Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2018 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 | 1,411 | 1,413 | theorem preimage_Ioi : ((↑) : ℤ → α) ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋ := by |
ext
simp [floor_lt]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 297 | 298 | theorem snormEssSup_const (c : F) (hμ : μ ≠ 0) :
snormEssSup (fun _ : α => c) μ = (‖c‖₊ : ℝ≥0∞) := by | rw [snormEssSup, essSup_const _ hμ]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.CompleteMetrizable
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathli... | Mathlib/Analysis/NormedSpace/Banach.lean | 88 | 157 | theorem exists_approx_preimage_norm_le (surj : Surjective f) :
∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by |
have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
refine Subset.antisymm (subset_univ _) fun y _ => ?_
rcases surj y with ⟨x, hx⟩
rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
refine mem_iUnion.2 ⟨n, subset_closure ?_⟩
refine (mem_image _ _ _).2 ⟨x, ⟨?_, hx⟩⟩
rwa [mem_ball, dist_eq_norm, sub... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 837 | 842 | theorem exists_nhds_split_inv {s : Set G} (hs : s ∈ 𝓝 (1 : G)) :
∃ V ∈ 𝓝 (1 : G), ∀ v ∈ V, ∀ w ∈ V, v / w ∈ s := by |
have : (fun p : G × G => p.1 * p.2⁻¹) ⁻¹' s ∈ 𝓝 ((1, 1) : G × G) :=
continuousAt_fst.mul continuousAt_snd.inv (by simpa)
simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage] using
this
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multis... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 158 | 159 | theorem pow_count [DecidableEq α] (a : α) : a ^ s.count a = (s.filter (Eq a)).prod := by |
rw [filter_eq, prod_replicate]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 259 | 262 | theorem list_prod_left {l : List ℤ} {n : ℕ} : J(l.prod | n) = (l.map fun a => J(a | n)).prod := by |
induction' l with n l' ih
· simp only [List.prod_nil, List.map_nil, one_left]
· rw [List.map, List.prod_cons, List.prod_cons, mul_left, ih]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import a... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 338 | 350 | theorem integral_gaussian_Ioi (b : ℝ) :
∫ x in Ioi (0 : ℝ), exp (-b * x ^ 2) = √(π / b) / 2 := by |
rcases le_or_lt b 0 with (hb | hb)
· rw [integral_undef, sqrt_eq_zero_of_nonpos, zero_div]
· exact div_nonpos_of_nonneg_of_nonpos pi_pos.le hb
· rwa [← IntegrableOn, integrableOn_Ioi_exp_neg_mul_sq_iff, not_lt]
rw [← RCLike.ofReal_inj (K := ℂ), ← integral_ofReal, ← RCLike.algebraMap_eq_ofReal,
coe_al... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 310 | 320 | theorem Algebra.isIntegral_trace [FiniteDimensional L F] {x : F} (hx : IsIntegral R x) :
IsIntegral R (Algebra.trace L F x) := by |
have hx' : IsIntegral L x := hx.tower_top
rw [← isIntegral_algebraMap_iff (algebraMap L (AlgebraicClosure F)).injective, trace_eq_sum_roots]
· refine (IsIntegral.multiset_sum ?_).nsmul _
intro y hy
rw [mem_roots_map (minpoly.ne_zero hx')] at hy
use minpoly R x, minpoly.monic hx
rw [← aeval_def] a... |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.Spec... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 203 | 206 | theorem Gamma1_in_Gamma0 (N : ℕ) : Gamma1 N ≤ Gamma0 N := by |
intro x HA
simp only [Gamma0_mem, Gamma1_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply] at *
exact HA.2.2
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,671 | 1,672 | theorem flip_injective_iff₂ [FiniteDimensional K V₂] : Injective B.flip ↔ Surjective B := by |
rw [← dualMap_injective_iff]; exact (evalEquiv K V₂).toEquiv.injective_comp B.dualMap
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 575 | 578 | theorem tendsto_measure_iUnion [Preorder ι] [IsDirected ι (· ≤ ·)] [Countable ι]
{s : ι → Set α} (hm : Monotone s) : Tendsto (μ ∘ s) atTop (𝓝 (μ (⋃ n, s n))) := by |
rw [measure_iUnion_eq_iSup hm.directed_le]
exact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm
|
/-
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.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Finite products... | Mathlib/Data/Prod/TProd.lean | 157 | 170 | theorem mk_preimage_tprod :
∀ (l : List ι) (t : ∀ i, Set (α i)), TProd.mk l ⁻¹' Set.tprod l t = { i | i ∈ l }.pi t
| [], t => by simp [Set.tprod]
| i :: l, t => by
ext f
have h : TProd.mk l f ∈ Set.tprod l t ↔ ∀ i : ι, i ∈ l → f i ∈ t i := by |
change f ∈ TProd.mk l ⁻¹' Set.tprod l t ↔ f ∈ { x | x ∈ l }.pi t
rw [mk_preimage_tprod l t]
-- `simp [Set.TProd, TProd.mk, this]` can close this goal but is slow.
rw [Set.tprod, TProd.mk, mem_preimage, mem_pi, prod_mk_mem_set_prod_eq]
simp_rw [mem_setOf_eq, mem_cons]
rw [forall_eq_or_imp, ... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
#align_import... | Mathlib/Algebra/Group/Invertible/Basic.lean | 77 | 82 | theorem Commute.invOf_left [Monoid α] {a b : α} [Invertible b] (h : Commute b a) :
Commute (⅟ b) a :=
calc
⅟ b * a = ⅟ b * (a * b * ⅟ b) := by | simp [mul_assoc]
_ = ⅟ b * (b * a * ⅟ b) := by rw [h.eq]
_ = a * ⅟ b := by simp [mul_assoc]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of options
... | Mathlib/Data/Option/NAry.lean | 195 | 197 | theorem map_map₂_antidistrib_right {g : γ → δ} {f' : β → α' → δ} {g' : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' b (g' a)) :
(map₂ f a b).map g = map₂ f' b (a.map g') := by | cases a <;> cases b <;> simp [h_antidistrib]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 643 | 649 | theorem MeasurableSet.image_inclusion' {s t : Set α} (h : s ⊆ t) {u : Set s}
(hs : MeasurableSet (Subtype.val ⁻¹' s : Set t)) (hu : MeasurableSet u) :
MeasurableSet (inclusion h '' u) := by |
rcases hu with ⟨u, hu, rfl⟩
convert (measurable_subtype_coe hu).inter hs
ext ⟨x, hx⟩
simpa [@and_comm _ (_ = x)] using and_comm
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community... | Mathlib/LinearAlgebra/StdBasis.lean | 175 | 200 | theorem linearIndependent_stdBasis [Ring R] [∀ i, AddCommGroup (Ms i)] [∀ i, Module R (Ms i)]
[DecidableEq η] (v : ∀ j, ιs j → Ms j) (hs : ∀ i, LinearIndependent R (v i)) :
LinearIndependent R fun ji : Σj, ιs j => stdBasis R Ms ji.1 (v ji.1 ji.2) := by |
have hs' : ∀ j : η, LinearIndependent R fun i : ιs j => stdBasis R Ms j (v j i) := by
intro j
exact (hs j).map' _ (ker_stdBasis _ _ _)
apply linearIndependent_iUnion_finite hs'
intro j J _ hiJ
have h₀ :
∀ j, span R (range fun i : ιs j => stdBasis R Ms j (v j i)) ≤
LinearMap.range (stdBasis R ... |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-commun... | Mathlib/SetTheory/Game/Ordinal.lean | 83 | 87 | theorem toPGame_moveLeft_hEq {o : Ordinal} :
have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o
HEq o.toPGame.moveLeft fun x : o.out.α => (typein (· < ·) x).toPGame := by |
rw [toPGame]
rfl
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 2,142 | 2,142 | theorem norm_div_eq_zero_iff : ‖a / b‖ = 0 ↔ a = b := by | rw [norm_eq_zero'', div_eq_one]
|
/-
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.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"0... | Mathlib/Algebra/Squarefree/Basic.lean | 120 | 126 | theorem squarefree_iff_multiplicity_le_one (r : R) :
Squarefree r ↔ ∀ x : R, multiplicity x r ≤ 1 ∨ IsUnit x := by |
refine forall_congr' fun a => ?_
rw [← sq, pow_dvd_iff_le_multiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]
norm_cast
rw [← one_add_one_eq_two]
simpa using PartENat.add_one_le_iff_lt (PartENat.natCast_ne_top 1)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 344 | 345 | theorem coeff_zero_eq_aeval_zero' (p : R[X]) : algebraMap R A (p.coeff 0) = aeval (0 : A) p := by |
simp [aeval_def]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 944 | 953 | theorem castNum_shiftLeft (m : Num) (n : Nat) : ↑(m <<< n) = (m : ℕ) <<< (n : ℕ) := by |
cases m <;> dsimp only [← shiftl_eq_shiftLeft, shiftl]
· symm
apply Nat.zero_shiftLeft
simp only [cast_pos]
induction' n with n IH
· rfl
simp [PosNum.shiftl_succ_eq_bit0_shiftl, Nat.shiftLeft_succ, IH,
Nat.bit0_val, pow_succ, ← mul_assoc, mul_comm,
-shiftl_eq_shiftLeft, -PosNum.shiftl_e... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 160 | 161 | theorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by |
simp [frontier, ha]
|
/-
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.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f... | Mathlib/Data/PNat/Xgcd.lean | 222 | 224 | theorem flip_isSpecial : (flip u).IsSpecial ↔ u.IsSpecial := by |
dsimp [IsSpecial, flip]
rw [mul_comm u.x, mul_comm u.zp, add_comm u.zp]
|
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5... | Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 85 | 88 | theorem log_div_sqrt_antitoneOn : AntitoneOn (fun x : ℝ => log x / √x) { x | exp 2 ≤ x } := by |
simp_rw [sqrt_eq_rpow]
convert @log_div_self_rpow_antitoneOn (1 / 2) (by norm_num)
norm_num
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_semin... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 227 | 263 | theorem snorm_le_snorm_mul_snorm_of_nnnorm {p q r : ℝ≥0∞}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (b : E → F → G)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hpqr : 1 / p = 1 / q + 1 / r) :
snorm (fun x => b (f x) (g x)) p μ ≤ snorm f q μ * snorm g r μ := by |
by_cases hp_zero : p = 0
· simp [hp_zero]
have hq_ne_zero : q ≠ 0 := by
intro hq_zero
simp only [hq_zero, hp_zero, one_div, ENNReal.inv_zero, top_add, ENNReal.inv_eq_top] at hpqr
have hr_ne_zero : r ≠ 0 := by
intro hr_zero
simp only [hr_zero, hp_zero, one_div, ENNReal.inv_zero, add_top, ENNReal... |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-co... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 114 | 123 | theorem strongEpi_of_strongEpi [StrongEpi (f ≫ g)] : StrongEpi g :=
{ epi := epi_of_epi f g
llp := fun {X Y} z _ => by
constructor
intro u v sq
have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v := by | simp only [Category.assoc, sq.w]
exact
CommSq.HasLift.mk'
⟨(CommSq.mk h₀).lift, by
simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w], by simp⟩ }
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,879 | 1,888 | theorem mulSingle_mem_pi [DecidableEq η] {I : Set η} {H : ∀ i, Subgroup (f i)} (i : η) (x : f i) :
Pi.mulSingle i x ∈ pi I H ↔ i ∈ I → x ∈ H i := by |
constructor
· intro h hi
simpa using h i hi
· intro h j hj
by_cases heq : j = i
· subst heq
simpa using h hj
· simp [heq, one_mem]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 159 | 165 | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by |
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 772 | 781 | theorem blockDiagonal'_mul [NonUnitalNonAssocSemiring α] [∀ i, Fintype (n' i)] [Fintype o]
(M : ∀ i, Matrix (m' i) (n' i) α) (N : ∀ i, Matrix (n' i) (p' i) α) :
(blockDiagonal' fun k => M k * N k) = blockDiagonal' M * blockDiagonal' N := by |
ext ⟨k, i⟩ ⟨k', j⟩
simp only [blockDiagonal'_apply, mul_apply, ← Finset.univ_sigma_univ, Finset.sum_sigma]
rw [Fintype.sum_eq_single k]
· simp only [if_pos, dif_pos] -- Porting note: added
split_ifs <;> simp
· intro j' hj'
exact Finset.sum_eq_zero fun _ _ => by rw [dif_neg hj'.symm, zero_mul]
|
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.D... | Mathlib/CategoryTheory/CofilteredSystem.lean | 238 | 256 | theorem IsMittagLeffler.toPreimages (h : F.IsMittagLeffler) : (F.toPreimages s).IsMittagLeffler :=
(isMittagLeffler_iff_subset_range_comp _).2 fun j => by
obtain ⟨j₁, g₁, f₁, -⟩ := IsCofilteredOrEmpty.cone_objs i j
obtain ⟨j₂, f₂, h₂⟩ := F.isMittagLeffler_iff_eventualRange.1 h j₁
refine ⟨j₂, f₂ ≫ f₁, fun ... |
rw [h₂]
exact ⟨_, rfl⟩
obtain ⟨y, hy, h₃⟩ := h.subset_image_eventualRange F (f₃ ≫ f₂) this
refine ⟨⟨y, mem_iInter.2 fun g₂ => ?_⟩, Subtype.ext ?_⟩
· obtain ⟨j₄, f₄, h₄⟩ := IsCofilteredOrEmpty.cone_maps g₂ ((f₃ ≫ f₂) ≫ g₁)
obtain ⟨y, rfl⟩ := F.mem_eventualRange_iff.1 hy f₄
rw [← map_... |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 113 | 116 | theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by |
cases hq
· assumption
· exact False.elim (CharP.char_ne_one R 1 rfl)
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 331 | 335 | theorem finsum_smul {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M]
(f : ι → R) (x : M) : (∑ᶠ i, f i) • x = ∑ᶠ i, f i • x := by |
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· exact ((smulAddHom R M).flip x).map_finsum_of_injective (smul_left_injective R hx) _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 894 | 897 | theorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) :
Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by |
obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h
exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩
|
/-
Copyright (c) 2018 Mitchell Rowett. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Rowett, Scott Morrison
-/
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Group... | Mathlib/GroupTheory/Coset.lean | 117 | 118 | theorem leftCoset_rightCoset (s : Set α) (a b : α) : op b • a • s = a • (op b • s) := by |
simp [← image_smul, (image_comp _ _ _).symm, Function.comp, mul_assoc]
|
/-
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 | 64 | 66 | theorem isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x := by |
dsimp only [IsOrtho]
rw [map_zero B, zero_apply]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated... | Mathlib/MeasureTheory/Measure/Dirac.lean | 87 | 92 | theorem map_eq_sum [Countable β] [MeasurableSingletonClass β] (μ : Measure α) (f : α → β)
(hf : Measurable f) : μ.map f = sum fun b : β => μ (f ⁻¹' {b}) • dirac b := by |
ext s
have : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}) := fun y _ => hf (measurableSet_singleton _)
simp [← tsum_measure_preimage_singleton (to_countable s) this, *,
tsum_subtype s fun b => μ (f ⁻¹' {b}), ← indicator_mul_right s fun b => μ (f ⁻¹' {b})]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,879 | 1,880 | theorem mulVec_one [Fintype n] (A : Matrix m n α) : A *ᵥ 1 = fun i => ∑ j, A i j := by |
ext; simp [mulVec, dotProduct]
|
/-
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.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.nor... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 98 | 103 | theorem exp_of_re_eq_zero (q : Quaternion ℝ) (hq : q.re = 0) :
exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q := by |
rw [exp_eq_tsum]
refine HasSum.tsum_eq ?_
simp_rw [← expSeries_apply_eq]
exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)
|
/-
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.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 69 | 73 | theorem permutationsAux2_snd_cons (t : α) (ts : List α) (r : List β) (y : α) (ys : List α)
(f : List α → β) :
(permutationsAux2 t ts r (y :: ys) f).2 =
f (t :: y :: ys ++ ts) :: (permutationsAux2 t ts r ys fun x : List α => f (y :: x)).2 := by |
simp [permutationsAux2, permutationsAux2_fst t _ _ ys]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,054 | 1,058 | theorem lintegral_sum {m : MeasurableSpace α} {ι} (f : α →ₛ ℝ≥0∞) (μ : ι → Measure α) :
f.lintegral (Measure.sum μ) = ∑' i, f.lintegral (μ i) := by |
simp only [lintegral, Measure.sum_apply, f.measurableSet_preimage, ← Finset.tsum_subtype, ←
ENNReal.tsum_mul_left]
apply ENNReal.tsum_comm
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 220 | 231 | theorem totient_le_degree_minpoly : Nat.totient n ≤ (minpoly ℤ μ).natDegree := by |
classical
let P : ℤ[X] := minpoly ℤ μ
-- minimal polynomial of `μ`
let P_K : K[X] := map (Int.castRingHom K) P
-- minimal polynomial of `μ` sent to `K[X]`
calc
n.totient = (primitiveRoots n K).card := h.card_primitiveRoots.symm
_ ≤ P_K.roots.toFinset.card := Finset.card_le_card (is_roots_of_minpoly... |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 291 | 294 | theorem twoCocycles_map_one_fst (f : twoCocycles A) (g : G) :
f.1 (1, g) = f.1 (1, 1) := by |
have := ((mem_twoCocycles_iff f.1).1 f.2 1 1 g).symm
simpa only [map_one, LinearMap.one_apply, one_mul, add_right_inj, this]
|
/-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Projectivization/Subspace.lean | 202 | 205 | theorem mem_span {S : Set (ℙ K V)} (u : ℙ K V) :
u ∈ span S ↔ ∀ W : Subspace K V, S ⊆ W → u ∈ W := by |
simp_rw [← span_le_subspace_iff]
exact ⟨fun hu W hW => hW hu, fun W => W (span S) (le_refl _)⟩
|
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.MinMax
import Mathlib.Order.WithBot
#align_import data.list.min_max from "leanprover-communi... | Mathlib/Data/List/MinMax.lean | 337 | 340 | theorem not_lt_maximum_of_mem' (ha : a ∈ l) : ¬maximum l < (a : WithBot α) := by |
cases h : l.maximum
· simp_all
· simp [not_lt_maximum_of_mem ha h, not_false_iff]
|
/-
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 | 673 | 695 | theorem wOppSide_iff_exists_wbtw {s : AffineSubspace R P} {x y : P} :
s.WOppSide x y ↔ ∃ p ∈ s, Wbtw R x p y := by |
refine ⟨fun h => ?_, fun ⟨p, hp, h⟩ => h.wOppSide₁₃ hp⟩
rcases h with ⟨p₁, hp₁, p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩
· rw [vsub_eq_zero_iff_eq] at h
rw [h]
exact ⟨p₁, hp₁, wbtw_self_left _ _ _⟩
· rw [vsub_eq_zero_iff_eq] at h
rw [← h]
exact ⟨p₂, hp₂, wbtw_self_right _ _ _⟩
· refine ⟨lineMa... |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Anal... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 69 | 80 | theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ := by |
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv
have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1
simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢
refine hp.mono fun x hx => ?_
by_cases h : x = 0
· convert hasSum_single (α := E) 0 _ <;> intros <;> simp [*]
· have hxx : ∀ n : ℕ,... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,116 | 1,116 | theorem Ici_diff_Ici : Ici a \ Ici b = Ico a b := by | rw [diff_eq, compl_Ici, Ici_inter_Iio]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import topology.algebra.module.multilinear from "leanprover-... | Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean | 332 | 337 | theorem _root_.ContinuousLinearMap.compContinuousMultilinearMap_coe (g : M₂ →L[R] M₃)
(f : ContinuousMultilinearMap R M₁ M₂) :
(g.compContinuousMultilinearMap f : (∀ i, M₁ i) → M₃) =
(g : M₂ → M₃) ∘ (f : (∀ i, M₁ i) → M₂) := by |
ext m
rfl
|
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.GroupTheory.PresentedGroup
import Mathlib.GroupTheory.Coxeter.Matrix
/-!
# Coxeter groups and Coxeter syst... | Mathlib/GroupTheory/Coxeter/Basic.lean | 371 | 378 | theorem wordProd_surjective : Surjective cs.wordProd := by |
intro w
apply cs.simple_induction_left w
· use []
rw [wordProd_nil]
· rintro _ i ⟨ω, rfl⟩
use i :: ω
rw [wordProd_cons]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 538 | 543 | theorem div_C_mul : p / (C a * q) = C a⁻¹ * (p / q) := by |
by_cases ha : a = 0
· simp [ha]
simp only [div_def, leadingCoeff_mul, mul_inv, leadingCoeff_C, C.map_mul, mul_assoc]
congr 3
rw [mul_left_comm q, ← mul_assoc, ← C.map_mul, mul_inv_cancel ha, C.map_one, one_mul]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 805 | 809 | theorem normalizedFactors_prod_of_prime [Nontrivial α] [Unique αˣ] {m : Multiset α}
(h : ∀ p ∈ m, Prime p) : normalizedFactors m.prod = m := by |
simpa only [← Multiset.rel_eq, ← associated_eq_eq] using
prime_factors_unique prime_of_normalized_factor h
(normalizedFactors_prod (m.prod_ne_zero_of_prime h))
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Ba... | Mathlib/RepresentationTheory/Basic.lean | 113 | 114 | theorem asAlgebraHom_of (g : G) : asAlgebraHom ρ (of k G g) = ρ g := by |
simp only [MonoidAlgebra.of_apply, asAlgebraHom_single, one_smul]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Yury Kudryashov
-/
import Mathlib.Topology.Homeomorph
import Mathlib.Data.Option.Basic
#align_import topology.paracompact from "leanprover-community/mathlib"@"2705404e70... | Mathlib/Topology/Compactness/Paracompact.lean | 196 | 244 | theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [WeaklyLocallyCompactSpace X]
[SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
{s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
∃ (α : Type v) (c : α → X) (r : ∀ a, ι... |
classical
-- For technical reasons we prepend two empty sets to the sequence `CompactExhaustion.choice X`
set K' : CompactExhaustion X := CompactExhaustion.choice X
set K : CompactExhaustion X := K'.shiftr.shiftr
set Kdiff := fun n ↦ K (n + 1) \ interior (K n)
-- Now we restate some properties of... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 477 | 480 | theorem preimage_coe_coe_diagonal (s : Set α) :
Prod.map (fun x : s => (x : α)) (fun x : s => (x : α)) ⁻¹' diagonal α = diagonal s := by |
ext ⟨⟨x, hx⟩, ⟨y, hy⟩⟩
simp [Set.diagonal]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-communit... | Mathlib/SetTheory/Cardinal/Finite.lean | 296 | 300 | theorem card_eq_zero_iff_empty (α : Type*) : card α = 0 ↔ IsEmpty α := by |
rw [← Cardinal.mk_eq_zero_iff]
conv_rhs => rw [← Nat.cast_zero]
simp only [← Cardinal.toPartENat_eq_natCast_iff]
simp only [PartENat.card, Nat.cast_zero]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 351 | 352 | theorem span_range_eq_iSup {ι : Sort*} {v : ι → M} : span R (range v) = ⨆ i, R ∙ v i := by |
rw [span_eq_iSup_of_singleton_spans, iSup_range]
|
/-
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.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"356447fe00e75e5... | Mathlib/Algebra/Lie/Semisimple/Basic.lean | 327 | 327 | theorem ad_ker_eq_bot_of_hasTrivialRadical [HasTrivialRadical R L] : (ad R L).ker = ⊥ := by | 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 Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 429 | 433 | theorem eval₂_gcd_eq_zero [CommSemiring k] [DecidableEq R]
{ϕ : R →+* k} {f g : R[X]} {α : k} (hf : f.eval₂ ϕ α = 0)
(hg : g.eval₂ ϕ α = 0) : (EuclideanDomain.gcd f g).eval₂ ϕ α = 0 := by |
rw [EuclideanDomain.gcd_eq_gcd_ab f g, Polynomial.eval₂_add, Polynomial.eval₂_mul,
Polynomial.eval₂_mul, hf, hg, zero_mul, zero_mul, zero_add]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib... | Mathlib/Data/Sum/Interval.lean | 76 | 88 | theorem sumLift₂_eq_empty :
sumLift₂ f g a b = ∅ ↔
(∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧
∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· constructor <;>
· rintro a b rfl rfl
exact map_eq_empty.1 h
cases a <;> cases b
· exact map_eq_empty.2 (h.1 _ _ rfl rfl)
· rfl
· rfl
· exact map_eq_empty.2 (h.2 _ _ rfl rfl)
|
/-
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.LinearAlgebra.PiTensorProduct
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.DirectSum.Algebra
#align_import linear_algebra.tensor_power from "leanpr... | Mathlib/LinearAlgebra/TensorPower.lean | 253 | 256 | theorem algebraMap₀_mul_algebraMap₀ (r s : R) :
cast R M (add_zero _) (algebraMap₀ r ₜ* algebraMap₀ s) = algebraMap₀ (r * s) := by |
rw [← smul_eq_mul, LinearEquiv.map_smul]
exact algebraMap₀_mul r (@algebraMap₀ R M _ _ _ s)
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_characte... | Mathlib/NumberTheory/MulChar/Basic.lean | 440 | 441 | theorem isNontrivial_iff (χ : MulChar R R') : χ.IsNontrivial ↔ χ ≠ 1 := by |
simp only [IsNontrivial, Ne, ext_iff, not_forall, one_apply_coe]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,223 | 1,224 | theorem mkPiAlgebraFin_apply_const (a : A) :
(MultilinearMap.mkPiAlgebraFin R n A fun _ => a) = a ^ n := by | simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 42 | 44 | theorem verschiebungFun_coeff (x : 𝕎 R) (n : ℕ) :
(verschiebungFun x).coeff n = if n = 0 then 0 else x.coeff (n - 1) := by |
simp only [verschiebungFun, ge_iff_le]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Eric Wieser
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.Star
import Mathlib.LinearAlgebra.... | Mathlib/Topology/Instances/Matrix.lean | 396 | 402 | theorem Matrix.blockDiagonal_tsum [DecidableEq p] [T2Space R] {f : X → p → Matrix m n R} :
blockDiagonal (∑' x, f x) = ∑' x, blockDiagonal (f x) := by |
by_cases hf : Summable f
· exact hf.hasSum.matrix_blockDiagonal.tsum_eq.symm
· have hft := summable_matrix_blockDiagonal.not.mpr hf
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft]
exact blockDiagonal_zero
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,413 | 1,413 | theorem neg_lf_iff {x y : PGame} : -y ⧏ x ↔ -x ⧏ y := by | rw [← neg_neg x, neg_lf_neg_iff, neg_neg]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 197 | 203 | theorem encode_lt_pair (cf cg) :
encode cf < encode (pair cf cg) ∧ encode cg < encode (pair cf cg) := by |
simp only [encodeCode_eq, encodeCode]
have := Nat.mul_le_mul_right (Nat.pair cf.encodeCode cg.encodeCode) (by decide : 1 ≤ 2 * 2)
rw [one_mul, mul_assoc] at this
have := lt_of_le_of_lt this (lt_add_of_pos_right _ (by decide : 0 < 4))
exact ⟨lt_of_le_of_lt (Nat.left_le_pair _ _) this, lt_of_le_of_lt (Nat.righ... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb3... | Mathlib/RingTheory/Polynomial/Opposites.lean | 95 | 99 | theorem coeff_opRingEquiv (p : R[X]ᵐᵒᵖ) (n : ℕ) :
(opRingEquiv R p).coeff n = op ((unop p).coeff n) := by |
induction' p using MulOpposite.rec' with p
cases p
rfl
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,370 | 1,377 | theorem integral_Iic_mul_deriv_eq_deriv_mul
(hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
(huv' : IntegrableOn (u * v') (Iic a)) (hu'v : IntegrableOn (u' * v) (Iic a))
(h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
∫ (x : ℝ)... |
rw [Pi.mul_def] at huv' hu'v
rw [eq_sub_iff_add_eq, ← integral_add huv' hu'v]
simpa only [add_comm] using integral_Iic_deriv_mul_eq_sub hu hv (hu'v.add huv') h_zero h_infty
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 114 | 123 | theorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by |
dsimp [destruct]
induction' f0 : s.1 0 with _ <;> intro h
· contradiction
· apply Subtype.eq
funext n
induction' n with n IH
· injection h with h'
rwa [h'] at f0
· exact s.2 IH
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,276 | 1,277 | theorem lt_succ_castPred {a : Fin (n + 1)} (ha : a ≠ last n) :
a < succ (a.castPred ha) := by | rw [lt_succ_castPred_iff, le_def]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 489 | 490 | theorem spanExt_inv_app_left : (spanExt iX iY iZ wf wg).inv.app WalkingSpan.left = iY.inv := by |
dsimp [spanExt]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Hom
#align_import algebra.algebra.equiv from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
/-!
# I... | Mathlib/Algebra/Algebra/Equiv.lean | 478 | 481 | theorem arrowCongr_symm (e₁ : A₁ ≃ₐ[R] A₁') (e₂ : A₂ ≃ₐ[R] A₂') :
(arrowCongr e₁ e₂).symm = arrowCongr e₁.symm e₂.symm := by |
ext
rfl
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 173 | 193 | theorem iterate_derivative_at_0 (n ν : ℕ) :
(Polynomial.derivative^[ν] (bernsteinPolynomial R n ν)).eval 0 =
(ascPochhammer R ν).eval ((n - (ν - 1) : ℕ) : R) := by |
by_cases h : ν ≤ n
· induction' ν with ν ih generalizing n
· simp [eval_at_0]
· have h' : ν ≤ n - 1 := le_tsub_of_add_le_right h
simp only [derivative_succ, ih (n - 1) h', iterate_derivative_succ_at_0_eq_zero,
Nat.succ_sub_succ_eq_sub, tsub_zero, sub_zero, iterate_derivative_sub,
iter... |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.Fin... | Mathlib/Analysis/Analytic/Polynomial.lean | 26 | 32 | theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt 𝕜 f z) (p : A[X]) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by |
refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_
· simp_rw [aeval_C]; apply analyticAt_const
· simp_rw [aeval_add]; exact hp.add hq
· convert hp.mul hf
simp_rw [pow_succ, aeval_mul, ← mul_assoc, aeval_X]
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 220 | 221 | theorem gcd_mul_lcm (i j : ℤ) : gcd i j * lcm i j = natAbs (i * j) := by |
rw [Int.gcd, Int.lcm, Nat.gcd_mul_lcm, natAbs_mul]
|
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
/-!
# Final functors wit... | Mathlib/CategoryTheory/Filtered/Final.lean | 56 | 72 | theorem isFiltered_structuredArrow_of_isFiltered_of_exists [IsFilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) (h₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),
∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t) (d : D) :
IsFiltered (StructuredArrow d F) := by |
have : Nonempty (StructuredArrow d F) := by
obtain ⟨c, ⟨f⟩⟩ := h₁ d
exact ⟨.mk f⟩
suffices IsFilteredOrEmpty (StructuredArrow d F) from IsFiltered.mk
refine ⟨fun f g => ?_, fun f g η μ => ?_⟩
· obtain ⟨c, ⟨t, ht⟩⟩ := h₂ (f.hom ≫ F.map (IsFiltered.leftToMax f.right g.right))
(g.hom ≫ F.map (IsFi... |
/-
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.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 312 | 320 | theorem outerMeasure_lt_top_of_isCompact [WeaklyLocallyCompactSpace G]
{K : Set G} (hK : IsCompact K) :
μ.outerMeasure K < ∞ := by |
rcases exists_compact_superset hK with ⟨F, h1F, h2F⟩
calc
μ.outerMeasure K ≤ μ.outerMeasure (interior F) := measure_mono h2F
_ ≤ μ ⟨F, h1F⟩ := by
apply μ.outerMeasure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset
_ < ⊤ := μ.lt_top _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kyle Miller
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align... | Mathlib/Data/Set/Finite.lean | 863 | 865 | theorem Finite.image {s : Set α} (f : α → β) (hs : s.Finite) : (f '' s).Finite := by |
have := hs.to_subtype
apply toFinite
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,208 | 1,212 | theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
AEStronglyMeasurable f (0 : Measure α) := by |
nontriviality α
inhabit α
exact ⟨fun _ => f default, stronglyMeasurable_const, rfl⟩
|
/-
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.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 570 | 572 | theorem cancel_iso_hom_right_assoc {W X X' Y Z : C} (f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X')
(g' : X' ⟶ Y) (h : Y ≅ Z) : f ≫ g ≫ h.hom = f' ≫ g' ≫ h.hom ↔ f ≫ g = f' ≫ g' := by |
simp only [← Category.assoc, cancel_mono]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 147 | 158 | theorem iff_quotient_mvPolynomial :
FiniteType R S ↔
∃ (s : Finset S) (f : MvPolynomial { x // x ∈ s } R →ₐ[R] S), Surjective f := by |
constructor
· rintro ⟨s, hs⟩
use s, MvPolynomial.aeval (↑)
intro x
have hrw : (↑s : Set S) = fun x : S => x ∈ s.val := rfl
rw [← Set.mem_range, ← AlgHom.coe_range, ← adjoin_eq_range, ← hrw, hs]
exact Set.mem_univ x
· rintro ⟨s, ⟨f, hsur⟩⟩
exact FiniteType.of_surjective (FiniteType.mvPolyn... |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 377 | 389 | theorem prod_leftInvSeq (ω : List B) : prod (lis ω) = (π ω)⁻¹ := by |
simp [leftInvSeq_eq_reverse_rightInvSeq_reverse, prod_reverse_noncomm]
have : List.map (fun x ↦ x⁻¹) (ris ω.reverse) = ris ω.reverse := calc
List.map (fun x ↦ x⁻¹) (ris ω.reverse)
_ = List.map id (ris ω.reverse) := by
apply List.map_congr
intro t ht
exact (cs.isReflectio... |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Finsupp.Defs
#align_import data.list.to_finsupp from "leanprover-community/mathlib"@"06a655b5fcfbda03502f9158bbf6c0f1400886f9"
/-!
# Lists as... | Mathlib/Data/List/ToFinsupp.lean | 147 | 156 | theorem toFinsupp_eq_sum_map_enum_single {R : Type*} [AddMonoid R] (l : List R)
[DecidablePred (getD l · 0 ≠ 0)] :
toFinsupp l = (l.enum.map fun nr : ℕ × R => Finsupp.single nr.1 nr.2).sum := by |
/- Porting note (#11215): TODO: `induction` fails to substitute `l = []` in
`[DecidablePred (getD l · 0 ≠ 0)]`, so we manually do some `revert`/`intro` as a workaround -/
revert l; intro l
induction l using List.reverseRecOn with
| nil => exact toFinsupp_nil
| append_singleton x xs ih =>
classical simp... |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.Order.Filter.ZeroAndBoundedAtFilter
... | Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean | 37 | 38 | theorem atImInfty_mem (S : Set ℍ) : S ∈ atImInfty ↔ ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → z ∈ S := by |
simp only [atImInfty_basis.mem_iff, true_and]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Topology.GDelta
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.Instances.EReal
import Mathlib.Analysis.Normed.G... | Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 339 | 346 | theorem measurable_of_isClosed' {f : δ → γ}
(hf : ∀ s, IsClosed s → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)) : Measurable f := by |
apply measurable_of_isClosed; intro s hs
rcases eq_empty_or_nonempty s with h1 | h1
· simp [h1]
by_cases h2 : s = univ
· simp [h2]
exact hf s hs h1 h2
|
/-
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 | 1,221 | 1,225 | theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
(hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial := by |
rcases hs with ⟨fx, hx, fy, hy, hxy⟩
rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩
exact ⟨x, hx, y, hy, mt (congr_arg f) hxy⟩
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheo... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 113 | 115 | theorem HasEigenvector.pow_apply {f : End R M} {μ : R} {v : M} (hv : f.HasEigenvector μ v) (n : ℕ) :
(f ^ n) v = μ ^ n • v := by |
induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,175 | 1,182 | theorem integral_eq_sub_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn f (Icc a b))
(hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' x) (Ioi x) x)
(f'int : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a := by |
refine (NormedSpace.eq_iff_forall_dual_eq ℝ).2 fun g => ?_
rw [← g.intervalIntegral_comp_comm f'int, g.map_sub]
exact integral_eq_sub_of_hasDeriv_right_of_le_real hab (g.continuous.comp_continuousOn hcont)
(fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx))
(g.integrable_comp ((intervalInte... |
/-
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 Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 732 | 739 | theorem Perm.kunion_right {l₁ l₂ : List (Sigma β)} (p : l₁ ~ l₂) (l) :
kunion l₁ l ~ kunion l₂ l := by |
induction p generalizing l with
| nil => rfl
| cons hd _ ih =>
simp [ih (List.kerase _ _), Perm.cons]
| swap s₁ s₂ l => simp [kerase_comm, Perm.swap]
| trans _ _ ih₁₂ ih₂₃ => exact Perm.trans (ih₁₂ l) (ih₂₃ l)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 836 | 839 | theorem ne_zero_iff {p : MvPolynomial σ R} : p ≠ 0 ↔ ∃ d, coeff d p ≠ 0 := by |
rw [Ne, eq_zero_iff]
push_neg
rfl
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mat... | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | 290 | 298 | theorem isSheaf_iff_isSheafPreservesLimitPairwiseIntersections :
F.IsSheaf ↔ F.IsSheafPreservesLimitPairwiseIntersections := by |
rw [isSheaf_iff_isSheafPairwiseIntersections]
constructor
· intro h ι U
exact ⟨preservesLimitOfPreservesLimitCone (Pairwise.coconeIsColimit U).op (h U).some⟩
· intro h ι U
haveI := (h U).some
exact ⟨PreservesLimit.preserves (Pairwise.coconeIsColimit U).op⟩
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import data.pi.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662d... | Mathlib/Data/Pi/Interval.lean | 52 | 53 | theorem card_Ioo : (Ioo a b).card = (∏ i, (Icc (a i) (b i)).card) - 2 := by |
rw [card_Ioo_eq_card_Icc_sub_two, card_Icc]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,441 | 1,442 | theorem mk_finsupp_of_fintype (α β : Type u) [Fintype α] [Zero β] :
#(α →₀ β) = #β ^ Fintype.card α := by | simp
|
/-
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.NumberTheory.NumberField.Embeddings
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb1... | Mathlib/NumberTheory/NumberField/Units/Basic.lean | 101 | 108 | theorem mem_torsion {x : (𝓞 K)ˣ} [NumberField K] :
x ∈ torsion K ↔ ∀ w : InfinitePlace K, w x = 1 := by |
rw [eq_iff_eq (x : K) 1, torsion, CommGroup.mem_torsion]
refine ⟨fun hx φ ↦ (((φ.comp $ algebraMap (𝓞 K) K).toMonoidHom.comp $
Units.coeHom _).isOfFinOrder hx).norm_eq_one, fun h ↦ isOfFinOrder_iff_pow_eq_one.2 ?_⟩
obtain ⟨n, hn, hx⟩ := Embeddings.pow_eq_one_of_norm_eq_one K ℂ x.val.isIntegral_coe h
exact... |
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Chain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Set.Subsingleton
#align_... | Mathlib/Order/Ideal.lean | 459 | 460 | theorem mem_sInf : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s := by |
simp_rw [← SetLike.mem_coe, coe_sInf, mem_iInter₂]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 709 | 711 | theorem term_ord_aux {i : I} (ho : ord I i < Ordinal.type ((·<·) : I → I → Prop)) :
term I ho = i := by |
simp only [term, ord, Ordinal.enum_typein]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 162 | 163 | theorem toNNReal_pos_iff : 0 < a.toNNReal ↔ 0 < a ∧ a < ∞ := by |
induction a <;> simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,468 | 1,473 | theorem IsOpen.mul_closure (hs : IsOpen s) (t : Set G) : s * closure t = s * t := by |
refine (mul_subset_iff.2 fun a ha b hb => ?_).antisymm (mul_subset_mul_left subset_closure)
rw [mem_closure_iff] at hb
have hbU : b ∈ s⁻¹ * {a * b} := ⟨a⁻¹, Set.inv_mem_inv.2 ha, a * b, rfl, inv_mul_cancel_left _ _⟩
obtain ⟨_, ⟨c, hc, d, rfl : d = _, rfl⟩, hcs⟩ := hb _ hs.inv.mul_right hbU
exact ⟨c⁻¹, hc, _,... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7... | Mathlib/Algebra/RingQuot.lean | 699 | 703 | theorem eq_liftAlgHom_comp_mkAlgHom {s : A → A → Prop} (f : RingQuot s →ₐ[S] B) :
f = liftAlgHom S ⟨f.comp (mkAlgHom S s), fun x y h ↦ congr_arg f (mkAlgHom_rel S h)⟩ := by |
conv_lhs => rw [← (liftAlgHom S).apply_symm_apply f]
rw [liftAlgHom]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,270 | 1,282 | theorem prod_dite_eq [DecidableEq α] (s : Finset α) (a : α) (b : ∀ x : α, a = x → β) :
∏ x ∈ s, (if h : a = x then b x h else 1) = ite (a ∈ s) (b a rfl) 1 := by |
split_ifs with h
· rw [Finset.prod_eq_single a, dif_pos rfl]
· intros _ _ h
rw [dif_neg]
exact h.symm
· simp [h]
· rw [Finset.prod_eq_one]
intros
rw [dif_neg]
rintro rfl
contradiction
|
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