rw [map_mul, map_mul, involute_ΞΉ, involute_ΞΉ, ihx, neg_mul_neg, mul_neg]
|
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 650 | 656 | theorem w : forkMap R P β« firstMap R P = forkMap R P β« secondMap R P := by |
apply limit.hom_ext
rintro β¨β¨Y, f, hfβ©, β¨Z, g, hgβ©β©
simp only [firstMap, secondMap, forkMap, limit.lift_Ο, limit.lift_Ο_assoc, assoc, Fan.mk_Ο_app,
Subtype.coe_mk]
rw [β P.map_comp, β op_comp, pullback.condition]
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.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,002 | 1,006 | theorem toIcoDiv_eq_floor (a b : Ξ±) : toIcoDiv hp a b = β(b - a) / pβ := by |
refine toIcoDiv_eq_of_sub_zsmul_mem_Ico hp ?_
rw [Set.mem_Ico, zsmul_eq_mul, β sub_nonneg, add_comm, sub_right_comm, β sub_lt_iff_lt_add,
sub_right_comm _ _ a]
exact β¨Int.sub_floor_div_mul_nonneg _ hp, Int.sub_floor_div_mul_lt _ hpβ©
|
/-
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.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 766 | 776 | theorem isSheafFor_arrows_iff : (ofArrows X Ο).IsSheafFor P β
(β (x : (i : I) β P.obj (op (X i))), Arrows.Compatible P Ο x β
β! t, β i, P.map (Ο i).op t = x i) := by |
refine β¨fun h x hx β¦ ?_, fun h x hx β¦ ?_β©
Β· obtain β¨t, htβ, htββ© := h _ hx.familyOfElements_compatible
refine β¨t, fun i β¦ ?_, fun t' ht' β¦ htβ _ fun _ _ β¨iβ© β¦ ?_β©
Β· rw [htβ _ (ofArrows.mk i), hx.familyOfElements_ofArrows_mk]
Β· rw [ht', hx.familyOfElements_ofArrows_mk]
Β· obtain β¨t, hA, htβ© := h (fun i... |
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 343 | 353 | theorem map_nullHomotopicMap' {W : Type*} [Category W] [Preadditive W] (G : V β₯€ W) [G.Additive]
(hom : β i j, c.Rel j i β (C.X i βΆ D.X j)) :
(G.mapHomologicalComplex c).map (nullHomotopicMap' hom) =
nullHomotopicMap' fun i j hij => by exact G.map (hom i j hij) := by |
ext n
erw [map_nullHomotopicMap]
congr
ext i j
split_ifs
Β· rfl
Β· rw [G.map_zero]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
#align_import algebra.direct_sum.ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Algebra/DirectSum/Ring.lean | 291 | 299 | theorem ofList_dProd {Ξ±} (l : List Ξ±) (fΞΉ : Ξ± β ΞΉ) (fA : β a, A (fΞΉ a)) :
of A _ (l.dProd fΞΉ fA) = (l.map fun a => of A (fΞΉ a) (fA a)).prod := by |
induction' l with head tail
Β· simp only [List.map_nil, List.prod_nil, List.dProd_nil]
rfl
Β· rename_i ih
simp only [List.map_cons, List.prod_cons, List.dProd_cons, β ih]
rw [DirectSum.of_mul_of (fA head)]
rfl
|
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 637 | 639 | theorem neg_pi_div_two_ne_zero : ((-Ο / 2 : β) : Angle) β 0 := by |
rw [β toReal_injective.ne_iff, toReal_neg_pi_div_two, toReal_zero]
exact div_ne_zero (neg_ne_zero.2 Real.pi_ne_zero) two_ne_zero
|
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 283 | 288 | theorem chromaticNumber_ne_top_iff_exists : G.chromaticNumber β β€ β β n, G.Colorable n := by |
rw [chromaticNumber]
convert_to β¨
n : {m | G.Colorable m}, (n : ββ) β β€ β _
Β· rw [iInf_subtype]
rw [β lt_top_iff_ne_top, ENat.iInf_coe_lt_top]
simp
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.Gro... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 182 | 191 | theorem mul_mem_nonZeroDivisors {a b : Mβ} : a * b β Mββ° β a β Mββ° β§ b β Mββ° := by |
constructor
Β· intro h
constructor <;> intro x h' <;> apply h
Β· rw [β mul_assoc, h', zero_mul]
Β· rw [mul_comm a b, β mul_assoc, h', zero_mul]
Β· rintro β¨ha, hbβ© x hx
apply ha
apply hb
rw [mul_assoc, hx]
|
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import a... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 252 | 269 | theorem continuousAt_gaussian_integral (b : β) (hb : 0 < re b) :
ContinuousAt (fun c : β => β« x : β, cexp (-c * (x : β) ^ 2)) b := by |
let f : β β β β β := fun (c : β) (x : β) => cexp (-c * (x : β) ^ 2)
obtain β¨d, hd, hd'β© := exists_between hb
have f_meas : β c : β, AEStronglyMeasurable (f c) volume := fun c => by
apply Continuous.aestronglyMeasurable
exact Complex.continuous_exp.comp (continuous_const.mul (continuous_ofReal.pow 2))
h... |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 491 | 501 | theorem lowerCentralSeries.map {H : Type*} [Group H] (f : G β* H) (n : β) :
Subgroup.map f (lowerCentralSeries G n) β€ lowerCentralSeries H n := by |
induction' n with d hd
Β· simp [Nat.zero_eq]
Β· rintro a β¨x, hx : x β lowerCentralSeries G d.succ, rflβ©
refine closure_induction hx ?_ (by simp [f.map_one, Subgroup.one_mem _])
(fun y z hy hz => by simp [MonoidHom.map_mul, Subgroup.mul_mem _ hy hz]) (fun y hy => by
rw [f.map_inv]; exact Subgroup.... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 153 | 161 | theorem evalβ_sum (p : T[X]) (g : β β T β R[X]) (x : S) :
(p.sum g).evalβ f x = p.sum fun n a => (g n a).evalβ f x := by |
let T : R[X] β+ S :=
{ toFun := evalβ f x
map_zero' := evalβ_zero _ _
map_add' := fun p q => evalβ_add _ _ }
have A : β y, evalβ f x y = T y := fun y => rfl
simp only [A]
rw [sum, map_sum, sum]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, 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 | 1,741 | 1,743 | theorem eventuallyLE_antisymm_iff [PartialOrder Ξ²] {l : Filter Ξ±} {f g : Ξ± β Ξ²} :
f =αΆ [l] g β f β€αΆ [l] g β§ g β€αΆ [l] f := by |
simp only [EventuallyEq, EventuallyLE, le_antisymm_iff, eventually_and]
|
/-
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.Data.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 1,272 | 1,291 | theorem pair_injective : Function.Injective2 pair := fun x x' y y' H => by
have ae := ext_iff.1 H
simp only [pair, mem_pair] at ae
obtain rfl : x = x' := by |
cases' (ae {x}).1 (by simp) with h h
Β· exact singleton_injective h
Β· have m : x' β ({x} : ZFSet) := by simp [h]
rw [mem_singleton.mp m]
have he : x = y β y = y' := by
rintro rfl
cases' (ae {x, y'}).2 (by simp only [eq_self_iff_true, or_true_iff]) with xy'x xy'xx
Β· rw [eq_comm, β mem_sin... |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Algebra.Lie.... | Mathlib/Algebra/Lie/Classical.lean | 122 | 130 | theorem sl_non_abelian [Fintype n] [Nontrivial R] (h : 1 < Fintype.card n) :
Β¬IsLieAbelian (sl n R) := by |
rcases Fintype.exists_pair_of_one_lt_card h with β¨j, i, hijβ©
let A := Eb R i j hij
let B := Eb R j i hij.symm
intro c
have c' : A.val * B.val = B.val * A.val := by
rw [β sub_eq_zero, β sl_bracket, c.trivial, ZeroMemClass.coe_zero]
simpa [A, B, stdBasisMatrix, Matrix.mul_apply, hij] using congr_fun (con... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
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 | 607 | 608 | theorem prod_sum_elim (s : Finset Ξ±) (t : Finset Ξ³) (f : Ξ± β Ξ²) (g : Ξ³ β Ξ²) :
β x β s.disjSum t, Sum.elim f g x = (β x β s, f x) * β x β t, g x := by | simp
|
/-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
import Mathlib.Analysis.Fourier.Inversion
/-!
# Mellin inversion formula
We derive the Mellin inversion formula as a consequence of the Fourier inversion formula.
## Mai... | Mathlib/Analysis/MellinInversion.lean | 89 | 121 | theorem mellin_inversion (Ο : β) (f : β β E) {x : β} (hx : 0 < x) (hf : MellinConvergent f Ο)
(hFf : VerticalIntegrable (mellin f) Ο) (hfx : ContinuousAt f x) :
mellinInv Ο (mellin f) x = f x := by |
let g := fun (u : β) => Real.exp (-Ο * u) β’ f (Real.exp (-u))
replace hf : Integrable g := by
rw [MellinConvergent, β rexp_neg_image_aux, integrableOn_image_iff_integrableOn_abs_deriv_smul
MeasurableSet.univ rexp_neg_deriv_aux rexp_neg_injOn_aux] at hf
replace hf : Integrable fun (x : β) β¦ cexp (-βΟ ... |
/-
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
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 40 | 41 | theorem gcd_add_mul_left_right (m n k : β) : gcd m (n + m * k) = gcd m n := by |
simp [gcd_rec m (n + m * k), gcd_rec m n]
|
/-
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.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import M... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 253 | 256 | theorem Inv_divided_by_X_pow_order_leftInv {f : kβ¦Xβ§} (hf : f β 0) :
(Inv_divided_by_X_pow_order hf) * (divided_by_X_pow_order hf) = 1 := by |
rw [mul_comm]
exact mul_invOfUnit (divided_by_X_pow_order hf) (firstUnitCoeff hf) rfl
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l 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 | 354 | 355 | theorem count_nth_succ {n : β} (hn : β hf : (setOf p).Finite, n < hf.toFinset.card) :
count p (nth p n + 1) = n + 1 := by | rw [count_succ, count_nth hn, if_pos (nth_mem _ hn)]
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanp... | Mathlib/RingTheory/Jacobson.lean | 426 | 452 | theorem isJacobson_polynomial_of_isJacobson (hR : IsJacobson R) : IsJacobson R[X] := by |
rw [isJacobson_iff_prime_eq]
intro I hI
let R' : Subring (R[X] β§Έ I) := ((Quotient.mk I).comp C).range
let i : R β+* R' := ((Quotient.mk I).comp C).rangeRestrict
have hi : Function.Surjective βi := ((Quotient.mk I).comp C).rangeRestrict_surjective
have hi' : RingHom.ker (mapRingHom i) β€ I := by
intro f ... |
/-
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 | 385 | 388 | theorem norm_eq_pow_val {x : β€_[p]} (hx : x β 0) : βxβ = (p : β) ^ (-x.valuation) := by |
refine @Padic.norm_eq_pow_val p hp x ?_
contrapose! hx
exact Subtype.val_injective hx
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 125 | 128 | theorem neg_one_pow_div_two_of_three_mod_four {n : β} (hn : n % 4 = 3) :
(-1 : β€) ^ (n / 2) = -1 := by |
rw [β Οβ_eq_neg_one_pow (Nat.odd_of_mod_four_eq_three hn), β natCast_mod, hn]
rfl
|
/-
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.Logic.Encodable.Basic
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.Subsingleton
#align_import logic.encodable.lattice from "leanprover-co... | Mathlib/Logic/Encodable/Lattice.lean | 30 | 33 | theorem iSup_decodeβ [CompleteLattice Ξ±] (f : Ξ² β Ξ±) :
β¨ (i : β) (b β decodeβ Ξ² i), f b = (β¨ b, f b) := by |
rw [iSup_comm]
simp only [mem_decodeβ, iSup_iSup_eq_right]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, YaΓ«l Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 573 | 576 | theorem measure_average_le_pos (hΞΌ : ΞΌ β 0) (hf : Integrable f ΞΌ) :
0 < ΞΌ {x | β¨ a, f a βΞΌ β€ f x} := by |
simpa using measure_setAverage_le_pos (Measure.measure_univ_ne_zero.2 hΞΌ) (measure_ne_top _ _)
hf.integrableOn
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, YaΓ«l Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 640 | 643 | theorem exists_not_mem_null_le_integral (hf : Integrable f ΞΌ) (hN : ΞΌ N = 0) :
β x, x β N β§ f x β€ β« a, f a βΞΌ := by |
simpa only [average_eq_integral] using
exists_not_mem_null_le_average (IsProbabilityMeasure.ne_zero ΞΌ) hf hN
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,601 | 1,605 | theorem Tendsto.prod_atTop [SemilatticeSup Ξ±] [SemilatticeSup Ξ³] {f g : Ξ± β Ξ³}
(hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
Tendsto (Prod.map f g) atTop atTop := by |
rw [β prod_atTop_atTop_eq]
exact hf.prod_map_prod_atTop hg
|
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Order.Hom.Basic
import Mathlib.Logic.Relation
#align_import order.antisymmetrization from "leanprover-community/mathlib"@"3353f661228bd27f632c600cd1a58b87... | Mathlib/Order/Antisymmetrization.lean | 153 | 156 | theorem antisymmetrization_fibration :
Relation.Fibration (Β· < Β·) (Β· < Β·) (@toAntisymmetrization Ξ± (Β· β€ Β·) _) := by |
rintro a β¨bβ© h
exact β¨b, h, rflβ©
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,085 | 1,086 | theorem mk_arrow_eq_zero_iff : #(Ξ± β Ξ²') = 0 β #Ξ± β 0 β§ #Ξ²' = 0 := by |
simp_rw [mk_eq_zero_iff, mk_ne_zero_iff, isEmpty_fun]
|
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomia... | Mathlib/Algebra/Polynomial/CancelLeads.lean | 52 | 71 | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree β€ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by |
by_cases hp : p = 0
Β· convert hq
simp [hp, cancelLeads]
rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
by_cases h0 :
C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
Β· exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
... |
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 628 | 629 | theorem toReal_eq_neg_pi_div_two_iff {ΞΈ : Angle} : ΞΈ.toReal = -Ο / 2 β ΞΈ = (-Ο / 2 : β) := by |
rw [β toReal_inj, toReal_neg_pi_div_two]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 1,533 | 1,589 | theorem union_support_maximal_linearIndependent_eq_range_basis {ΞΉ : Type w} (b : Basis ΞΉ R M)
{ΞΊ : Type w'} (v : ΞΊ β M) (i : LinearIndependent R v) (m : i.Maximal) :
β k, ((b.repr (v k)).support : Set ΞΉ) = Set.univ := by |
-- If that's not the case,
by_contra h
simp only [β Ne.eq_def, ne_univ_iff_exists_not_mem, mem_iUnion, not_exists_not,
Finsupp.mem_support_iff, Finset.mem_coe] at h
-- We have some basis element `b b'` which is not in the support of any of the `v i`.
obtain β¨b', wβ© := h
-- Using this, we'll construct a... |
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 604 | 604 | theorem toReal_eq_pi_iff {ΞΈ : Angle} : ΞΈ.toReal = Ο β ΞΈ = Ο := by | rw [β toReal_inj, toReal_pi]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# The... | Mathlib/CategoryTheory/Comma/Arrow.lean | 86 | 88 | theorem mk_eq (f : Arrow T) : Arrow.mk f.hom = f := by |
cases f
rfl
|
/-
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 SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 193 | 199 | theorem rpow_eq_nhds_of_pos {p : β Γ β} (hp_fst : 0 < p.fst) :
(fun x : β Γ β => x.1 ^ x.2) =αΆ [π p] fun x => exp (log x.1 * x.2) := by |
suffices βαΆ x : β Γ β in π p, 0 < x.1 from
this.mono fun x hx β¦ by
dsimp only
rw [rpow_def_of_pos hx]
exact IsOpen.eventually_mem (isOpen_lt continuous_const continuous_fst) hp_fst
|
/-
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 | 210 | 210 | theorem IsBigOWith.isBigO (h : IsBigOWith c l f g) : f =O[l] g := by | rw [IsBigO_def]; exact β¨c, hβ©
|
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,316 | 1,316 | theorem eq_div_iff_mul_eq'' : a = b / c β c * a = b := by | rw [eq_div_iff_mul_eq', mul_comm]
|
/-
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 | 2,961 | 2,966 | theorem comap_map_eq (H : Subgroup G) : comap f (map f H) = H β f.ker := by |
refine le_antisymm ?_ (sup_le (le_comap_map _ _) (ker_le_comap _ _))
intro x hx; simp only [exists_prop, mem_map, mem_comap] at hx
rcases hx with β¨y, hy, hy'β©
rw [β mul_inv_cancel_left y x]
exact mul_mem_sup hy (by simp [mem_ker, hy'])
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 265 | 265 | theorem relindex_bot_right : H.relindex β₯ = 1 := by | rw [relindex, subgroupOf_bot_eq_top, index_top]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 457 | 458 | theorem derivative_sq (p : R[X]) : derivative (p ^ 2) = C 2 * p * derivative p := by |
rw [derivative_pow_succ, Nat.cast_one, one_add_one_eq_two, pow_one]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 28 | 29 | theorem zipWith_distrib_tail : (zipWith f l l').tail = zipWith f l.tail l'.tail := by |
rw [β drop_one]; simp [zipWith_distrib_drop]
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-com... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 70 | 71 | theorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} :
G.dist u v = 0 β u = v β¨ Β¬G.Reachable u v := by | simp [dist, Nat.sInf_eq_zero, Reachable]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, 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 | 262 | 262 | theorem degree_one_le : degree (1 : R[X]) β€ (0 : WithBot β) := by | rw [β C_1]; exact degree_C_le
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Order.Lattice
import Mathlib.Order.ULift
import Mathlib.Tactic.PushNeg
#align_import order.bounded_order from "leanprover-community/mathlib"@"70d50ecf... | Mathlib/Order/BoundedOrder.lean | 192 | 197 | theorem OrderTop.ext_top {Ξ±} {hA : PartialOrder Ξ±} (A : OrderTop Ξ±) {hB : PartialOrder Ξ±}
(B : OrderTop Ξ±) (H : β x y : Ξ±, (haveI := hA; x β€ y) β x β€ y) :
(@Top.top Ξ± (@OrderTop.toTop Ξ± hA.toLE A)) = (@Top.top Ξ± (@OrderTop.toTop Ξ± hB.toLE B)) := by |
cases PartialOrder.ext H
apply top_unique
exact @le_top _ _ A _
|
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