Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2019 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 947 | 952 | theorem iteratedFDeriv_clm_apply_const_apply
{n : โโ} {c : E โ F โL[๐] G} (hc : ContDiff ๐ n c)
{i : โ} (hi : i โค n) {x : E} {u : F} {m : Fin i โ E} :
(iteratedFDeriv ๐ i (fun y โฆ (c y) u) x) m = (iteratedFDeriv ๐ i c x) m u := by |
simp only [โ iteratedFDerivWithin_univ]
exact iteratedFDerivWithin_clm_apply_const_apply uniqueDiffOn_univ hc.contDiffOn hi (mem_univ _)
|
/-
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.BigOperators.Group.List
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.Group.Action.Defs
import Mathli... | Mathlib/Algebra/GradedMonoid.lean | 461 | 464 | theorem GradedMonoid.list_prod_ofFn_eq_dProd {n : โ} (f : Fin n โ GradedMonoid A) :
(List.ofFn f).prod =
GradedMonoid.mk _ ((List.finRange n).dProd (fun i => (f i).1) fun i => (f i).2) := by |
rw [List.ofFn_eq_map, GradedMonoid.list_prod_map_eq_dProd]
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl
-/
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 | 2,147 | 2,149 | theorem prod_erase_eq_div {a : ฮฑ} (h : a โ s) :
โ x โ s.erase a, f x = (โ x โ s, f x) / f a := by |
rw [eq_div_iff_mul_eq', prod_erase_mul _ _ h]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.m... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 201 | 209 | theorem index_pos (K : PositiveCompacts G) {V : Set G} (hV : (interior V).Nonempty) :
0 < index (K : Set G) V := by |
unfold index; rw [Nat.sInf_def, Nat.find_pos, mem_image]
ยท rintro โจt, h1t, h2tโฉ; rw [Finset.card_eq_zero] at h2t; subst h2t
obtain โจg, hgโฉ := K.interior_nonempty
show g โ (โ
: Set G)
convert h1t (interior_subset hg); symm
simp only [Finset.not_mem_empty, iUnion_of_empty, iUnion_empty]
ยท exact ind... |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 54 | 58 | theorem le_max_left (a b : ฮฑ) : a โค max a b := by |
-- Porting note: no `min_tac` tactic
if h : a โค b
then simp [max_def, if_pos h]; exact h
else simp [max_def, if_neg h, le_refl]
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.LinearAlgebra.TensorProduct.Graded.Internal
import Mathlib.LinearAlgebra.QuadraticForm.Prod
/-!
# Cliff... | Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean | 101 | 104 | theorem map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one
(hmโ : mโ โ evenOdd Qโ 1) (hmโ : mโ โ evenOdd Qโ 1) :
map fโ mโ * map fโ mโ = - map fโ mโ * map fโ mโ := by |
simp [map_mul_map_of_isOrtho_of_mem_evenOdd _ _ hf _ _ hmโ hmโ]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 592 | 592 | theorem start_mem_support {u v : V} (p : G.Walk u v) : u โ p.support := by | cases p <;> simp
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import pr... | Mathlib/Probability/Martingale/Convergence.lean | 186 | 190 | theorem Submartingale.upcrossings_ae_lt_top [IsFiniteMeasure ฮผ] (hf : Submartingale f โฑ ฮผ)
(hbdd : โ n, snorm (f n) 1 ฮผ โค R) : โแต ฯ โฮผ, โ a b : โ, a < b โ upcrossings a b f ฯ < โ := by |
simp only [ae_all_iff, eventually_imp_distrib_left]
rintro a b hab
exact hf.upcrossings_ae_lt_top' hbdd (Rat.cast_lt.2 hab)
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 78 | 80 | theorem bernoulli'_def (n : โ) :
bernoulli' n = 1 - โ k โ range n, n.choose k / (n - k + 1) * bernoulli' k := by |
rw [bernoulli'_def', โ Fin.sum_univ_eq_sum_range]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theo... | Mathlib/MeasureTheory/Constructions/Pi.lean | 278 | 284 | theorem pi'_pi [โ i, SigmaFinite (ฮผ i)] (s : โ i, Set (ฮฑ i)) :
pi' ฮผ (pi univ s) = โ i, ฮผ i (s i) := by |
rw [pi']
rw [โ MeasurableEquiv.piMeasurableEquivTProd_symm_apply, MeasurableEquiv.map_apply,
MeasurableEquiv.piMeasurableEquivTProd_symm_apply, elim_preimage_pi, tprod_tprod _ ฮผ, โ
List.prod_toFinset, sortedUniv_toFinset] <;>
exact sortedUniv_nodup ฮน
|
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 258 | 276 | theorem tendsto_abs_re_smul {p : Fin 2 โ โค} (hp : IsCoprime (p 0) (p 1)) :
Tendsto
(fun g : { g : SL(2, โค) // (โโg) 1 = p } => |((g : SL(2, โค)) โข z).re|) cofinite atTop := by |
suffices
Tendsto (fun g : (fun g : SL(2, โค) => (โโg) 1) โปยน' {p} => ((g : SL(2, โค)) โข z).re) cofinite
(cocompact โ)
by exact tendsto_norm_cocompact_atTop.comp this
have : ((p 0 : โ) ^ 2 + (p 1 : โ) ^ 2)โปยน โ 0 := by
apply inv_ne_zero
exact mod_cast hp.sq_add_sq_ne_zero
let f := Homeomorph.mul... |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaรซl Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 783 | 787 | theorem ball_add_ball_subset (p : Seminorm ๐ E) (rโ rโ : โ) (xโ xโ : E) :
p.ball (xโ : E) rโ + p.ball (xโ : E) rโ โ p.ball (xโ + xโ) (rโ + rโ) := by |
rintro x โจyโ, hyโ, yโ, hyโ, rflโฉ
rw [mem_ball, add_sub_add_comm]
exact (map_add_le_add p _ _).trans_lt (add_lt_add hyโ hyโ)
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e31... | Mathlib/Data/QPF/Multivariate/Basic.lean | 164 | 177 | theorem mem_supp {ฮฑ : TypeVec n} (x : F ฮฑ) (i) (u : ฮฑ i) :
u โ supp x i โ โ a f, abs โจa, fโฉ = x โ u โ f i '' univ := by |
rw [supp]; dsimp; constructor
ยท intro h a f haf
have : LiftP (fun i u => u โ f i '' univ) x := by
rw [liftP_iff]
refine โจa, f, haf.symm, ?_โฉ
intro i u
exact mem_image_of_mem _ (mem_univ _)
exact h this
intro h p; rw [liftP_iff]
rintro โจa, f, xeq, h'โฉ
rcases h a f xeq.symm with... |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 100 | 106 | theorem mult_log_place_eq_zero {x : (๐ K)หฃ} {w : InfinitePlace K} :
mult w * Real.log (w x) = 0 โ w x = 1 := by |
rw [mul_eq_zero, or_iff_right, Real.log_eq_zero, or_iff_right, or_iff_left]
ยท linarith [(apply_nonneg _ _ : 0 โค w x)]
ยท simp only [ne_eq, map_eq_zero, coe_ne_zero x, not_false_eq_true]
ยท refine (ne_of_gt ?_)
rw [mult]; split_ifs <;> norm_num
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hรถlzl, 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 | 476 | 477 | theorem degree_X_pow_le (n : โ) : degree (X ^ n : R[X]) โค n := by |
simpa only [C_1, one_mul] using degree_C_mul_X_pow_le n (1 : R)
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 1,609 | 1,611 | theorem map_ofDual_min (s : Finset ฮฑแตแต) : s.min.map ofDual = (s.image ofDual).max := by |
rw [max_eq_sup_withBot, sup_image]
exact congr_fun Option.map_id _
|
/-
Copyright (c) 2018 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel, Johannes Hรถlzl, Rรฉmy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 1,107 | 1,108 | theorem blimsup_or_eq_sup : (blimsup u f fun x => p x โจ q x) = blimsup u f p โ blimsup u f q := by |
simp only [blimsup_eq_limsup, โ limsup_sup_filter, โ inf_sup_left, sup_principal, setOf_or]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Connected.Basic
/-!
# Totally disconnected and totally separated topological spaces
## Main definitions
We define the... | Mathlib/Topology/Connected/TotallyDisconnected.lean | 123 | 128 | theorem totallyDisconnectedSpace_iff_connectedComponent_singleton :
TotallyDisconnectedSpace ฮฑ โ โ x : ฮฑ, connectedComponent x = {x} := by |
rw [totallyDisconnectedSpace_iff_connectedComponent_subsingleton]
refine forall_congr' fun x => ?_
rw [subsingleton_iff_singleton]
exact mem_connectedComponent
|
/-
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 | 1,068 | 1,070 | theorem toSet_sInter {x : ZFSet.{u}} (h : x.Nonempty) : (โโ x).toSet = โโ (toSet '' x.toSet) := by |
ext
simp [mem_sInter h]
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68a... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 156 | 158 | theorem mem_compression :
a โ ๐ u v s โ a โ s โง compress u v a โ s โจ a โ s โง โ b โ s, compress u v b = a := by |
simp_rw [compression, mem_union, mem_filter, mem_image, and_comm]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 378 | 382 | theorem isReal_iff {w : InfinitePlace K} :
IsReal w โ ComplexEmbedding.IsReal (embedding w) := by |
refine โจ?_, fun h => โจembedding w, h, mk_embedding wโฉโฉ
rintro โจฯ, โจhฯ, rflโฉโฉ
rwa [embedding_mk_eq_of_isReal hฯ]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 104 | 107 | theorem diff_mem_nhdsWithin_diff {x : ฮฑ} {s t : Set ฮฑ} (hs : s โ ๐[t] x) (t' : Set ฮฑ) :
s \ t' โ ๐[t \ t'] x := by |
rw [nhdsWithin, diff_eq, diff_eq, โ inf_principal, โ inf_assoc]
exact inter_mem_inf hs (mem_principal_self _)
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 319 | 322 | theorem map_comap_bracket_eq {Jโ Jโ : LieIdeal R L'} (h : f.IsIdealMorphism) :
map f โ
comap f Jโ, comap f Jโโ = โ
f.idealRange โ Jโ, f.idealRange โ Jโโ := by |
rw [โ map_sup_ker_eq_map, โ comap_bracket_eq h, map_comap_eq h, inf_eq_right]
exact le_trans (LieSubmodule.lie_le_left _ _) inf_le_left
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 326 | 330 | theorem wSameSide_smul_vsub_vadd_left {s : AffineSubspace R P} {pโ pโ : P} (x : P) (hpโ : pโ โ s)
(hpโ : pโ โ s) {t : R} (ht : 0 โค t) : s.WSameSide (t โข (x -แตฅ pโ) +แตฅ pโ) x := by |
refine โจpโ, hpโ, pโ, hpโ, ?_โฉ
rw [vadd_vsub]
exact SameRay.sameRay_nonneg_smul_left _ ht
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Support
#align_import algebra.monoid_algebra.degree from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 149 | 153 | theorem sup_support_pow_le (degb0 : degb 0 โค 0) (degbm : โ a b, degb (a + b) โค degb a + degb b)
(n : โ) (f : R[A]) : (f ^ n).support.sup degb โค n โข f.support.sup degb := by |
rw [โ List.prod_replicate, โ List.sum_replicate]
refine (sup_support_list_prod_le degb0 degbm _).trans_eq ?_
rw [List.map_replicate]
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, 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 | 813 | 814 | theorem add_eq_right_iff {a b : Cardinal} : a + b = b โ max โตโ a โค b โจ a = 0 := by |
rw [add_comm, add_eq_left_iff]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 129 | 131 | theorem neg_one_lt_goldConj : -1 < ฯ := by |
rw [neg_lt, โ inv_gold]
exact inv_lt_one one_lt_gold
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sรฉbastien Gouรซzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 600 | 603 | theorem norm_fderiv_le_of_lipschitzOn {f : E โ F} {xโ : E} {s : Set E} (hs : s โ ๐ xโ)
{C : โโฅ0} (hlip : LipschitzOnWith C f s) : โfderiv ๐ f xโโ โค C := by |
refine norm_fderiv_le_of_lip' ๐ C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,350 | 1,352 | theorem lTensor_comp_rTensor (f : M โโ[R] P) (g : N โโ[R] Q) :
(g.lTensor P).comp (f.rTensor N) = map f g := by |
simp only [lTensor, rTensor, โ map_comp, id_comp, comp_id]
|
/-
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 Sรถnne, Sรฉbastien Gouรซzel,
Rรฉmy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 108 | 109 | theorem rpow_inv_rpow_self {y : โ} (hy : y โ 0) (x : โโฅ0) : (x ^ y) ^ (1 / y) = x := by |
field_simp [โ rpow_mul]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 544 | 546 | theorem ppow_apply {f : ArithmeticFunction R} {k x : โ} (kpos : 0 < k) : f.ppow k x = f x ^ k := by |
rw [ppow, dif_neg (Nat.ne_of_gt kpos)]
rfl
|
/-
Copyright (c) 2022 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 258 | 261 | theorem erase_mem_compression_of_mem_compression : s โ ๐ a ๐ โ s.erase a โ ๐ a ๐ := by |
simp_rw [mem_compression, erase_idem]
refine Or.imp (fun h => โจh.2, h.2โฉ) fun h => ?_
rwa [erase_eq_of_not_mem (insert_ne_self.1 <| ne_of_mem_of_not_mem h.2 h.1)]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#ali... | Mathlib/SetTheory/Ordinal/Notation.lean | 372 | 378 | theorem NF.of_dvd_omega_opow {b e n a} (h : NF (ONote.oadd e n a))
(d : ฯ ^ b โฃ repr (ONote.oadd e n a)) :
b โค repr e โง ฯ ^ b โฃ repr a := by |
have := mt repr_inj.1 (fun h => by injection h : ONote.oadd e n a โ 0)
have L := le_of_not_lt fun l => not_le_of_lt (h.below_of_lt l).repr_lt (le_of_dvd this d)
simp only [repr] at d
exact โจL, (dvd_add_iff <| (opow_dvd_opow _ L).mul_right _).1 dโฉ
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 247 | 249 | theorem GammaSeq_eq_betaIntegral_of_re_pos {s : โ} (hs : 0 < re s) (n : โ) :
GammaSeq s n = (n : โ) ^ s * betaIntegral s (n + 1) := by |
rw [GammaSeq, betaIntegral_eval_nat_add_one_right hs n, โ mul_div_assoc]
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 988 | 990 | theorem lowerBounds_insert (a : ฮฑ) (s : Set ฮฑ) :
lowerBounds (insert a s) = Iic a โฉ lowerBounds s := by |
rw [insert_eq, lowerBounds_union, lowerBounds_singleton]
|
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.D... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 493 | 497 | theorem le_iff_mem_closure (x y : ProjectiveSpectrum ๐) :
x โค y โ y โ closure ({x} : Set (ProjectiveSpectrum ๐)) := by |
rw [โ as_ideal_le_as_ideal, โ zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
vanishingIdeal_singleton]
simp only [as_ideal_le_as_ideal, coe_subset_coe]
|
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4... | Mathlib/Algebra/CubicDiscriminant.lean | 391 | 392 | theorem natDegree_of_a_eq_zero (ha : P.a = 0) : P.toPoly.natDegree โค 2 := by |
simpa only [of_a_eq_zero ha] using natDegree_quadratic_le
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernรกndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 266 | 267 | theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :
log b x = pred (sInf { o | x < b ^ o }) := by | simp only [log, dif_pos h]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 888 | 890 | theorem le_ncard_diff (s t : Set ฮฑ) (hs : s.Finite := by | toFinite_tac) :
t.ncard - s.ncard โค (t \ s).ncard :=
tsub_le_iff_left.mpr (by rw [add_comm]; apply ncard_le_ncard_diff_add_ncard _ _ hs)
|
/-
Copyright (c) 2022 Sebastian Monnet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Monnet
-/
import Mathlib.FieldTheory.Galois
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
#align_... | Mathlib/FieldTheory/KrullTopology.lean | 203 | 209 | theorem IntermediateField.fixingSubgroup_isOpen {K L : Type*} [Field K] [Field L] [Algebra K L]
(E : IntermediateField K L) [FiniteDimensional K E] :
IsOpen (E.fixingSubgroup : Set (L โโ[K] L)) := by |
have h_basis : E.fixingSubgroup.carrier โ galGroupBasis K L :=
โจE.fixingSubgroup, โจE, โน_โบ, rflโฉ, rflโฉ
have h_nhd := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
exact Subgroup.isOpen_of_mem_nhds _ h_nhd
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 148 | 155 | theorem range_inl : range (inl R M Mโ) = ker (snd R M Mโ) := by |
ext x
simp only [mem_ker, mem_range]
constructor
ยท rintro โจy, rflโฉ
rfl
ยท intro h
exact โจx.fst, Prod.ext rfl h.symmโฉ
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 1,458 | 1,467 | theorem Basis.mem_span_iff_repr_mem (m : M) :
m โ span R (Set.range b) โ โ i, b.repr m i โ Set.range (algebraMap R S) := by |
refine
โจfun hm i => โจ(b.restrictScalars R).repr โจm, hmโฉ i, b.restrictScalars_repr_apply R โจm, hmโฉ iโฉ,
fun h => ?_โฉ
rw [โ b.total_repr m, Finsupp.total_apply S _]
refine sum_mem fun i _ => ?_
obtain โจ_, hโฉ := h i
simp_rw [โ h, algebraMap_smul]
exact smul_mem _ _ (subset_span (Set.mem_range_self i)... |
/-
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.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 571 | 574 | theorem Wbtw.sameRay_vsub_right {x y z : P} (h : Wbtw R x y z) : SameRay R (z -แตฅ x) (z -แตฅ y) := by |
rcases h with โจt, โจ_, ht1โฉ, rflโฉ
simpa [lineMap_apply, vsub_vadd_eq_vsub_sub, sub_smul] using
SameRay.sameRay_nonneg_smul_right (z -แตฅ x) (sub_nonneg.2 ht1)
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 828 | 830 | theorem Orthonormal.inner_finsupp_eq_sum_right {v : ฮน โ E} (hv : Orthonormal ๐ v) (lโ lโ : ฮน โโ ๐) :
โชFinsupp.total ฮน E ๐ v lโ, Finsupp.total ฮน E ๐ v lโโซ = lโ.sum fun i y => conj (lโ i) * y := by |
simp only [lโ.total_apply _, Finsupp.inner_sum, hv.inner_left_finsupp, mul_comm, smul_eq_mul]
|
/-
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.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import ... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 225 | 229 | theorem contDiffAt_finsum {s : Set E} (f : SmoothPartitionOfUnity ฮน ๐(โ, E) E s) {xโ : E}
{g : ฮน โ E โ F} (hฯ : โ i, xโ โ tsupport (f i) โ ContDiffAt โ n (g i) xโ) :
ContDiffAt โ n (fun x โฆ โแถ i, f i x โข g i x) xโ := by |
simp only [โ contMDiffAt_iff_contDiffAt] at *
exact f.contMDiffAt_finsum hฯ
|
/-
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 | 334 | 337 | theorem Fin.cast_eq_cast' {n m : โ} (h : Fin n = Fin m) :
_root_.cast h = Fin.cast (fin_injective h) := by |
cases fin_injective h
rfl
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 364 | 365 | theorem finAddFlip_apply_natAdd (k : Fin n) (m : โ) :
finAddFlip (Fin.natAdd m k) = Fin.castAdd m k := by | simp [finAddFlip]
|
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Marginals of multivariate functions
In this ... | Mathlib/MeasureTheory/Integral/Marginal.lean | 202 | 212 | theorem lmarginal_image [DecidableEq ฮด'] {e : ฮด' โ ฮด} (he : Injective e) (s : Finset ฮด')
{f : (โ i, ฯ (e i)) โ โโฅ0โ} (hf : Measurable f) (x : โ i, ฯ i) :
(โซโฏโซโป_s.image e, f โ (ยท โ' e) โฮผ) x = (โซโฏโซโป_s, f โฮผ โ' e) (x โ' e) := by |
have h : Measurable ((ยท โ' e) : (โ i, ฯ i) โ _) :=
measurable_pi_iff.mpr <| fun i โฆ measurable_pi_apply (e i)
induction s using Finset.induction generalizing x with
| empty => simp
| insert hi ih =>
rw [image_insert, lmarginal_insert _ (hf.comp h) (he.mem_finset_image.not.mpr hi),
lmarginal_inser... |
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 816 | 817 | theorem one_div_le_of_neg (ha : a < 0) (hb : b < 0) : 1 / a โค b โ 1 / b โค a := by |
simpa using inv_le_of_neg ha hb
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,485 | 1,488 | theorem round_sub_int (x : ฮฑ) (y : โค) : round (x - y) = round x - y := by |
rw [sub_eq_add_neg]
norm_cast
rw [round_add_int, sub_eq_add_neg]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 424 | 427 | theorem cancel_counitInv_right_assoc' {W X X' Y Y' Z : D} (f : W โถ X) (g : X โถ Y) (h : Y โถ Z)
(f' : W โถ X') (g' : X' โถ Y') (h' : Y' โถ Z) :
f โซ g โซ h โซ e.counitInv.app Z = f' โซ g' โซ h' โซ e.counitInv.app Z โ
f โซ g โซ h = f' โซ g' โซ h' := by | simp only [โ Category.assoc, cancel_mono]
|
/-
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
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 559 | 561 | theorem nodal_eq_mul_nodal_erase [DecidableEq ฮน] {i : ฮน} (hi : i โ s) :
nodal s v = (X - C (v i)) * nodal (s.erase i) v := by |
simp_rw [nodal, Finset.mul_prod_erase _ (fun x => X - C (v x)) hi]
|
/-
Copyright (c) 2022 Violeta Hernรกndez Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernรกndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 208 | 234 | theorem enumOrd_isNormal_iff_isClosed (hs : s.Unbounded (ยท < ยท)) :
IsNormal (enumOrd s) โ IsClosed s := by |
have Hs := enumOrd_strictMono hs
refine
โจfun h => isClosed_iff_sup.2 fun {ฮน} hฮน f hf => ?_, fun h =>
(isNormal_iff_strictMono_limit _).2 โจHs, fun a ha o H => ?_โฉโฉ
ยท let g : ฮน โ Ordinal.{u} := fun i => (enumOrdOrderIso hs).symm โจ_, hf iโฉ
suffices enumOrd s (sup.{u, u} g) = sup.{u, u} f by
rw [... |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.pfunctor.multivariate.M from ... | Mathlib/Data/PFunctor/Multivariate/M.lean | 195 | 198 | theorem M.dest'_eq_dest' {ฮฑ : TypeVec n} {x : P.last.M} {aโ : P.A}
{fโ : P.last.B aโ โ P.last.M} (hโ : PFunctor.M.dest x = โจaโ, fโโฉ) {aโ : P.A}
{fโ : P.last.B aโ โ P.last.M} (hโ : PFunctor.M.dest x = โจaโ, fโโฉ) (f' : M.Path P x โน ฮฑ) :
M.dest' P hโ f' = M.dest' P hโ f' := by | cases hโ.symm.trans hโ; rfl
|
/-
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.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,527 | 1,532 | theorem preimage_injective : Injective (preimage f) โ Surjective f := by |
refine โจfun h y => ?_, Surjective.preimage_injectiveโฉ
obtain โจx, hxโฉ : (f โปยน' {y}).Nonempty := by
rw [h.nonempty_apply_iff preimage_empty]
apply singleton_nonempty
exact โจx, hxโฉ
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaรซl Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 167 | 169 | theorem csInf_div (hsโ : s.Nonempty) (hsโ : BddBelow s) (htโ : t.Nonempty) (htโ : BddAbove t) :
sInf (s / t) = sInf s / sSup t := by |
rw [div_eq_mul_inv, csInf_mul hsโ hsโ htโ.inv htโ.inv, csInf_inv htโ htโ, div_eq_mul_inv]
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 549 | 552 | theorem nhds_prod_eq {x : X} {y : Y} : ๐ (x, y) = ๐ x รหข ๐ y := by |
dsimp only [SProd.sprod]
rw [Filter.prod, instTopologicalSpaceProd, nhds_inf (tโ := TopologicalSpace.induced Prod.fst _)
(tโ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hรถlzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 448 | 451 | theorem degree_mul_leadingCoeff_inv (p : K[X]) {q : K[X]} (h : q โ 0) :
degree (p * C (leadingCoeff q)โปยน) = degree p := by |
have hโ : (leadingCoeff q)โปยน โ 0 := inv_ne_zero (mt leadingCoeff_eq_zero.1 h)
rw [degree_mul_C hโ]
|
/-
Copyright (c) 2021 Kalle Kytรถlรค. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytรถlรค
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 536 | 539 | theorem cthickening_eq_iInter_thickening'' (ฮด : โ) (E : Set ฮฑ) :
cthickening ฮด E = โ (ฮต : โ) (_ : max 0 ฮด < ฮต), thickening ฮต E := by |
rw [โ cthickening_max_zero, cthickening_eq_iInter_thickening]
exact le_max_left _ _
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoid... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 179 | 190 | theorem rightUnitor_naturality {M N : ModuleCat R} (f : M โถ N) :
tensorHom f (๐ (ModuleCat.of R R)) โซ (rightUnitor N).hom = (rightUnitor M).hom โซ f := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply LinearMap.ext; intro x
apply LinearMap.ext_ring
-- Porting note (#10934): used to be dsimp
change ((rightUnitor N).hom) ((tensorHom f (๐ (of R R))) (x โโ[R] (1 : R))) =
f (((rightUnitor M).hom) (x โโ[R] 1))
erw [TensorProduct.rid_tmu... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hรถlzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 758 | 758 | theorem restrict_univ (f : ฮฑ โโ ฮฒ) : restrict f univ = f := by | simp [restrict]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hรถlzl, 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 | 279 | 280 | theorem natDegree_natCast (n : โ) : natDegree (n : R[X]) = 0 := by |
simp only [โ C_eq_natCast, natDegree_C]
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 299 | 301 | theorem ฮน_range_map_lift (f : M โโ[R] A) (cond : โ m, f m * f m = algebraMap _ _ (Q m)) :
(ฮน Q).range.map (lift Q โจf, condโฉ).toLinearMap = LinearMap.range f := by |
rw [โ LinearMap.range_comp, ฮน_comp_lift]
|
/-
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.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_impo... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 99 | 110 | theorem frequently_lt_rpow_neg (h : LiouvilleWith p x) (hlt : q < p) :
โแถ n : โ in atTop, โ m : โค, x โ m / n โง |x - m / n| < n ^ (-q) := by |
rcases h.exists_pos with โจC, _hCโ, hCโฉ
have : โแถ n : โ in atTop, C < n ^ (p - q) := by
simpa only [(ยท โ ยท), neg_sub, one_div] using
((tendsto_rpow_atTop (sub_pos.2 hlt)).comp tendsto_natCast_atTop_atTop).eventually
(eventually_gt_atTop C)
refine (this.and_frequently hC).mono ?_
rintro n โจhnC,... |
/-
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 | 1,405 | 1,406 | theorem mem_iota {m n : Nat} : m โ iota n โ 1 โค m โง m โค n := by |
simp [iota_eq_reverse_range', Nat.add_comm, Nat.lt_succ]
|
/-
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.Logic.Equiv.Nat
import Mathlib.Data.PNat.Basic
import Mathlib.Order.Directed
import Mathlib.Data.Countable.Defs
import Mathli... | Mathlib/Logic/Encodable/Basic.lean | 522 | 523 | theorem up_down {a : ฮฑ} : (down a).up = a := by |
simp [up, down,Equiv.left_inv _ _, Equiv.symm_apply_apply]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# The... | Mathlib/CategoryTheory/Sites/Plus.lean | 243 | 270 | theorem plusMap_toPlus : J.plusMap (J.toPlus P) = J.toPlus (J.plusObj P) := by |
ext X : 2
refine colimit.hom_ext (fun S => ?_)
dsimp only [plusMap, toPlus]
let e : S.unop โถ โค := homOfLE (OrderTop.le_top _)
rw [ฮน_colimMap, โ colimit.w _ e.op, โ Category.assoc, โ Category.assoc]
congr 1
refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)
erw [Multiequalizer.lift_ฮน]
simp only [unop_o... |
/-
Copyright (c) 2020 Frรฉdรฉric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frรฉdรฉric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 454 | 460 | theorem liftAux.smul {ฯ : MultilinearMap R s E} (r : R) (x : โจ[R] i, s i) :
liftAux ฯ (r โข x) = r โข liftAux ฯ x := by |
refine PiTensorProduct.induction_on' x ?_ ?_
ยท intro z f
rw [smul_tprodCoeff' r z f, liftAux_tprodCoeff, liftAux_tprodCoeff, smul_assoc]
ยท intro z y ihz ihy
rw [smul_add, (liftAux ฯ).map_add, ihz, ihy, (liftAux ฯ).map_add, smul_add]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Sites.CoverLifting
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_th... | Mathlib/CategoryTheory/Sites/DenseSubsite.lean | 245 | 248 | theorem appHom_valid_glue {X : D} {Y : C} (f : op X โถ op (G.obj Y)) :
appHom ฮฑ X โซ โฑ'.val.map f = โฑ.map f โซ ฮฑ.app (op Y) := by |
ext
apply appHom_restrict
|
/-
Copyright (c) 2021 Yaรซl Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies
-/
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "lea... | Mathlib/Data/Sigma/Order.lean | 89 | 96 | theorem lt_def [โ i, LT (ฮฑ i)] {a b : ฮฃi, ฮฑ i} : a < b โ โ h : a.1 = b.1, h.rec a.2 < b.2 := by |
constructor
ยท rintro โจi, a, b, hโฉ
exact โจrfl, hโฉ
ยท obtain โจi, aโฉ := a
obtain โจj, bโฉ := b
rintro โจrfl : i = j, hโฉ
exact lt.fiber _ _ _ h
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 58 | 70 | theorem formPerm_disjoint_iff (hl : Nodup l) (hl' : Nodup l') (hn : 2 โค l.length)
(hn' : 2 โค l'.length) : Perm.Disjoint (formPerm l) (formPerm l') โ l.Disjoint l' := by |
rw [disjoint_iff_eq_or_eq, List.Disjoint]
constructor
ยท rintro h x hx hx'
specialize h x
rw [formPerm_apply_mem_eq_self_iff _ hl _ hx, formPerm_apply_mem_eq_self_iff _ hl' _ hx'] at h
omega
ยท intro h x
by_cases hx : x โ l
on_goal 1 => by_cases hx' : x โ l'
ยท exact (h hx hx').elim
al... |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `ฮผ` on a type `ฮฑ` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 1,093 | 1,101 | theorem ae_eq_restrict_iff_indicator_ae_eq {g : ฮฑ โ ฮฒ} (hs : MeasurableSet s) :
f =แต[ฮผ.restrict s] g โ s.indicator f =แต[ฮผ] s.indicator g := by |
rw [Filter.EventuallyEq, ae_restrict_iff' hs]
refine โจfun h => ?_, fun h => ?_โฉ <;> filter_upwards [h] with x hx
ยท by_cases hxs : x โ s
ยท simp [hxs, hx hxs]
ยท simp [hxs]
ยท intro hxs
simpa [hxs] using hx
|
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mango... | Mathlib/NumberTheory/VonMangoldt.lean | 79 | 79 | theorem vonMangoldt_apply_one : ฮ 1 = 0 := by | simp [vonMangoldt_apply]
|
/-
Copyright (c) 2020 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 686 | 691 | theorem length_pos_of_mem_splitWrtComposition {l l' : List ฮฑ} {c : Composition l.length}
(h : l' โ l.splitWrtComposition c) : 0 < length l' := by |
have : l'.length โ (l.splitWrtComposition c).map List.length :=
List.mem_map_of_mem List.length h
rw [map_length_splitWrtComposition] at this
exact c.blocks_pos this
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 480 | 485 | theorem hasMellin_cpow_Ioc (a : โ) {s : โ} (hs : 0 < re s + re a) :
HasMellin (indicator (Ioc 0 1) (fun t => โt ^ a : โ โ โ)) s (1 / (s + a)) := by |
have := hasMellin_one_Ioc (by rwa [add_re] : 0 < (s + a).re)
simp_rw [HasMellin, โ MellinConvergent.cpow_smul, โ mellin_cpow_smul, โ indicator_smul,
smul_eq_mul, mul_one] at this
exact this
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 675 | 678 | theorem ncard_preimage_of_injective_subset_range {s : Set ฮฒ} (H : f.Injective)
(hs : s โ Set.range f) :
(f โปยน' s).ncard = s.ncard := by |
rw [โ ncard_image_of_injective _ H, image_preimage_eq_iff.mpr hs]
|
/-
Copyright (c) 2019 Calle Sรถnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sรถnne
-/
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 | 530 | 533 | theorem toReal_coe_eq_self_iff {ฮธ : โ} : (ฮธ : Angle).toReal = ฮธ โ -ฯ < ฮธ โง ฮธ โค ฯ := by |
rw [toReal_coe, toIocMod_eq_self two_pi_pos]
ring_nf
rfl
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 744 | 748 | theorem IsRefl.nondegenerate_of_separatingLeft {B : M โโ[R] M โโ[R] Mโ} (hB : B.IsRefl)
(hB' : B.SeparatingLeft) : B.Nondegenerate := by |
refine โจhB', ?_โฉ
rw [separatingRight_iff_flip_ker_eq_bot, hB.ker_eq_bot_iff_ker_flip_eq_bot.mp]
rwa [โ separatingLeft_iff_ker_eq_bot]
|
/-
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.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 133 | 135 | theorem mem_vsub_const_affineSegment {x y z : P} (p : P) :
z -แตฅ p โ affineSegment R (x -แตฅ p) (y -แตฅ p) โ z โ affineSegment R x y := by |
rw [โ affineSegment_vsub_const_image, (vsub_left_injective p).mem_set_image]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat... | Mathlib/Data/Nat/Multiplicity.lean | 61 | 77 | theorem multiplicity_eq_card_pow_dvd {m n b : โ} (hm : m โ 1) (hn : 0 < n) (hb : log m n < b) :
multiplicity m n = โ((Finset.Ico 1 b).filter fun i => m ^ i โฃ n).card :=
calc
multiplicity m n = โ(Ico 1 <| (multiplicity m n).get (finite_nat_iff.2 โจhm, hnโฉ) + 1).card := by |
simp
_ = โ((Finset.Ico 1 b).filter fun i => m ^ i โฃ n).card :=
congr_arg _ <|
congr_arg card <|
Finset.ext fun i => by
rw [mem_filter, mem_Ico, mem_Ico, Nat.lt_succ_iff, โ @PartENat.coe_le_coe i,
PartENat.natCast_get, โ pow_dvd_iff_le_multiplicity, and_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
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community... | Mathlib/Analysis/Complex/Schwarz.lean | 65 | 88 | theorem schwarz_aux {f : โ โ โ} (hd : DifferentiableOn โ f (ball c Rโ))
(h_maps : MapsTo f (ball c Rโ) (ball (f c) Rโ)) (hz : z โ ball c Rโ) :
โdslope f c zโ โค Rโ / Rโ := by |
have hRโ : 0 < Rโ := nonempty_ball.1 โจz, hzโฉ
suffices โแถ r in ๐[<] Rโ, โdslope f c zโ โค Rโ / r by
refine ge_of_tendsto ?_ this
exact (tendsto_const_nhds.div tendsto_id hRโ.ne').mono_left nhdsWithin_le_nhds
rw [mem_ball] at hz
filter_upwards [Ioo_mem_nhdsWithin_Iio โจhz, le_rflโฉ] with r hr
have hrโ : ... |
/-
Copyright (c) 2024 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
import Mathlib.MeasureTheory.SetSemiring
/-!
# Algebra of sets
in this file we define the notion of algebra of sets ang give its basic properties. An algebra
of set... | Mathlib/MeasureTheory/SetAlgebra.lean | 122 | 134 | theorem generateFrom_generateSetAlgebra_eq :
generateFrom (generateSetAlgebra ๐) = generateFrom ๐ := by |
refine le_antisymm (fun s ms โฆ ?_) (generateFrom_mono self_subset_generateSetAlgebra)
refine @generateFrom_induction _ _ (generateSetAlgebra ๐) (fun t ht โฆ ?_)
(@MeasurableSet.empty _ (generateFrom ๐))
(fun t โฆ MeasurableSet.compl)
(fun f hf โฆ MeasurableSet.iUnion hf)
s ms
induction ht with
|... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 387 | 393 | theorem expand_card (f : K[X]) : expand K q f = f ^ q := by |
cases' CharP.exists K with p hp
letI := hp
rcases FiniteField.card K p with โจโจn, nposโฉ, โจhp, hnโฉโฉ
haveI : Fact p.Prime := โจhpโฉ
dsimp at hn
rw [hn, โ map_expand_pow_char, frobenius_pow hn, RingHom.one_def, map_id]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 260 | 276 | theorem matPolyEquiv_eq_X_pow_sub_C {K : Type*} (k : โ) [Field K] (M : Matrix n n K) :
matPolyEquiv ((expand K k : K[X] โ+* K[X]).mapMatrix (charmatrix (M ^ k))) =
X ^ k - C (M ^ k) := by |
-- Porting note: `i` and `j` are used later on, but were not mentioned in mathlib3
ext m i j
rw [coeff_sub, coeff_C, matPolyEquiv_coeff_apply, RingHom.mapMatrix_apply, Matrix.map_apply,
AlgHom.coe_toRingHom, DMatrix.sub_apply, coeff_X_pow]
by_cases hij : i = j
ยท rw [hij, charmatrix_apply_eq, AlgHom.map_s... |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,131 | 1,133 | theorem Subtype.dense_iff {s : Set X} {t : Set s} : Dense t โ s โ closure ((โ) '' t) := by |
rw [inducing_subtype_val.dense_iff, SetCoe.forall]
rfl
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 703 | 704 | theorem rescale_neg_one_X : rescale (-1 : A) X = -X := by |
rw [rescale_X, map_neg, map_one, neg_one_mul]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,765 | 2,768 | theorem length_eq_length_filter_add {l : List (ฮฑ)} (f : ฮฑ โ Bool) :
l.length = (l.filter f).length + (l.filter (! f ยท)).length := by |
simp_rw [โ List.countP_eq_length_filter, l.length_eq_countP_add_countP f, Bool.not_eq_true,
Bool.decide_eq_false]
|
/-
Copyright (c) 2022 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel
-/
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 | 164 | 174 | theorem eq_zero_iff (f : ฮฑ โ E) {s : Set ฮฑ} :
eVariationOn f s = 0 โ โ x โ s, โ y โ s, edist (f x) (f y) = 0 := by |
constructor
ยท rintro h x xs y ys
rw [โ le_zero_iff, โ h]
exact edist_le f xs ys
ยท rintro h
dsimp only [eVariationOn]
rw [ENNReal.iSup_eq_zero]
rintro โจn, u, um, usโฉ
exact Finset.sum_eq_zero fun i _ => h _ (us i.succ) _ (us i)
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sรฉbastien Gouรซzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 411 | 413 | theorem hasFDerivWithinAt_of_mem_nhds (h : s โ ๐ x) :
HasFDerivWithinAt f f' s x โ HasFDerivAt f f' x := by |
rw [HasFDerivAt, HasFDerivWithinAt, nhdsWithin_eq_nhds.mpr h]
|
/-
Copyright (c) 2020 Nicolรฒ Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolรฒ Cavalleri, Yury Kudryashov
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
#align_import geometry.manifold.diffeo... | Mathlib/Geometry/Manifold/Diffeomorph.lean | 322 | 331 | theorem contMDiffWithinAt_comp_diffeomorph_iff {m} (h : M โโ^nโฎI, Jโฏ N) {f : N โ M'} {s x}
(hm : m โค n) :
ContMDiffWithinAt I I' m (f โ h) s x โ ContMDiffWithinAt J I' m f (h.symm โปยน' s) (h x) := by |
constructor
ยท intro Hfh
rw [โ h.symm_apply_apply x] at Hfh
simpa only [(ยท โ ยท), h.apply_symm_apply] using
Hfh.comp (h x) (h.symm.contMDiffWithinAt.of_le hm) (mapsTo_preimage _ _)
ยท rw [โ h.image_eq_preimage]
exact fun hf => hf.comp x (h.contMDiffWithinAt.of_le hm) (mapsTo_image _ _)
|
/-
Copyright (c) 2020 Nicolรฒ Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolรฒ Cavalleri
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.algebra.monoid from "leanprover-community/mathlib"@"e354e865255654389cc46e603216023... | Mathlib/Geometry/Manifold/Algebra/Monoid.lean | 354 | 357 | theorem ContMDiffAt.prod (h : โ i โ t, ContMDiffAt I' I n (f i) xโ) :
ContMDiffAt I' I n (fun x โฆ โ i โ t, f i x) xโ := by |
simp only [โ contMDiffWithinAt_univ] at *
exact ContMDiffWithinAt.prod h
|
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro
-/
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 129 | 138 | theorem nhds_mkOfNhds_single [DecidableEq ฮฑ] {aโ : ฮฑ} {l : Filter ฮฑ} (h : pure aโ โค l) (b : ฮฑ) :
@nhds ฮฑ (TopologicalSpace.mkOfNhds (update pure aโ l)) b =
(update pure aโ l : ฮฑ โ Filter ฮฑ) b := by |
refine nhds_mkOfNhds _ _ (le_update_iff.mpr โจh, fun _ _ => le_rflโฉ) fun a s hs => ?_
rcases eq_or_ne a aโ with (rfl | ha)
ยท filter_upwards [hs] with b hb
rcases eq_or_ne b a with (rfl | hb)
ยท exact hs
ยท rwa [update_noteq hb]
ยท simpa only [update_noteq ha, mem_pure, eventually_pure] using hs
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 750 | 759 | theorem Algebra.adjoin.powerBasis'_minpoly_gen [IsDomain R] [IsDomain S] [NoZeroSMulDivisors R S]
[IsIntegrallyClosed R] {x : S} (hx' : IsIntegral R x) :
minpoly R x = minpoly R (Algebra.adjoin.powerBasis' hx').gen := by |
haveI := isDomain_of_prime (prime_of_isIntegrallyClosed hx')
haveI :=
noZeroSMulDivisors_of_prime_of_degree_ne_zero (prime_of_isIntegrallyClosed hx')
(ne_of_lt (degree_pos hx')).symm
rw [โ minpolyGen_eq, adjoin.powerBasis', minpolyGen_map, minpolyGen_eq,
AdjoinRoot.powerBasis'_gen, โ isAdjoinRootMo... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 76 | 80 | theorem traverse_map (f : ฮฒ โ F ฮณ) (g : ฮฑ โ ฮฒ) (x : t ฮฑ) :
traverse f (g <$> x) = traverse (f โ g) x := by |
rw [@map_eq_traverse_id t _ _ _ _ g]
refine (comp_traverse (G := Id) f (pure โ g) x).symm.trans ?_
congr; apply Comp.applicative_id_comp
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.m... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 461 | 470 | theorem chaar_mono {Kโ : PositiveCompacts G} {Kโ Kโ : Compacts G} (h : (Kโ : Set G) โ Kโ) :
chaar Kโ Kโ โค chaar Kโ Kโ := by |
let eval : (Compacts G โ โ) โ โ := fun f => f Kโ - f Kโ
have : Continuous eval := (continuous_apply Kโ).sub (continuous_apply Kโ)
rw [โ sub_nonneg]; show chaar Kโ โ eval โปยน' Ici (0 : โ)
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar Kโ โค)
unfold clPrehaar; rw [IsClosed.closure_subset_iff]
ยท rintro _ โจU,... |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rรฉmy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,385 | 1,391 | theorem Icc_union_Ici' (hโ : c โค b) : Icc a b โช Ici c = Ici (min a c) := by |
ext1 x
simp_rw [mem_union, mem_Icc, mem_Ici, min_le_iff]
by_cases hc : c โค x
ยท simp only [hc, or_true] -- Porting note: restore `tauto`
ยท have hxb : x โค b := (le_of_not_ge hc).trans hโ
simp only [hxb, and_true] -- Porting note: restore `tauto`
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hรถlzl, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 410 | 413 | theorem FiniteDimensional.finrank_pos_iff_exists_ne_zero [NoZeroSMulDivisors R M] :
0 < finrank R M โ โ x : M, x โ 0 := by |
rw [โ @rank_pos_iff_exists_ne_zero R M, โ finrank_eq_rank]
norm_cast
|
/-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.Polynomial.Eval
#align_import data.mv_polynomial.polynomial from "leanprover-community/mathlib"@"0b8... | Mathlib/Algebra/MvPolynomial/Polynomial.lean | 30 | 40 | theorem eval_polynomial_eval_finSuccEquiv {n : โ} {x : Fin n โ R}
[CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R) :
(eval x) (Polynomial.eval q (finSuccEquiv R n f)) = eval (Fin.cases (eval x q) x) f := by |
simp only [finSuccEquiv_apply, coe_evalโHom, polynomial_eval_evalโ, eval_evalโ]
conv in RingHom.comp _ _ =>
refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_
simp
simp only [evalโ_id]
congr
funext i
refine Fin.cases (by simp) (by simp) i
|
/-
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.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 168 | 173 | theorem Subpresheaf.family_of_elements_compatible {U : Cแตแต} (s : F.obj U) :
(G.familyOfElementsOfSection s).Compatible := by |
intro Yโ Yโ Z gโ gโ fโ fโ hโ hโ e
refine Subtype.ext ?_ -- Porting note: `ext1` does not work here
change F.map gโ.op (F.map fโ.op s) = F.map gโ.op (F.map fโ.op s)
rw [โ FunctorToTypes.map_comp_apply, โ FunctorToTypes.map_comp_apply, โ op_comp, โ op_comp, e]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e1... | Mathlib/FieldTheory/Minpoly/Basic.lean | 67 | 70 | theorem algHom_eq (f : B โโ[A] B') (hf : Function.Injective f) (x : B) :
minpoly A (f x) = minpoly A x := by |
refine dif_ctx_congr (isIntegral_algHom_iff _ hf) (fun _ => ?_) fun _ => rfl
simp_rw [โ Polynomial.aeval_def, aeval_algHom, AlgHom.comp_apply, _root_.map_eq_zero_iff f hf]
|
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