Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
|---|---|---|---|---|---|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 608 | 609 | theorem I_sq : I ^ 2 = -1 := by | rw [sq, I_mul_I] |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Complex.Norm
/-!
# The partial order on the complex numbers
This order is defined by `z β€ w β z.re β€ w.re β§ z.im = w.im`.
This is a natural order o... | Mathlib/Data/Complex/Order.lean | 87 | 88 | theorem not_lt_iff {z w : β} : Β¬z < w β w.re β€ z.re β¨ z.im β w.im := by | rw [lt_def, not_and_or, not_lt] |
/-
Copyright (c) 2022 JoΓ«l Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
/-!
# The Karoubi envelope ... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 85 | 85 | theorem p_comm {P Q : Karoubi C} (f : Hom P Q) : P.p β« f.f = f.f β« Q.p := by | rw [p_comp, comp_p] |
/-
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.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | Mathlib/Algebra/GeomSum.lean | 162 | 166 | theorem geom_sumβ_mul_of_ge [CommSemiring R] [PartialOrder R] [AddLeftReflectLE R] [AddLeftMono R]
[ExistsAddOfLE R] [Sub R] [OrderedSub R] {x y : R} (hxy : y β€ x) (n : β) :
(β i β range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n := by | apply eq_tsub_of_add_eq
simpa only [tsub_add_cancel_of_le hxy] using geom_sumβ_mul_add (x - y) y n |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... | Mathlib/Data/List/FinRange.lean | 44 | 47 | theorem pairwise_lt_finRange (n : β) : Pairwise (Β· < Β·) (finRange n) := by | rw [finRange_eq_pmap_range]
exact List.pairwise_lt_range.pmap (by simp) (by simp) |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 213 | 217 | theorem dist_eq_abs_sub_dist_iff_angle_eq_zero {pβ pβ pβ : P} (hpβpβ : pβ β pβ) (hpβpβ : pβ β pβ) :
dist pβ pβ = |dist pβ pβ - dist pβ pβ| β β pβ pβ pβ = 0 := by | rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, β vsub_sub_vsub_cancel_right]
exact
norm_sub_eq_abs_sub_norm_iff_angle_eq_zero (fun he => hpβpβ (vsub_eq_zero_iff_eq.1 he)) |
/-
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, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Algebra.Group.Nat.Range
import Mathlib.Data.Set.Finite.Basic
/-!
# Counting on β
Thi... | Mathlib/Data/Nat/Count.lean | 86 | 88 | theorem count_one : count p 1 = if p 0 then 1 else 0 := by | simp [count_succ]
theorem count_succ' (n : β) : |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | Mathlib/Data/List/Sublists.lean | 169 | 173 | theorem length_sublists (l : List Ξ±) : length (sublists l) = 2 ^ length l := by | simp only [sublists_eq_sublists', length_map, length_sublists', length_reverse]
theorem map_pure_sublist_sublists (l : List Ξ±) : map pure l <+ sublists l := by
induction' l using reverseRecOn with l a ih <;> simp only [map, map_append, sublists_concat] |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | Mathlib/LinearAlgebra/Lagrange.lean | 569 | 571 | theorem nodalWeight_ne_zero (hvs : Set.InjOn v s) (hi : i β s) : nodalWeight s v i β 0 := by | rw [nodalWeight, prod_ne_zero_iff]
intro j hj |
/-
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.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 133 | 136 | theorem IsEquivalent.tendsto_const {c : Ξ²} (hu : u ~[l] const _ c) : Tendsto u l (π c) := by | rcases em <| c = 0 with rfl | h
Β· exact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds
Β· exact (isEquivalent_const_iff_tendsto h).mp hu |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... | Mathlib/Data/List/Permutation.lean | 244 | 246 | theorem permutations_append (is ts : List Ξ±) :
permutations (is ++ ts) = (permutations is).map (Β· ++ ts) ++ permutationsAux ts is.reverse := by | simp [permutations, permutationsAux_append] |
/-
Copyright (c) 2021 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 305 | 311 | theorem tendsto_iff_forall_lintegral_tendsto {Ξ³ : Type*} {F : Filter Ξ³}
{ΞΌs : Ξ³ β ProbabilityMeasure Ξ©} {ΞΌ : ProbabilityMeasure Ξ©} :
Tendsto ΞΌs F (π ΞΌ) β
β f : Ξ© βα΅ ββ₯0,
Tendsto (fun i β¦ β«β» Ο, f Ο β(ΞΌs i : Measure Ξ©)) F (π (β«β» Ο, f Ο β(ΞΌ : Measure Ξ©))) := by | rw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]
exact FiniteMeasure.tendsto_iff_forall_lintegral_tendsto |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
/-!
# Angles between vectors
This fil... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 49 | 54 | theorem angle_smul_smul {c : β} (hc : c β 0) (x y : V) : angle (c β’ x) (c β’ y) = angle x y := by | have : c * c β 0 := mul_ne_zero hc hc
rw [angle, angle, real_inner_smul_left, inner_smul_right, norm_smul, norm_smul, Real.norm_eq_abs,
mul_mul_mul_comm _ βxβ, abs_mul_abs_self, β mul_assoc c c, mul_div_mul_left _ _ this]
@[simp] |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Calculus.Deriv.Basic
/-!
# One-dimensional derivatives on pi-types.
-/
varia... | Mathlib/Analysis/Calculus/Deriv/Pi.lean | 15 | 22 | theorem hasDerivAt_update (x : ΞΉ β π) (i : ΞΉ) (y : π) :
HasDerivAt (Function.update x i) (Pi.single i (1 : π)) y := by | convert (hasFDerivAt_update x y).hasDerivAt
ext z j
rw [Pi.single, Function.update_apply]
split_ifs with h
Β· simp [h]
Β· simp [Pi.single_eq_of_ne h] |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 101 | 103 | theorem exists_rat_eq_nth_num : β q : β, (of v).nums n = (q : K) := by | rcases exists_gcf_pair_rat_eq_nth_conts v n with β¨β¨a, _β©, nth_cont_eqβ©
use a |
/-
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.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | Mathlib/Analysis/Convex/Side.lean | 431 | 433 | theorem sOppSide_iff_exists_left {s : AffineSubspace R P} {x y pβ : P} (h : pβ β s) :
s.SOppSide x y β x β s β§ y β s β§ β pβ β s, SameRay R (x -α΅₯ pβ) (pβ -α΅₯ y) := by | rw [SOppSide, and_comm, wOppSide_iff_exists_left h, and_assoc, and_congr_right_iff] |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.NumberField.Embeddings
/-!
# Cyclotomic extensions of `β` are totally complex nu... | Mathlib/NumberTheory/Cyclotomic/Embeddings.lean | 41 | 60 | theorem nrComplexPlaces_eq_totient_div_two [h : IsCyclotomicExtension {n} β K] :
haveI := IsCyclotomicExtension.numberField {n} β K
nrComplexPlaces K = Ο n / 2 := by | have := IsCyclotomicExtension.numberField {n} β K
by_cases hn : 2 < n
Β· obtain β¨k, hk : Ο n = k + kβ© := totient_even hn
have key := card_add_two_mul_card_eq_rank K
rw [nrRealPlaces_eq_zero K hn, zero_add, IsCyclotomicExtension.finrank (n := n) K
(cyclotomic.irreducible_rat n.pos), hk, β two_mul, Nat.m... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 735 | 743 | theorem map_think' {Ξ± Ξ²} : β (f : Ξ± β Ξ²) (s), f <$> think s = think (f <$> s) :=
map_think
theorem mem_map (f : Ξ± β Ξ²) {a} {s : Computation Ξ±} (m : a β s) : f a β map f s := by | rw [β bind_pure]; apply mem_bind m; apply ret_mem
theorem exists_of_mem_map {f : Ξ± β Ξ²} {b : Ξ²} {s : Computation Ξ±} (h : b β map f s) :
β a, a β s β§ f a = b := by
rw [β bind_pure] at h |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 343 | 344 | theorem continuousOn_log : ContinuousOn log {0}αΆ := by | simp +unfoldPartialApp only [continuousOn_iff_continuous_restrict, |
/-
Copyright (c) 2021 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 325 | 331 | theorem gauge_eq_zero (hs : Absorbent β s) (hb : Bornology.IsVonNBounded β s) :
gauge s x = 0 β x = 0 := by | refine β¨fun hβ β¦ by_contra fun (hne : x β 0) β¦ ?_, fun h β¦ h.symm βΈ gauge_zeroβ©
have : {x}αΆ β comap (gauge s) (π 0) :=
comap_gauge_nhds_zero_le hs hb (isOpen_compl_singleton.mem_nhds hne.symm)
rcases ((nhds_basis_zero_abs_lt _).comap _).mem_iff.1 this with β¨r, hrβ, hrβ©
exact hr (by simpa [hβ]) rfl |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Jens Wagemaker, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.... | Mathlib/Algebra/BigOperators/Associated.lean | 159 | 168 | theorem finset_prod_mk {p : Finset Ξ²} {f : Ξ² β Ξ±} :
(β i β p, Associates.mk (f i)) = Associates.mk (β i β p, f i) := by | rw [Finset.prod_eq_multiset_prod, β Function.comp_def, β Multiset.map_map, prod_mk,
β Finset.prod_eq_multiset_prod]
theorem rel_associated_iff_map_eq_map {p q : Multiset Ξ±} :
Multiset.Rel Associated p q β p.map Associates.mk = q.map Associates.mk := by
rw [β Multiset.rel_eq, Multiset.rel_map]
simp only [mk... |
/-
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.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 157 | 160 | theorem tangentMap_id : tangentMap I I (id : M β M) = id := by | ext1 β¨x, vβ©; simp [tangentMap]
theorem tangentMapWithin_id {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.proj) :
tangentMapWithin I I (id : M β M) s p = p := by |
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 41 | 44 | theorem evalβ_one_cyclotomic_prime_pow {R S : Type*} [CommRing R] [Semiring S] (f : R β+* S)
{p : β} (k : β) [Fact p.Prime] : evalβ f 1 (cyclotomic (p ^ (k + 1)) R) = p := by | simp
private theorem cyclotomic_neg_one_pos {n : β} (hn : 2 < n) {R} |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.NumberField.Embeddings
/-!
# Cyclotomic extensions of `β` are totally complex nu... | Mathlib/NumberTheory/Cyclotomic/Embeddings.lean | 30 | 35 | theorem nrRealPlaces_eq_zero [IsCyclotomicExtension {n} β K]
(hn : 2 < n) :
haveI := IsCyclotomicExtension.numberField {n} β K
nrRealPlaces K = 0 := by | have := IsCyclotomicExtension.numberField {n} β K
apply (IsCyclotomicExtension.zeta_spec n β K).nrRealPlaces_eq_zero_of_two_lt hn |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 172 | 173 | theorem mem_support_ofFinset_iff (a : Ξ±) : a β (ofFinset f s h h').support β a β s β§ f a β 0 := by | simp |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | Mathlib/Data/Set/Prod.lean | 889 | 892 | theorem univ_pi_ite (s : Set ΞΉ) [DecidablePred (Β· β s)] (t : β i, Set (Ξ± i)) :
(pi univ fun i => if i β s then t i else univ) = s.pi t := by | ext
simp_rw [mem_univ_pi] |
/-
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, Yury Kudryashov
-/
import Mathlib.Order.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | Mathlib/Topology/Irreducible.lean | 202 | 217 | theorem Subtype.irreducibleSpace (h : IsIrreducible s) : IrreducibleSpace s where
isPreirreducible_univ :=
(Subtype.preirreducibleSpace h.isPreirreducible).isPreirreducible_univ
toNonempty := h.nonempty.to_subtype
/-- An infinite type with cofinite topology is an irreducible topological space. -/
instance (pri... | haveI : Infinite (CofiniteTopology X) := βΉ_βΊ
simp only [CofiniteTopology.isOpen_iff, univ_inter]
intro hu hv hu' hv'
simpa only [compl_union, compl_compl] using ((hu hu').union (hv hv')).infinite_compl.nonempty
toNonempty := (inferInstance : Nonempty X)
theorem irreducibleComponents_eq_singleton [Irreduc... |
/-
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.Real
/-!
# Power function... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 821 | 824 | theorem rpow_lt_one {x : ββ₯0β} {z : β} (hx : x < 1) (hz : 0 < z) : x ^ z < 1 := by | lift x to ββ₯0 using ne_of_lt (lt_of_lt_of_le hx le_top)
simp only [coe_lt_one_iff] at hx
simp [β coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.rpow_lt_one hx hz] |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | Mathlib/CategoryTheory/Sites/Sieves.lean | 246 | 254 | theorem functorPushforward_comp (R : Presieve X) :
R.functorPushforward (F β G) = (R.functorPushforward F).functorPushforward G := by | funext x
ext f
constructor
Β· rintro β¨X, fβ, gβ, hβ, rflβ©
exact β¨F.obj X, F.map fβ, gβ, β¨X, fβ, π _, hβ, by simpβ©, rflβ©
Β· rintro β¨X, fβ, gβ, β¨X', fβ, gβ, hβ, rflβ©, rflβ©
exact β¨X', fβ, gβ β« G.map gβ, hβ, by simpβ© |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | Mathlib/Algebra/Polynomial/Expand.lean | 127 | 128 | theorem expand_eq_C {p : β} (hp : 0 < p) {f : R[X]} {r : R} : expand R p f = C r β f = C r := by | rw [β expand_C, expand_inj hp, expand_C] |
/-
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.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 743 | 747 | theorem map_lt_add_floor_translationNumber_add_one (x : β) : f x < x + βΟ fβ + 1 := by | rw [add_assoc]
norm_cast
refine map_lt_of_translationNumber_lt_int _ ?_ _
push_cast |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | Mathlib/Probability/Distributions/Uniform.lean | 305 | 306 | theorem toOuterMeasure_uniformOfFintype_apply [Fintype s] :
(uniformOfFintype Ξ±).toOuterMeasure s = Fintype.card s / Fintype.card Ξ± := by | |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | Mathlib/GroupTheory/SemidirectProduct.lean | 205 | 207 | theorem lift_inl (n : N) : lift fn fg h (inl n) = fn n := by | simp [lift]
@[simp] |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | Mathlib/NumberTheory/PythagoreanTriples.lean | 218 | 225 | theorem isPrimitiveClassified_of_coprime_of_zero_left (hc : Int.gcd x y = 1) (hx : x = 0) :
h.IsPrimitiveClassified := by | subst x
change Nat.gcd 0 (Int.natAbs y) = 1 at hc
rw [Nat.gcd_zero_left (Int.natAbs y)] at hc
rcases Int.natAbs_eq y with hy | hy
Β· use 1, 0
rw [hy, hc, Int.gcd_zero_right] |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 471 | 474 | theorem oangle_add_cyc3 {x y z : V} (hx : x β 0) (hy : y β 0) (hz : z β 0) :
o.oangle x y + o.oangle y z + o.oangle z x = 0 := by | simp [hx, hy, hz]
/-- Given three nonzero vectors, adding the angles between them in cyclic order, with the first |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Spectrum mapping theorem
This file develops proves the spectral mappi... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 143 | 145 | theorem nonempty_of_isAlgClosed_of_finiteDimensional [IsAlgClosed π] [Nontrivial A]
[I : FiniteDimensional π A] (a : A) : (Ο a).Nonempty := by | obtain β¨p, β¨h_mon, h_eval_pβ©β© := isIntegral_of_noetherian (IsNoetherian.iff_fg.2 I) a |
/-
Copyright (c) 2023 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne, Peter Pfaffelhuber, YaΓ«l Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 237 | 244 | theorem disjoint_cylinder_iff [Nonempty (β i, Ξ± i)] {s t : Finset ΞΉ} {S : Set (β i : s, Ξ± i)}
{T : Set (β i : t, Ξ± i)} [DecidableEq ΞΉ] :
Disjoint (cylinder s S) (cylinder t T) β
Disjoint
(Finset.restrictβ Finset.subset_union_left β»ΒΉ' S)
(Finset.restrictβ Finset.subset_union_right β»ΒΉ' T) :=... | simp_rw [Set.disjoint_iff, subset_empty_iff, inter_cylinder, cylinder_eq_empty_iff] |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 144 | 146 | theorem le_degrees_add_left (h : Disjoint p.degrees q.degrees) : p.degrees β€ (p + q).degrees := by | classical
apply Finset.sup_le |
/-
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.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 107 | 108 | theorem polar_neg (f : M β N) (x y : M) : polar (-f) x y = -polar f x y := by | simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add] |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# LindelΓΆf sets and LindelΓΆf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 196 | 206 | theorem IsLindelof.disjoint_nhdsSet_left {l : Filter X} [CountableInterFilter l]
(hs : IsLindelof s) :
Disjoint (πΛ’ s) l β β x β s, Disjoint (π x) l := by | refine β¨fun h x hx β¦ h.mono_left <| nhds_le_nhdsSet hx, fun H β¦ ?_β©
choose! U hxU hUl using fun x hx β¦ (nhds_basis_opens x).disjoint_iff_left.1 (H x hx)
choose hxU hUo using hxU
rcases hs.elim_nhds_subcover U fun x hx β¦ (hUo x hx).mem_nhds (hxU x hx) with β¨t, htc, hts, hstβ©
refine (hasBasis_nhdsSet _).disjoint_... |
/-
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
/-!
# Closure, interior, and frontier of preimages under `re`... | Mathlib/Analysis/Complex/ReImTopology.lean | 164 | 165 | theorem frontier_setOf_le_re_and_im_le (a b : β) :
frontier { z | a β€ re z β§ im z β€ b } = { z | a β€ re z β§ im z = b β¨ re z = a β§ im z β€ b } := by | |
/-
Copyright (c) 2018 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Johannes HΓΆlzl, RΓ©my Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 911 | 921 | theorem liminf_le_iff' [DenselyOrdered Ξ²] {x : Ξ²}
(hβ : f.IsCoboundedUnder (Β· β₯ Β·) u := by | isBoundedDefault)
(hβ : f.IsBoundedUnder (Β· β₯ Β·) u := by isBoundedDefault) :
liminf u f β€ x β β y > x, βαΆ a in f, u a β€ y := le_limsup_iff' (Ξ² := Ξ²α΅α΅) hβ hβ
lemma liminf_le_limsup_of_frequently_le {v : Ξ± β Ξ²} (h : βαΆ x in f, u x β€ v x)
(hβ : f.IsBoundedUnder (Β· β₯ Β·) u := by isBoundedDefault)
(hβ : f.Is... |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 533 | 534 | theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = β G' β s, edgeSet G' := by | ext e |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A βΆ B` and `g : B βΆ C` satisfy `w : f β« g = 0`,
we have `image_le_ker... | Mathlib/Algebra/Homology/ImageToKernel.lean | 127 | 132 | theorem imageToKernel_comp_hom_inv_comp [HasEqualizers V] [HasImages V] {Z : V} {i : B β
Z} (w) :
imageToKernel (f β« i.hom) (i.inv β« g) w =
(imageSubobjectCompIso _ _).hom β«
imageToKernel f g (by simpa using w) β« (kernelSubobjectIsoComp i.inv g).inv := by | ext
simp |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, YaΓ«l Dillies
-/
import Mathlib.Data.Set.Image
/-!
# Directed indexed families and sets
This file defines directed indexed families and directed sets. An indexed famil... | Mathlib/Order/Directed.lean | 58 | 60 | theorem directedOn_range {f : ΞΉ β Ξ±} : Directed r f β DirectedOn r (Set.range f) := by | simp_rw [Directed, DirectedOn, Set.forall_mem_range, Set.exists_range_iff] |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# LindelΓΆf sets and LindelΓΆf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 382 | 406 | theorem isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis (b : ΞΉ β Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : β i, IsLindelof (b i)) (U : Set X) :
IsLindelof U β§ IsOpen U β β s : Set ΞΉ, s.Countable β§ U = β i β s, b i := by | constructor
Β· rintro β¨hβ, hββ©
obtain β¨Y, f, rfl, hfβ© := hb.open_eq_iUnion hβ
choose f' hf' using hf
have : b β f' = f := funext hf'
subst this
obtain β¨t, htβ© :=
hβ.elim_countable_subcover (b β f') (fun i => hb.isOpen (Set.mem_range_self _)) Subset.rfl
refine β¨t.image f', Countable.image ... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... | Mathlib/Algebra/Ring/Units.lean | 77 | 78 | theorem divp_sub (a b : Ξ±) (u : Ξ±Λ£) : a /β u - b = (a - b * u) /β u := by | simp only [divp, sub_mul, sub_right_inj] |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | Mathlib/Analysis/Convex/Between.lean | 219 | 222 | theorem sbtw_const_vsub_iff {x y z : P} (p : P) :
Sbtw R (p -α΅₯ x) (p -α΅₯ y) (p -α΅₯ z) β Sbtw R x y z := by | rw [Sbtw, Sbtw, wbtw_const_vsub_iff, (vsub_right_injective p).ne_iff,
(vsub_right_injective p).ne_iff] |
/-
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.Order.Interval.Set.IsoIoo
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.UrysohnsBounded
/-!
# Tietze extension theore... | Mathlib/Topology/TietzeExtension.lean | 220 | 262 | theorem exists_extension_norm_eq_of_isClosedEmbedding' (f : X βα΅ β) (e : C(X, Y))
(he : IsClosedEmbedding e) : β g : Y βα΅ β, βgβ = βfβ β§ g.compContinuous e = f := by | /- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
between the previous approximation and `f`. -/
choose F hF_norm hF_dist using fun f : X βα΅ β => tietze_extension_step f e he
set g : β β Y βα΅ β := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
have g0 : g ... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | Mathlib/Topology/UniformSpace/Separation.lean | 142 | 146 | theorem t0Space_iff_uniformity' :
T0Space Ξ± β Pairwise fun x y β¦ β r β π€ Ξ±, (x, y) β r := by | simp [t0Space_iff_not_inseparable, inseparable_iff_ker_uniformity]
theorem t0Space_iff_ker_uniformity : T0Space Ξ± β (π€ Ξ±).ker = diagonal Ξ± := by |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
/-!
# Equivalences and sets
In this file we pr... | Mathlib/Logic/Equiv/Set.lean | 143 | 145 | theorem prod_assoc_image {Ξ± Ξ² Ξ³} {s : Set Ξ±} {t : Set Ξ²} {u : Set Ξ³} :
Equiv.prodAssoc Ξ± Ξ² Ξ³ '' (s ΓΛ’ t) ΓΛ’ u = s ΓΛ’ t ΓΛ’ u := by | simpa only [Equiv.image_eq_preimage] using prod_assoc_symm_preimage |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Birkhoff sums
In this file we define `birkhoffSum f g n x` ... | Mathlib/Dynamics/BirkhoffSum/Basic.lean | 55 | 57 | theorem Function.IsFixedPt.birkhoffSum_eq {f : Ξ± β Ξ±} {x : Ξ±} (h : IsFixedPt f x) (g : Ξ± β M)
(n : β) : birkhoffSum f g n x = n β’ g x := by | simp [birkhoffSum, (h.iterate _).eq] |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | Mathlib/RingTheory/Coprime/Lemmas.lean | 213 | 216 | theorem IsRelPrime.prod_left : (β i β t, IsRelPrime (s i) x) β IsRelPrime (β i β t, s i) x := by | classical
refine Finset.induction_on t (fun _ β¦ isRelPrime_one_left) fun b t hbt ih H β¦ ?_
rw [Finset.prod_insert hbt] |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, YaΓ«l Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | Mathlib/MeasureTheory/Integral/Average.lean | 149 | 150 | theorem laverage_lt_top (hf : β«β» x, f x βΞΌ β β) : β¨β» x, f x βΞΌ < β := by | obtain rfl | hΞΌ := eq_or_ne ΞΌ 0 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | Mathlib/Data/Set/Card.lean | 693 | 697 | theorem ncard_preimage_of_injective_subset_range {s : Set Ξ²} (H : f.Injective)
(hs : s β Set.range f) :
(f β»ΒΉ' s).ncard = s.ncard := by | rw [β ncard_image_of_injective _ H, image_preimage_eq_iff.mpr hs] |
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
/-!
# Theorems about membership of elements in vectors
This file contains theorems for membership in a `v.toList` for a vector `v`.
Having ... | Mathlib/Data/Vector/Mem.lean | 38 | 41 | theorem not_mem_zero (v : Vector Ξ± 0) : a β v.toList :=
(Vector.eq_nil v).symm βΈ not_mem_nil a
theorem mem_cons_iff (v : Vector Ξ± n) : a' β (a ::α΅₯ v).toList β a' = a β¨ a' β v.toList := by | |
/-
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.NNReal
/-!
# Limits and a... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 303 | 307 | theorem isLittleO_zpow_exp_pos_mul_atTop (k : β€) {b : β} (hb : 0 < b) :
(fun x : β => x ^ k) =o[atTop] fun x => exp (b * x) := by | simpa only [Real.rpow_intCast] using isLittleO_rpow_exp_pos_mul_atTop k hb
/-- `x ^ k = o(exp(b * x))` as `x β β` for any natural `k` and positive `b`. -/ |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 159 | 167 | theorem div2_val : β n, div2 n = n / 2
| (n : β) => congr_arg ofNat n.div2_val
| -[n+1] => congr_arg negSucc n.div2_val
theorem bit_val (b n) : bit b n = 2 * n + cond b 1 0 := by | cases b
Β· apply (add_zero _).symm
Β· rfl |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Floris van Doorn
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Basic
/-!
# Explicit least witnesses to existentials on positive natural numbers
Implement... | Mathlib/Data/PNat/Find.lean | 91 | 92 | theorem one_le_find : 1 < PNat.find h β Β¬p 1 := by | simp |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 464 | 466 | theorem affineCombination_map (e : ΞΉβ βͺ ΞΉ) (w : ΞΉ β k) (p : ΞΉ β P) :
(sβ.map e).affineCombination k p w = sβ.affineCombination k (p β e) (w β e) := by | simp_rw [affineCombination_apply, weightedVSubOfPoint_map] |
/-
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
-/
import Mathlib.Data.Set.Countable
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.FunProp.Attr
import Mathlib.Tactic.Mea... | Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 247 | 248 | theorem measurableSet_insert {a : Ξ±} {s : Set Ξ±} :
MeasurableSet (insert a s) β MeasurableSet s := by | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Topology.Algebra.Module.Equiv
import Mathlib.Topology.Algebra.Module.UniformConvergence
import Mathlib.Topology.Algebra.Separation... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 96 | 101 | theorem topologicalSpace_eq [UniformSpace F] [IsUniformAddGroup F] (π : Set (Set E)) :
instTopologicalSpace Ο F π = TopologicalSpace.induced (UniformOnFun.ofFun π β DFunLike.coe)
(UniformOnFun.topologicalSpace E F π) := by | rw [instTopologicalSpace]
congr
exact IsUniformAddGroup.toUniformSpace_eq |
/-
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 Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | Mathlib/Data/Option/NAry.lean | 140 | 143 | theorem map_mapβ_distrib_left {g : Ξ³ β Ξ΄} {f' : Ξ±' β Ξ² β Ξ΄} {g' : Ξ± β Ξ±'}
(h_distrib : β a b, g (f a b) = f' (g' a) b) :
(mapβ f a b).map g = mapβ f' (a.map g') b := by | cases a <;> cases b <;> simp [h_distrib] |
/-
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.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `Ξ±` is given by `Nat.card Ξ±`. This function has
the "junk val... | Mathlib/Data/Finite/Card.lean | 72 | 75 | theorem one_lt_card_iff_nontrivial [Finite Ξ±] : 1 < Nat.card Ξ± β Nontrivial Ξ± := by | haveI := Fintype.ofFinite Ξ±
simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial] |
/-
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.Basis.Submodule
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 204 | 208 | theorem basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix
[Fintype ΞΊ'] [DecidableEq ΞΉ] [DecidableEq ΞΉ'] :
c.toMatrix c' * LinearMap.toMatrix b' c' f * b'.toMatrix b = LinearMap.toMatrix b c f := by | cases nonempty_fintype ΞΊ
rw [basis_toMatrix_mul_linearMap_toMatrix, linearMap_toMatrix_mul_basis_toMatrix] |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | Mathlib/Algebra/CubicDiscriminant.lean | 137 | 138 | theorem of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly = C P.d := by | rw [of_b_eq_zero ha hb, hc, C_0, zero_mul, zero_add] |
/-
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.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 308 | 310 | 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 _ |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on β
This file defines a simple dista... | Mathlib/Data/Nat/Dist.lean | 92 | 96 | theorem dist_pos_of_ne {i j : Nat} (h : i β j) : 0 < dist i j := by | cases h.lt_or_lt with
| inl h => rw [dist_eq_sub_of_le h.le]; apply tsub_pos_of_lt h
| inr h => rw [dist_eq_sub_of_le_right h.le]; apply tsub_pos_of_lt h |
/-
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.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 593 | 595 | theorem hasStrictFDerivAt_sub_const_iff (c : F) :
HasStrictFDerivAt (f Β· - c) f' x β HasStrictFDerivAt f f' x := by | simp only [sub_eq_add_neg, hasStrictFDerivAt_add_const_iff] |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 | theorem pow (H : IsRelPrime x y) : IsRelPrime (x ^ m) (y ^ n) :=
H.pow_left.pow_right
theorem pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y β IsRelPrime x y := by | refine β¨fun h β¦ ?_, IsRelPrime.pow_leftβ©
rw [β Finset.card_range m, β Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm)
theorem pow_right_iff (hm : 0 < m) : IsRelPrime x (y ^ m) β IsRelPrime x y := |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E ββ.[R] F` is a linear map from a submodule of `E` to... | Mathlib/LinearAlgebra/LinearPMap.lean | 772 | 775 | theorem mem_domain_iff {f : E ββ.[R] F} {x : E} : x β f.domain β β y : F, (x, y) β f.graph := by | constructor <;> intro h
Β· use f β¨x, hβ©
exact f.mem_graph β¨x, hβ© |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 651 | 658 | theorem prod_withDensity_rightβ {g : Ξ² β ββ₯0β} (hg : AEMeasurable g Ξ½) :
ΞΌ.prod (Ξ½.withDensity g) = (ΞΌ.prod Ξ½).withDensity (fun z β¦ g z.2) := by | refine ext_of_lintegral _ fun Ο hΟ β¦ ?_
rw [lintegral_prod _ hΟ.aemeasurable, lintegral_withDensity_eq_lintegral_mulβ _ hΟ.aemeasurable,
lintegral_prod]
Β· refine lintegral_congr (fun x β¦ ?_)
rw [lintegral_withDensity_eq_lintegral_mulβ hg (by fun_prop)]
simp |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Extra lemmas about intervals
This file contains lemma... | Mathlib/Order/Interval/Set/Disjoint.lean | 143 | 145 | theorem eq_of_Ico_disjoint {xβ xβ yβ yβ : Ξ±} (h : Disjoint (Ico xβ xβ) (Ico yβ yβ)) (hx : xβ < xβ)
(h2 : xβ β Ico yβ yβ) : yβ = xβ := by | rw [Ico_disjoint_Ico, min_eq_left (le_of_lt h2.2), le_max_iff] at h |
/-
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, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Algebra.Group.Nat.Range
import Mathlib.Data.Set.Finite.Basic
/-!
# Counting on β
Thi... | Mathlib/Data/Nat/Count.lean | 120 | 122 | theorem count_le_card (hp : (setOf p).Finite) (n : β) : count p n β€ #hp.toFinset := by | rw [count_eq_card_filter_range]
exact Finset.card_mono fun x hx β¦ hp.mem_toFinset.2 (mem_filter.1 hx).2 |
/-
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.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 464 | 465 | theorem HasStrictFDerivAt.sub (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
HasStrictFDerivAt (fun x => f x - g x) (f' - g') x := by | |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | Mathlib/GroupTheory/OrderOfElement.lean | 279 | 280 | theorem exists_pow_eq_self_of_coprime (h : n.Coprime (orderOf x)) : β m : β, (x ^ n) ^ m = x := by | by_cases h0 : orderOf x = 0 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 83 | 85 | theorem smeval_one : (1 : R[X]).smeval x = 1 β’ x ^ 0 := by | rw [β C_1, smeval_C]
simp only [Nat.cast_one, one_smul] |
/-
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, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 407 | 410 | theorem exists_Icc_mem_subset_of_mem_nhdsLE [OrderTopology Ξ±] {a : Ξ±} {s : Set Ξ±}
(hs : s β π[β€] a) : β b β€ a, Icc b a β π[β€] a β§ Icc b a β s := by | simpa only [Icc_toDual, toDual.surjective.exists] using
exists_Icc_mem_subset_of_mem_nhdsGE (Ξ± := Ξ±α΅α΅) (a := toDual a) hs |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 129 | 130 | theorem associated_generator_span_self [IsPrincipalIdealRing R] [IsDomain R] (r : R) :
Associated (generator <| Ideal.span {r}) r := by | |
/-
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.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | Mathlib/LinearAlgebra/Determinant.lean | 411 | 413 | theorem LinearEquiv.det_mul_det_symm {A : Type*} [CommRing A] [Module A M] (f : M ββ[A] M) :
LinearMap.det (f : M ββ[A] M) * LinearMap.det (f.symm : M ββ[A] M) = 1 := by | simp [β LinearMap.det_comp] |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | Mathlib/Analysis/Convex/Topology.lean | 296 | 301 | theorem Convex.interior_closure_eq_interior_of_nonempty_interior {s : Set E} (hs : Convex π s)
(hs' : (interior s).Nonempty) : interior (closure s) = interior s := by | refine subset_antisymm ?_ (interior_mono subset_closure)
intro y hy
rcases hs' with β¨x, hxβ©
have h := AffineMap.lineMap_apply_one (k := π) x y |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 559 | 605 | theorem ContDiffOn.congr_mono (hf : ContDiffOn π n f s) (hβ : β x β sβ, fβ x = f x) (hs : sβ β s) :
ContDiffOn π n fβ sβ :=
(hf.mono hs).congr hβ
/-- If a function is `C^n` on a set with `n β₯ 1`, then it is differentiable there. -/
theorem ContDiffOn.differentiableOn (h : ContDiffOn π n f s) (hn : 1 β€ n) :
... | intro x xs
rcases h x xs with β¨u, u_open, xu, huβ©
apply (contDiffWithinAt_inter _).1 (hu x β¨xs, xuβ©)
exact IsOpen.mem_nhds u_open xu
/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/
theorem contDiffOn_succ_iff_hasFDerivWithinAt (hn : n β β) :
ContDiffOn π (n + 1)... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | Mathlib/Data/List/Sigma.lean | 311 | 315 | theorem dlookup_append (lβ lβ : List (Sigma Ξ²)) (a : Ξ±) :
(lβ ++ lβ).dlookup a = (lβ.dlookup a).or (lβ.dlookup a) := by | induction lβ with
| nil => rfl
| cons x lβ IH => |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Group.Measure
/-!
# Bochner Integration on Groups
We develop properties of inte... | Mathlib/MeasureTheory/Group/Integral.lean | 150 | 154 | theorem integrable_comp_div_left (f : G β F) [IsInvInvariant ΞΌ] [IsMulLeftInvariant ΞΌ] (g : G) :
Integrable (fun t => f (g / t)) ΞΌ β Integrable f ΞΌ := by | refine β¨fun h => ?_, fun h => h.comp_div_left gβ©
convert h.comp_inv.comp_mul_left gβ»ΒΉ
simp_rw [div_inv_eq_mul, mul_inv_cancel_left] |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes HΓΆlzl, YaΓ«l Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 407 | 412 | theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 β βaβ = 0 := by | rw [Metric.inseparable_iff, dist_one_right]
@[to_additive]
lemma dist_one_left (a : E) : dist 1 a = βaβ := by rw [dist_comm, dist_one_right] |
/-
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
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 221 | 222 | theorem restrict_inter_add_diffβ (s : Set Ξ±) (ht : NullMeasurableSet t ΞΌ) :
ΞΌ.restrict (s β© t) + ΞΌ.restrict (s \ t) = ΞΌ.restrict s := by | |
/-
Copyright (c) 2024 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.MeasureTheory.Integral.PeakFunction
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
/-!
# Fourier inversion formula
In a fin... | Mathlib/Analysis/Fourier/Inversion.lean | 177 | 181 | theorem MeasureTheory.Integrable.fourier_inversion_inv
(hf : Integrable f) (h'f : Integrable (π f)) {v : V}
(hv : ContinuousAt f v) : π (πβ» f) v = f v := by | rw [fourierIntegralInv_comm]
exact fourier_inversion hf h'f hv |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | Mathlib/Algebra/Polynomial/Div.lean | 424 | 430 | theorem sum_modByMonic_coeff (hq : q.Monic) {n : β} (hn : q.degree β€ n) :
(β i : Fin n, monomial i ((p %β q).coeff i)) = p %β q := by | nontriviality R
exact
(sum_fin (fun i c => monomial i c) (by simp) ((degree_modByMonic_lt _ hq).trans_le hn)).trans
(sum_monomial_eq _) |
/-
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.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | Mathlib/Data/Multiset/Powerset.lean | 154 | 158 | theorem powersetCardAux_eq_map_coe {n} {l : List Ξ±} :
powersetCardAux n l = (sublistsLen n l).map (β) := by | rw [powersetCardAux, sublistsLenAux_eq, append_nil]
@[simp] |
/-
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.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `π : Finset (Fins... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 56 | 57 | theorem mem_memberSubfamily : s β π.memberSubfamily a β insert a s β π β§ a β s := by | simp_rw [memberSubfamily, mem_image, mem_filter] |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | Mathlib/GroupTheory/OrderOfElement.lean | 641 | 648 | theorem orderOf_dvd_of_mem_zpowers (h : y β Subgroup.zpowers x) : orderOf y β£ orderOf x := by | obtain β¨k, rflβ© := Subgroup.mem_zpowers_iff.mp h
rw [orderOf_dvd_iff_pow_eq_one]
exact zpow_pow_orderOf
theorem smul_eq_self_of_mem_zpowers {Ξ± : Type*} [MulAction G Ξ±] (hx : x β Subgroup.zpowers y)
{a : Ξ±} (hs : y β’ a = a) : x β’ a = a := by
obtain β¨k, rflβ© := Subgroup.mem_zpowers_iff.mp hx |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 581 | 583 | theorem empty_toList {s : Finset Ξ±} : s.toList.isEmpty β s = β
:= by | simp
@[simp] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Kim Morrison, Jens Wagemaker, Andrew Yang, Yuyang Zhao
-/
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# The... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 44 | 45 | theorem integralNormalization_C {x : R} (hx : x β 0) : integralNormalization (C x) = 1 := by | simp [integralNormalization, sum_def, support_C hx, degree_C hx] |
/-
Copyright (c) 2022 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 193 | 202 | theorem sum_sq_le_of_mem_box (hx : x β box n d) : β i : Fin n, x i ^ 2 β€ n * (d - 1) ^ 2 := by | rw [mem_box] at hx
have : β i, x i ^ 2 β€ (d - 1) ^ 2 := fun i =>
Nat.pow_le_pow_left (Nat.le_sub_one_of_lt (hx i)) _
exact (sum_le_card_nsmul univ _ _ fun i _ => this i).trans (by rw [card_fin, smul_eq_mul])
theorem sum_eq : (β i : Fin n, d * (2 * d + 1) ^ (i : β)) = ((2 * d + 1) ^ n - 1) / 2 := by
refine (N... |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 296 | 299 | theorem associator_naturality_left {f f' : a βΆ b} (Ξ· : f βΆ f') (g : b βΆ c) (h : c βΆ d) :
Ξ· β· g β· h β« (Ξ±_ f' g h).hom = (Ξ±_ f g h).hom β« Ξ· β· (g β« h) := by | simp
@[reassoc] |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 82 | 87 | theorem fixedBy_subset_fixedBy_zpow (g : G) (j : β€) :
fixedBy Ξ± g β fixedBy Ξ± (g ^ j) := by | intro a a_in_fixedBy
rw [mem_fixedBy, zpow_smul_eq_iff_minimalPeriod_dvd,
minimalPeriod_eq_one_iff_fixedBy.mpr a_in_fixedBy, Int.natCast_one]
exact one_dvd j |
/-
Copyright (c) 2021 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | Mathlib/Order/Interval/Finset/Fin.lean | 161 | 163 | theorem finsetImage_val_Ioi : (Ioi a).image val = Ioo (a : β) n := by | simp [β coe_inj]
@[simp] |
/-
Copyright (c) 2021 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.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
/-!
# Finite intervals of positive naturals
This file proves that `β+` is a `LocallyFiniteOrder` ... | Mathlib/Data/PNat/Interval.lean | 67 | 72 | theorem card_Ioo : #(Ioo a b) = b - a - 1 := by | rw [β Nat.card_Ioo, β map_subtype_embedding_Ioo, card_map]
@[simp]
theorem card_uIcc : #(uIcc a b) = (b - a : β€).natAbs + 1 := by
rw [β Nat.card_uIcc, β map_subtype_embedding_uIcc, card_map] |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, JoΓ«l Riou
-/
import Mathlib.CategoryTheory.Adjunction.Mates
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 64 | 66 | theorem leftAdjointUniq_hom_app_counit {F F' : C β₯€ D} {G : D β₯€ C} (adj1 : F β£ G) (adj2 : F' β£ G)
(x : D) :
(leftAdjointUniq adj1 adj2).hom.app (G.obj x) β« adj2.counit.app x = adj1.counit.app x := by | |
/-
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.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | Mathlib/Order/Heyting/Basic.lean | 641 | 642 | theorem le_compl_iff_disjoint_right : a β€ bαΆ β Disjoint a b := by | rw [β himp_bot, le_himp_iff, disjoint_iff_inf_le] |
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