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) 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.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topo... | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | 94 | 106 | theorem exists_locallyFinite_subset_iUnion_ball_radius_lt (hs : IsClosed s) {R : α → ℝ}
(hR : ∀ x ∈ s, 0 < R x) :
∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
(∀ i, c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
(LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) := by |
have : ∀ x ∈ s, (𝓝 x).HasBasis (fun r : ℝ => 0 < r ∧ r < R x) fun r => ball x r := fun x hx =>
nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx))
rcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs this with
⟨ι, c, r', hr', hsub', hfin⟩
rcases exists_subset_iUnion_ball_radius_p... |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
#align_import ca... | Mathlib/CategoryTheory/Functor/EpiMono.lean | 325 | 334 | theorem strongEpi_map_iff_strongEpi_of_isEquivalence [IsEquivalence F] :
StrongEpi (F.map f) ↔ StrongEpi f := by |
constructor
· intro
have e : Arrow.mk f ≅ Arrow.mk (F.inv.map (F.map f)) :=
Arrow.isoOfNatIso F.asEquivalence.unitIso (Arrow.mk f)
rw [StrongEpi.iff_of_arrow_iso e]
infer_instance
· intro
infer_instance
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 248 | 253 | theorem ne_and_ne_of_swap_mul_apply_ne_self {f : Perm α} {x y : α} (hy : (swap x (f x) * f) y ≠ y) :
f y ≠ y ∧ y ≠ x := by |
simp only [swap_apply_def, mul_apply, f.injective.eq_iff] at *
by_cases h : f y = x
· constructor <;> intro <;> simp_all only [if_true, eq_self_iff_true, not_true, Ne]
· split_ifs at hy with h h <;> try { simp [*] at * }
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 173 | 175 | theorem Q_idem (q : ℕ) : (Q q : K[X] ⟶ K[X]) ≫ Q q = Q q := by |
ext n
exact Q_f_idem q n
|
/-
Copyright (c) 2019 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.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6e... | Mathlib/Data/Rat/Cast/Lemmas.lean | 55 | 57 | theorem cast_ofScientific {K} [DivisionRing K] (m : ℕ) (s : Bool) (e : ℕ) :
(OfScientific.ofScientific m s e : ℚ) = (OfScientific.ofScientific m s e : K) := by |
rw [← NNRat.cast_ofScientific (K := K), ← NNRat.cast_ofScientific, cast_nnratCast]
|
/-
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.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 1,148 | 1,222 | theorem preimage_connectedComponent_connected [TopologicalSpace β] {f : α → β}
(connected_fibers : ∀ t : β, IsConnected (f ⁻¹' {t}))
(hcl : ∀ T : Set β, IsClosed T ↔ IsClosed (f ⁻¹' T)) (t : β) :
IsConnected (f ⁻¹' connectedComponent t) := by |
-- The following proof is essentially https://stacks.math.columbia.edu/tag/0377
-- although the statement is slightly different
have hf : Surjective f := Surjective.of_comp fun t : β => (connected_fibers t).1
refine ⟨Nonempty.preimage connectedComponent_nonempty hf, ?_⟩
have hT : IsClosed (f ⁻¹' connectedCom... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 478 | 480 | theorem finrank_eq_rank [Module.Finite R M] :
↑(FiniteDimensional.finrank R M) = Module.rank R M := by |
rw [FiniteDimensional.finrank, cast_toNat_of_lt_aleph0 (rank_lt_aleph0 R M)]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 117 | 123 | theorem normed_le_div_measure_closedBall_rIn (x : E) :
f.normed μ x ≤ 1 / (μ (closedBall c f.rIn)).toReal := by |
rw [normed_def]
gcongr
· exact ENNReal.toReal_pos (measure_closedBall_pos _ _ f.rIn_pos).ne' measure_closedBall_lt_top.ne
· exact f.le_one
· exact f.measure_closedBall_le_integral μ
|
/-
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.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 155 | 159 | theorem toBaseChange_comp_ofBaseChange (Q : QuadraticForm R V) :
(toBaseChange A Q).comp (ofBaseChange A Q) = AlgHom.id _ _ := by |
ext v
change toBaseChange A Q (ofBaseChange A Q (1 ⊗ₜ[R] ι Q v)) = 1 ⊗ₜ[R] ι Q v
rw [ofBaseChange_tmul_ι, toBaseChange_ι]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,510 | 1,542 | theorem succ_ok {q s n} {c d : List Γ'} :
Reaches₁ (TM2.step tr) ⟨some (Λ'.succ q), s, K'.elim (trList [n]) [] c d⟩
⟨some q, none, K'.elim (trList [n.succ]) [] c d⟩ := by |
simp only [TM2.step, trList, trNat.eq_1, Nat.cast_succ, Num.add_one]
cases' (n : Num) with a
· refine TransGen.head rfl ?_
simp only [Option.mem_def, TM2.stepAux, elim_main, decide_False, elim_update_main, ne_eq,
Function.update_noteq, elim_rev, elim_update_rev, decide_True, Function.update_same,
... |
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 1,316 | 1,317 | theorem finTwoProd_zero (R : Type*) [Semiring R] : Basis.finTwoProd R 0 = (1, 0) := by |
simp [Basis.finTwoProd, LinearEquiv.finTwoArrow]
|
/-
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 | 705 | 746 | theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
(hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) :
Tendsto f atTop (𝓝 (limUnder atTop f)) := by |
suffices ∃ a, Tendsto f atTop (𝓝 a) from tendsto_nhds_limUnder this
suffices CauchySeq f from cauchySeq_tendsto_of_complete this
apply Metric.cauchySeq_iff'.2 (fun ε εpos ↦ ?_)
have A : ∀ᶠ (n : ℕ) in atTop, ∫ (x : ℝ) in Ici ↑n, ‖f' x‖ < ε := by
have L : Tendsto (fun (n : ℕ) ↦ ∫ x in Ici (n : ℝ), ‖f' x‖) a... |
/-
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.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/... | Mathlib/Probability/IdentDistrib.lean | 311 | 312 | theorem variance_eq {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) :
variance f μ = variance g ν := by | rw [variance, h.evariance_eq]; rfl
|
/-
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.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,173 | 1,175 | theorem mem_idealRange (x : L) : f x ∈ idealRange f := by |
rw [idealRange_eq_map]
exact LieIdeal.mem_map (LieSubmodule.mem_top x)
|
/-
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.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
... | Mathlib/GroupTheory/Perm/Option.lean | 76 | 77 | theorem Equiv.Perm.decomposeOption_symm_of_none_apply {α : Type*} [DecidableEq α] (e : Perm α)
(i : Option α) : Equiv.Perm.decomposeOption.symm (none, e) i = i.map e := by | simp
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 372 | 374 | theorem toIcoDiv_sub_eq_toIcoDiv_add' (a b c : α) :
toIcoDiv hp (a - c) b = toIcoDiv hp a (b + c) := by |
rw [← sub_neg_eq_add, toIcoDiv_sub_eq_toIcoDiv_add, sub_eq_add_neg]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 229 | 230 | theorem neg_normalize (n d z) : -normalize n d z = normalize (-n) d z := by |
simp [normalize]; rfl
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 103 | 105 | theorem pullbackIsoProdSubtype_inv_fst (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullbackIsoProdSubtype f g).inv ≫ pullback.fst = pullbackFst f g := by |
simp [pullbackCone, pullbackIsoProdSubtype]
|
/-
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.Logic.Equiv.TransferInstance
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb... | Mathlib/Algebra/Group/AddChar.lean | 310 | 312 | theorem pow_mulShift (ψ : AddChar R M) (n : ℕ) : ψ ^ n = mulShift ψ n := by |
ext x
rw [pow_apply, ← mulShift_spec']
|
/-
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.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/Multiset.lean | 71 | 79 | theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAd... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Yaël Dillies
-/
import Mathlib.Data.Real.NNReal
import Mathlib.Tactic.GCongr.Core
#align_import analysis.normed.group.sem... | Mathlib/Analysis/Normed/Group/Seminorm.lean | 148 | 150 | theorem map_sub_le_max : f (x - y) ≤ max (f x) (f y) := by |
rw [sub_eq_add_neg, ← NonarchAddGroupSeminormClass.map_neg_eq_map' f y]
exact map_add_le_max _ _ _
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | Mathlib/Analysis/MeanInequalities.lean | 358 | 384 | theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) :
∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) := by |
by_cases hF_zero : ∑ i ∈ s, f i ^ p = 0
· exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero
by_cases hG_zero : ∑ i ∈ s, g i ^ q = 0
· calc
∑ i ∈ s, f i * g i = ∑ i ∈ s, g i * f i := by
congr with i
rw [mul_comm]
_ ≤ (∑ i ∈ s, g i ^ q) ^ (1 / q) * (∑ i ∈ s, f i ^ p) ^ (1 / p... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import... | Mathlib/Topology/List.lean | 74 | 75 | theorem nhds_cons (a : α) (l : List α) : 𝓝 (a::l) = List.cons <$> 𝓝 a <*> 𝓝 l := by |
rw [nhds_list, List.traverse_cons _, ← nhds_list]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 399 | 413 | theorem equiv_fst_eq_iff_leftCosetEquivalence {g₁ g₂ : G} :
(hSK.equiv g₁).fst = (hSK.equiv g₂).fst ↔ LeftCosetEquivalence K g₁ g₂ := by |
rw [LeftCosetEquivalence, leftCoset_eq_iff]
constructor
· intro h
rw [← hSK.equiv_fst_mul_equiv_snd g₂, ← hSK.equiv_fst_mul_equiv_snd g₁, ← h,
mul_inv_rev, ← mul_assoc, inv_mul_cancel_right, ← coe_inv, ← coe_mul]
exact Subtype.property _
· intro h
apply (mem_leftTransversals_iff_existsUnique_... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,749 | 1,751 | theorem tendsto_Iic_atBot [SemilatticeInf α] {a : α} {f : β → Iic a} {l : Filter β} :
Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l atBot := by |
rw [atBot_Iic_eq, tendsto_comap_iff, Function.comp_def]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 458 | 463 | theorem Antiperiodic.add_zsmul_eq [AddGroup α] [AddGroup β] (h : Antiperiodic f c) (n : ℤ) :
f (x + n • c) = (n.negOnePow : ℤ) • f x := by |
rcases Int.even_or_odd' n with ⟨k, rfl | rfl⟩
· rw [h.even_zsmul_periodic, Int.negOnePow_two_mul, Units.val_one, one_zsmul]
· rw [h.odd_zsmul_antiperiodic, Int.negOnePow_two_mul_add_one, Units.val_neg,
Units.val_one, neg_zsmul, one_zsmul]
|
/-
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.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab... | Mathlib/Data/Complex/Determinant.lean | 24 | 26 | theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by |
rw [← LinearMap.det_toMatrix basisOneI, toMatrix_conjAe, Matrix.det_fin_two_of]
simp
|
/-
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 | 480 | 489 | theorem LinearMap.continuous_of_isClosed_graph (hg : IsClosed (g.graph : Set <| E × F)) :
Continuous g := by |
letI : CompleteSpace g.graph := completeSpace_coe_iff_isComplete.mpr hg.isComplete
let φ₀ : E →ₗ[𝕜] E × F := LinearMap.id.prod g
have : Function.LeftInverse Prod.fst φ₀ := fun x => rfl
let φ : E ≃ₗ[𝕜] g.graph :=
(LinearEquiv.ofLeftInverse this).trans (LinearEquiv.ofEq _ _ g.graph_eq_range_prod.symm)
le... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 147 | 153 | theorem prime_of_coprime (n : ℕ) (h1 : 1 < n) (h : ∀ m < n, m ≠ 0 → n.Coprime m) : Prime n := by |
refine prime_def_lt.mpr ⟨h1, fun m mlt mdvd => ?_⟩
have hm : m ≠ 0 := by
rintro rfl
rw [zero_dvd_iff] at mdvd
exact mlt.ne' mdvd
exact (h m mlt hm).symm.eq_one_of_dvd mdvd
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 155 | 157 | theorem exp_unop [T2Space 𝔸] (x : 𝔸ᵐᵒᵖ) :
exp 𝕂 (MulOpposite.unop x) = MulOpposite.unop (exp 𝕂 x) := by |
simp_rw [exp, expSeries_sum_eq, ← MulOpposite.unop_pow, ← MulOpposite.unop_smul, tsum_unop]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 650 | 658 | theorem natCast_eq_iff (p : ℕ) (n : ℕ) (z : ZMod p) [NeZero p] :
↑n = z ↔ ∃ k, n = z.val + p * k := by |
constructor
· rintro rfl
refine ⟨n / p, ?_⟩
rw [val_natCast, Nat.mod_add_div]
· rintro ⟨k, rfl⟩
rw [Nat.cast_add, natCast_zmod_val, Nat.cast_mul, natCast_self, zero_mul,
add_zero]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,079 | 1,080 | theorem iUnion_Ioc_zsmul : ⋃ n : ℤ, Ioc (n • p) ((n + 1) • p) = univ := by |
simpa only [zero_add] using iUnion_Ioc_add_zsmul hp 0
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
... | Mathlib/Algebra/AddTorsor.lean | 265 | 267 | theorem vadd_eq_vadd_iff_sub_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
v₁ +ᵥ p₁ = v₂ +ᵥ p₂ ↔ v₂ - v₁ = p₁ -ᵥ p₂ := by |
rw [vadd_eq_vadd_iff_neg_add_eq_vsub, neg_add_eq_sub]
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 531 | 533 | theorem rootSet_neg (p : T[X]) (S) [CommRing S] [IsDomain S] [Algebra T S] :
(-p).rootSet S = p.rootSet S := by |
rw [rootSet, aroots_neg, rootSet]
|
/-
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.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 200 | 201 | theorem filter_eq' (l : List α) (a : α) : l.filter (a = ·) = replicate (count a l) a := by |
simpa only [eq_comm] using filter_eq l a
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
/-!
# Some finiteness results of tensor product
This file contains some finiteness results of... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 52 | 60 | theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
|
/-
Copyright (c) 2022 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
import Mathlib.Topology.Separation
/-!
# Perfect Sets
In this file we define perfect subsets of a topological space, and prove some basic properties,
including a... | Mathlib/Topology/Perfect.lean | 132 | 144 | theorem preperfect_iff_perfect_closure [T1Space α] : Preperfect C ↔ Perfect (closure C) := by |
constructor <;> intro h
· exact h.perfect_closure
intro x xC
have H : AccPt x (𝓟 (closure C)) := h.acc _ (subset_closure xC)
rw [accPt_iff_frequently] at *
have : ∀ y, y ≠ x ∧ y ∈ closure C → ∃ᶠ z in 𝓝 y, z ≠ x ∧ z ∈ C := by
rintro y ⟨hyx, yC⟩
simp only [← mem_compl_singleton_iff, and_comm, ← fre... |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 1,128 | 1,130 | theorem model_nonemptyTheory_iff : M ⊨ L.nonemptyTheory ↔ Nonempty M := by |
simp only [nonemptyTheory, Theory.model_iff, Set.mem_singleton_iff, forall_eq,
Sentence.realize_cardGe, Nat.cast_one, one_le_iff_ne_zero, mk_ne_zero_iff]
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 286 | 289 | theorem IsUpperSet.image (hs : IsUpperSet s) (f : α ≃o β) : IsUpperSet (f '' s : Set β) := by |
change IsUpperSet ((f : α ≃ β) '' s)
rw [Set.image_equiv_eq_preimage_symm]
exact hs.preimage f.symm.monotone
|
/-
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.Topology.Gluing
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
#align_import algebraic... | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | 137 | 151 | theorem pullback_base (i j k : D.J) (S : Set (D.V (i, j)).carrier) :
(π₂ i, j, k) '' ((π₁ i, j, k) ⁻¹' S) = D.f i k ⁻¹' (D.f i j '' S) := by |
have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _
have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _
rw [← eq₁, ← eq₂]
-- Porting note: `rw` to `erw` on `coe_comp`
erw [coe_comp]
rw [Set.image_comp]
-- Porting note: `rw` to `erw` on `coe_comp`
... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 409 | 411 | theorem finrank_eq_nat_card_basis (h : Basis ι R M) :
finrank R M = Nat.card ι := by |
rw [Nat.card, ← toNat_lift.{v}, h.mk_eq_rank, toNat_lift, finrank]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.t... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 219 | 224 | theorem indicator_le_thickenedIndicator {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
(E.indicator fun _ => (1 : ℝ≥0)) ≤ thickenedIndicator δ_pos E := by |
intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicator_one δ_pos E h, le_refl]
· simp only [h, indicator_of_not_mem, not_false_iff, zero_le]
|
/-
Copyright (c) 2023 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.Lemmas
/-!
# `compute_degree` and `monicity`: tactics for explicit polynomials
This file defines two related tactics: `comp... | Mathlib/Tactic/ComputeDegree.lean | 101 | 103 | theorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}
(h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :
(f + g).coeff n = a + b := by | subst ‹_› ‹_›; apply coeff_add
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,097 | 2,101 | theorem isLittleO_norm_pow_id {n : ℕ} (h : 1 < n) :
(fun x : E' => ‖x‖ ^ n) =o[𝓝 0] fun x => x := by |
have := @isLittleO_norm_pow_norm_pow E' _ _ _ h
simp only [pow_one] at this
exact isLittleO_norm_right.mp this
|
/-
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, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... | Mathlib/RingTheory/Localization/Integral.lean | 223 | 239 | theorem isIntegral_localization [Algebra.IsIntegral R S] :
(map Sₘ (algebraMap R S)
(show _ ≤ (Algebra.algebraMapSubmonoid S M).comap _ from M.le_comap_map) :
Rₘ →+* _).IsIntegral := by |
intro x
obtain ⟨⟨s, ⟨u, hu⟩⟩, hx⟩ := surj (Algebra.algebraMapSubmonoid S M) x
obtain ⟨v, hv⟩ := hu
obtain ⟨v', hv'⟩ := isUnit_iff_exists_inv'.1 (map_units Rₘ ⟨v, hv.1⟩)
refine @IsIntegral.of_mul_unit Rₘ _ _ _ (localizationAlgebra M S) x (algebraMap S Sₘ u) v' ?_ ?_
· replace hv' := congr_arg (@algebraMap R... |
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
/-!
# Integration with respect to a finite product of measures
... | Mathlib/MeasureTheory/Integral/Pi.lean | 95 | 98 | theorem integral_fintype_prod_eq_pow {E : Type*} (ι : Type*) [Fintype ι] (f : E → 𝕜)
[MeasureSpace E] [SigmaFinite (volume : Measure E)] :
∫ x : ι → E, ∏ i, f (x i) = (∫ x, f x) ^ (card ι) := by |
rw [integral_fintype_prod_eq_prod, Finset.prod_const, card]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 270 | 272 | theorem uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} :
UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) := by |
simp only [UniformEquicontinuous, eventually_all, @forall_swap _ ι]; rfl
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 144 | 145 | theorem isOpen_B {K : Set (E →L[𝕜] F)} {r s ε : ℝ} : IsOpen (B f K r s ε) := by |
simp [B, isOpen_biUnion, IsOpen.inter, isOpen_A]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "lean... | Mathlib/GroupTheory/SemidirectProduct.lean | 228 | 228 | theorem lift_inl (n : N) : lift f₁ f₂ h (inl n) = f₁ n := by | simp [lift]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 547 | 547 | theorem inv_mul_lt_iff_lt_mul' : c⁻¹ * a < b ↔ a < b * c := by | rw [inv_mul_lt_iff_lt_mul, mul_comm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 347 | 349 | theorem guard_eq_some' {p : Prop} [Decidable p] (u) : _root_.guard p = some u ↔ p := by |
cases u
by_cases h : p <;> simp [_root_.guard, h]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 326 | 329 | theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : ℕ) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) := by |
rw [descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, ← C_eq_natCast,
eval_C_mul, Nat.cast_comm, ← mul_sub]
|
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b34... | Mathlib/Data/Seq/Seq.lean | 974 | 977 | theorem ret_bind (a : α) (f : α → Seq1 β) : bind (ret a) f = f a := by |
simp only [bind, map, ret.eq_1, map_nil]
cases' f a with a s
apply recOn s <;> intros <;> simp
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 487 | 493 | theorem realize_relabelEquiv {g : α ≃ β} {k} {φ : L.BoundedFormula α k} {v : β → M}
{xs : Fin k → M} : (relabelEquiv g φ).Realize v xs ↔ φ.Realize (v ∘ g) xs := by |
simp only [relabelEquiv, mapTermRelEquiv_apply, Equiv.coe_refl]
refine realize_mapTermRel_id (fun n t xs => ?_) fun _ _ _ => rfl
simp only [relabelEquiv_apply, Term.realize_relabel]
refine congr (congr rfl ?_) rfl
ext (i | i) <;> rfl
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 306 | 307 | theorem card_subtype (p : α → Prop) [DecidablePred p] (s : Finset α) :
(s.subtype p).card = (s.filter p).card := by | simp [Finset.subtype]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@... | Mathlib/Data/Real/Basic.lean | 322 | 322 | theorem mk_add {f g : CauSeq ℚ abs} : mk (f + g) = mk f + mk g := by | simp [mk, ← ofCauchy_add]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,307 | 1,311 | theorem prodComparison_inv_natural (f : A ⟶ A') (g : B ⟶ B') [IsIso (prodComparison F A B)]
[IsIso (prodComparison F A' B')] :
inv (prodComparison F A B) ≫ F.map (prod.map f g) =
prod.map (F.map f) (F.map g) ≫ inv (prodComparison F A' B') := by |
rw [IsIso.eq_comp_inv, Category.assoc, IsIso.inv_comp_eq, prodComparison_natural]
|
/-
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.Finsupp
import Mathlib.Data.Finsupp.Order
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finsupp.interval from "leanprover-comm... | Mathlib/Data/Finsupp/Interval.lean | 133 | 135 | theorem card_uIcc :
(uIcc f g).card = ∏ i ∈ f.support ∪ g.support, (uIcc (f i) (g i)).card := by |
rw [← support_inf_union_support_sup]; exact card_Icc (_ : ι →₀ α) _
|
/-
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 | 126 | 130 | theorem card_of_subtype {p : α → Prop} (s : Finset α) (H : ∀ x : α, x ∈ s ↔ p x)
[Fintype { x // p x }] : card { x // p x } = s.card := by |
rw [← subtype_card s H]
congr
apply Subsingleton.elim
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Sqrt
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed... | Mathlib/Data/Complex/Abs.lean | 323 | 324 | theorem lim_re (f : CauSeq ℂ Complex.abs) : lim (cauSeqRe f) = (lim f).re := by |
rw [lim_eq_lim_im_add_lim_re]; simp [ofReal']
|
/-
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.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_su... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 119 | 125 | theorem quadraticChar_odd_prime [DecidableEq F] (hF : ringChar F ≠ 2) {p : ℕ} [Fact p.Prime]
(hp₁ : p ≠ 2) (hp₂ : ringChar F ≠ p) :
quadraticChar F p = quadraticChar (ZMod p) (χ₄ (Fintype.card F) * Fintype.card F) := by |
rw [← quadraticChar_neg_one hF]
have h := quadraticChar_card_card hF (ne_of_eq_of_ne (ringChar_zmod_n p) hp₁)
(ne_of_eq_of_ne (ringChar_zmod_n p) hp₂.symm)
rwa [card p] at h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 313 | 315 | theorem single_eq_update [DecidableEq α] (a : α) (b : M) :
⇑(single a b) = Function.update (0 : _) a b := by |
classical rw [single_eq_set_indicator, ← Set.piecewise_eq_indicator, Set.piecewise_singleton]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | 300 | 301 | theorem isoPullback_inv_fst (h : IsPullback fst snd f g) [HasPullback f g] :
h.isoPullback.inv ≫ fst = pullback.fst := by | simp [Iso.inv_comp_eq]
|
/-
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 | 118 | 120 | theorem empty : μ ⊥ = 0 := by |
have := μ.sup_disjoint' ⊥ ⊥
simpa [apply_eq_coe_toFun] using this
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 773 | 789 | theorem Integrable.bdd_mul {F : Type*} [NormedDivisionRing F] {f g : α → F} (hint : Integrable g μ)
(hm : AEStronglyMeasurable f μ) (hfbdd : ∃ C, ∀ x, ‖f x‖ ≤ C) :
Integrable (fun x => f x * g x) μ := by |
cases' isEmpty_or_nonempty α with hα hα
· rw [μ.eq_zero_of_isEmpty]
exact integrable_zero_measure
· refine ⟨hm.mul hint.1, ?_⟩
obtain ⟨C, hC⟩ := hfbdd
have hCnonneg : 0 ≤ C := le_trans (norm_nonneg _) (hC hα.some)
have : (fun x => ‖f x * g x‖₊) ≤ fun x => ⟨C, hCnonneg⟩ * ‖g x‖₊ := by
intro ... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 290 | 292 | theorem fiber_card_ne_zero_iff_mem_image (s : Finset α) (f : α → β) [DecidableEq β] (y : β) :
(s.filter fun x => f x = y).card ≠ 0 ↔ y ∈ s.image f := by |
rw [← Nat.pos_iff_ne_zero, card_pos, fiber_nonempty_iff_mem_image]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
/-!
# Spaces where indicators of closed sets have decreasing approx... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 95 | 105 | theorem measure_of_cont_bdd_of_tendsto_indicator [OpensMeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E)
(fs : ℕ → Ω →ᵇ ℝ≥0) (fs_bdd : ∀ n ω, fs n ω ≤ c)
(fs_lim : Tendsto (fun n ω ↦ fs n ω) atTop (𝓝 (indicator E fun _ ↦ (1 : ℝ≥0)))) :
Tendsto (fun ... |
have fs_lim' :
∀ ω, Tendsto (fun n : ℕ ↦ (fs n ω : ℝ≥0)) atTop (𝓝 (indicator E (fun _ ↦ (1 : ℝ≥0)) ω)) := by
rw [tendsto_pi_nhds] at fs_lim
exact fun ω ↦ fs_lim ω
apply measure_of_cont_bdd_of_tendsto_filter_indicator μ E_mble fs
(eventually_of_forall fun n ↦ eventually_of_forall (fs_bdd n)) (event... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 787 | 793 | theorem Arrows.pullbackCompatible_iff (x : (i : I) → P.obj (op (X i))) :
Compatible P π x ↔ PullbackCompatible P π x := by |
refine ⟨fun t i j ↦ ?_, fun t i j Z gi gj comm ↦ ?_⟩
· apply t
exact pullback.condition
· rw [← pullback.lift_fst _ _ comm, op_comp, FunctorToTypes.map_comp_apply, t i j,
← FunctorToTypes.map_comp_apply, ← op_comp, pullback.lift_snd]
|
/-
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.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecf... | Mathlib/Data/PNat/Interval.lean | 94 | 99 | theorem card_Ioo : (Ioo a b).card = b - a - 1 := by |
rw [← Nat.card_Ioo]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Ioo _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
|
/-
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 | 1,062 | 1,064 | theorem one_lt_ncard (hs : s.Finite := by | toFinite_tac) :
1 < s.ncard ↔ ∃ a ∈ s, ∃ b ∈ s, a ≠ b := by
simp_rw [ncard_eq_toFinset_card _ hs, Finset.one_lt_card, Finite.mem_toFinset]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Data.Set.UnionLift
import Mathlib.Topology.Homeomorph
#align_import topology.continuous_function.basic from "leanprover-community/mathlib"@"55d771... | Mathlib/Topology/ContinuousFunction/Basic.lean | 466 | 468 | theorem liftCover_restrict {i : ι} : (liftCover S φ hφ hS).restrict (S i) = φ i := by |
ext
simp only [coe_restrict, Function.comp_apply, liftCover_coe]
|
/-
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.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 82 | 85 | theorem associator_hom_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).hom = (α_ X.V Y.V Z.V).hom := by |
dsimp
simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 201 | 203 | theorem pmap_map (g : γ → α) (x : Option γ) (H) :
pmap f (x.map g) H = pmap (fun a h ↦ f (g a) h) x fun a h ↦ H _ (mem_map_of_mem _ h) := by |
cases x <;> simp only [map_none', map_some', pmap]
|
/-
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 | 1,565 | 1,584 | theorem round_eq (x : α) : round x = ⌊x + 1 / 2⌋ := by |
simp_rw [round, (by simp only [lt_div_iff', two_pos] : 2 * fract x < 1 ↔ fract x < 1 / 2)]
cases' lt_or_le (fract x) (1 / 2) with hx hx
· conv_rhs => rw [← fract_add_floor x, add_assoc, add_left_comm, floor_int_add]
rw [if_pos hx, self_eq_add_right, floor_eq_iff, cast_zero, zero_add]
constructor
· li... |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.f... | Mathlib/Topology/FiberBundle/Trivialization.lean | 141 | 142 | theorem mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.baseSet := by |
rw [e.target_eq, prod_univ, mem_preimage]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import M... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 309 | 331 | theorem lastStep_nonempty :
{i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty := by |
by_contra h
suffices H : Function.Injective p.index from not_injective_of_ordinal p.index H
intro x y hxy
wlog x_le_y : x ≤ y generalizing x y
· exact (this hxy.symm (le_of_not_le x_le_y)).symm
rcases eq_or_lt_of_le x_le_y with (rfl | H); · rfl
simp only [nonempty_def, not_exists, exists_prop, not_and, n... |
/-
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 | 458 | 459 | theorem map_toPoly : (map φ P).toPoly = Polynomial.map φ P.toPoly := by |
simp only [map, toPoly, map_C, map_X, Polynomial.map_add, Polynomial.map_mul, Polynomial.map_pow]
|
/-
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 | 69 | 69 | theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by | simp [eigenspace]
|
/-
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 | 181 | 192 | theorem mul_one {n} (a : ⨂[R]^n M) : cast R M (add_zero _) (a ₜ* ₜ1) = a := by |
rw [gMul_def, gOne_def]
induction a using PiTensorProduct.induction_on with
| smul_tprod r a =>
rw [← TensorProduct.smul_tmul', LinearEquiv.map_smul, LinearEquiv.map_smul, ← gMul_def,
tprod_mul_tprod R a _, cast_tprod]
congr 2 with i
rw [Fin.append_elim0]
refine congr_arg a (Fin.ext ?_)
... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594b... | Mathlib/Algebra/Polynomial/BigOperators.lean | 92 | 111 | theorem coeff_list_prod_of_natDegree_le (l : List S[X]) (n : ℕ) (hl : ∀ p ∈ l, natDegree p ≤ n) :
coeff (List.prod l) (l.length * n) = (l.map fun p => coeff p n).prod := by |
induction' l with hd tl IH
· simp
· have hl' : ∀ p ∈ tl, natDegree p ≤ n := fun p hp => hl p (List.mem_cons_of_mem _ hp)
simp only [List.prod_cons, List.map, List.length]
rw [add_mul, one_mul, add_comm, ← IH hl', mul_comm tl.length]
have h : natDegree tl.prod ≤ n * tl.length := by
refine (natDe... |
/-
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.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 332 | 333 | theorem braiding_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).hom = (λ_ X).hom ≫ (ρ_ X).inv := by |
simp [← braiding_rightUnitor]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 614 | 616 | theorem range_add (a b : ℕ) : range (a + b) = range a ∪ (range b).map (addLeftEmbedding a) := by |
rw [← val_inj, union_val]
exact Multiset.range_add_eq_union a b
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 642 | 649 | theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
(hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
IntegrableOn (fun x => f x • g x) K μ := by |
rw [IntegrableOn, ← integrable_norm_iff]
· simp_rw [norm_smul]
refine IntegrableOn.continuousOn_mul ?_ hg.norm hK
exact continuous_norm.comp_continuousOn hf
· exact (hf.aestronglyMeasurable hK.measurableSet).smul hg.1
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 270 | 273 | theorem natSepDegree_le_natDegree : f.natSepDegree ≤ f.natDegree := by |
have := f.map (algebraMap F f.SplittingField) |>.card_roots'
rw [← aroots_def, natDegree_map] at this
exact (f.aroots f.SplittingField).toFinset_card_le.trans this
|
/-
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.Defs
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.LinearAlgebra.Tenso... | Mathlib/Algebra/Algebra/Bilinear.lean | 230 | 232 | theorem mulRight_one : mulRight R (1 : A) = LinearMap.id := by |
ext
simp
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 1,137 | 1,149 | theorem model_distinctConstantsTheory {M : Type w} [L[[α]].Structure M] (s : Set α) :
M ⊨ L.distinctConstantsTheory s ↔ Set.InjOn (fun i : α => (L.con i : M)) s := by |
simp only [distinctConstantsTheory, Theory.model_iff, Set.mem_image, Set.mem_inter,
Set.mem_prod, Set.mem_compl, Prod.exists, forall_exists_index, and_imp]
refine ⟨fun h a as b bs ab => ?_, ?_⟩
· contrapose! ab
have h' := h _ a b ⟨⟨as, bs⟩, ab⟩ rfl
simp only [Sentence.Realize, Formula.realize_not, Fo... |
/-
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.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 252 | 252 | theorem boundary_zero : c.boundary 0 = 0 := by | simp [boundary, Fin.ext_iff]
|
/-
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 | 57 | 59 | theorem opRingEquiv_op_C_mul_X_pow (r : R) (n : ℕ) :
opRingEquiv R (op (C r * X ^ n : R[X])) = C (op r) * X ^ n := by |
simp only [X_pow_mul, op_mul, op_pow, map_mul, map_pow, opRingEquiv_op_X, opRingEquiv_op_C]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 844 | 858 | theorem integral_sin_sq_mul_cos_sq :
∫ x in a..b, sin x ^ 2 * cos x ^ 2 = (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 := by |
convert integral_sin_pow_even_mul_cos_pow_even 1 1 using 1
have h1 : ∀ c : ℝ, (↑1 - c) / ↑2 * ((↑1 + c) / ↑2) = (↑1 - c ^ 2) / 4 := fun c => by ring
have h2 : Continuous fun x => cos (2 * x) ^ 2 := by continuity
have h3 : ∀ x, cos x * sin x = sin (2 * x) / 2 := by intro; rw [sin_two_mul]; ring
have h4 : ∀ d ... |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.denoms_clearable from "leanprover-community/mathlib"@"... | Mathlib/Algebra/Polynomial/DenomsClearable.lean | 90 | 107 | theorem one_le_pow_mul_abs_eval_div {K : Type*} [LinearOrderedField K] {f : ℤ[X]} {a b : ℤ}
(b0 : 0 < b) (fab : eval ((a : K) / b) (f.map (algebraMap ℤ K)) ≠ 0) :
(1 : K) ≤ (b : K) ^ f.natDegree * |eval ((a : K) / b) (f.map (algebraMap ℤ K))| := by |
obtain ⟨ev, bi, bu, hF⟩ :=
denomsClearable_natDegree (b := b) (algebraMap ℤ K) f a
(by
rw [eq_intCast, one_div_mul_cancel]
rw [Int.cast_ne_zero]
exact b0.ne.symm)
obtain Fa := _root_.congr_arg abs hF
rw [eq_one_div_of_mul_eq_one_left bu, eq_intCast, eq_intCast, abs_mul] at Fa
... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 131 | 134 | theorem removeZero_applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
(c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by |
ext v i
simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a"
/... | Mathlib/CategoryTheory/Sites/Canonical.lean | 125 | 150 | theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X)
(hR : Presieve.IsSheafFor P (R : Presieve X))
(hR' : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (_ : S f), Presieve.IsSeparatedFor P (R.pullback f : Presieve Y))
(hS : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (_ : R f), Presieve.IsSheafFor P (S.pullback f : Presieve Y)) :
Presieve.IsSheafFor... |
have : (bind R fun Y f _ => S.pullback f : Presieve X) ≤ S := by
rintro Z f ⟨W, f, g, hg, hf : S _, rfl⟩
apply hf
apply Presieve.isSheafFor_subsieve_aux P this
· apply isSheafFor_bind _ _ _ hR hS
intro Y f hf Z g
rw [← pullback_comp]
apply (hS (R.downward_closed hf _)).isSeparatedFor
· intr... |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 749 | 751 | theorem lift_comp_reindex (e : ι ≃ ι₂) (φ : MultilinearMap R (fun i ↦ s (e.symm i)) E) :
lift φ ∘ₗ (reindex R s e) = lift ((domDomCongrLinearEquiv' R R s _ e).symm φ) := by |
ext; simp [reindex]
|
/-
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.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-c... | Mathlib/ModelTheory/Definability.lean | 187 | 211 | theorem definable_iff_finitely_definable :
A.Definable L s ↔ ∃ (A0 : Finset M), (A0 : Set M) ⊆ A ∧
(A0 : Set M).Definable L s := by |
letI := Classical.decEq M
letI := Classical.decEq α
constructor
· simp only [definable_iff_exists_formula_sum]
rintro ⟨φ, rfl⟩
let A0 := (φ.freeVarFinset.preimage Sum.inl
(Function.Injective.injOn Sum.inl_injective)).image Subtype.val
have hA0 : (A0 : Set M) ⊆ A := by simp [A0]
refine ⟨A0... |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 290 | 293 | theorem normAtPlace_apply_isComplex {w : InfinitePlace K} (hw : IsComplex w) (x : E K) :
normAtPlace w x = ‖x.2 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
dif_neg (not_isReal_iff_isComplex.mpr hw)]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... | Mathlib/Data/Nat/Fib/Basic.lean | 121 | 124 | theorem fib_add_two_strictMono : StrictMono fun n => fib (n + 2) := by |
refine strictMono_nat_of_lt_succ fun n => ?_
rw [add_right_comm]
exact fib_lt_fib_succ (self_le_add_left _ _)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 448 | 452 | theorem norm_sub_le_of_geometric_bound_of_hasSum (hr : r < 1) (hf : ∀ n, ‖f n‖ ≤ C * r ^ n) {a : α}
(ha : HasSum f a) (n : ℕ) : ‖(∑ x ∈ Finset.range n, f x) - a‖ ≤ C * r ^ n / (1 - r) := by |
rw [← dist_eq_norm]
apply dist_le_of_le_geometric_of_tendsto r C hr (dist_partial_sum_le_of_le_geometric hf)
exact ha.tendsto_sum_nat
|
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
#align_import computability.reduce from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
/-!
#... | Mathlib/Computability/Reduce.lean | 131 | 136 | theorem computable_of_manyOneReducible {p : α → Prop} {q : β → Prop} (h₁ : p ≤₀ q)
(h₂ : ComputablePred q) : ComputablePred p := by |
rcases h₁ with ⟨f, c, hf⟩
rw [show p = fun a => q (f a) from Set.ext hf]
rcases computable_iff.1 h₂ with ⟨g, hg, rfl⟩
exact ⟨by infer_instance, by simpa using hg.comp c⟩
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Ba... | Mathlib/RingTheory/AlgebraicIndependent.lean | 103 | 106 | theorem algebraMap_injective : Injective (algebraMap R A) := by |
simpa [Function.comp] using
(Injective.of_comp_iff (algebraicIndependent_iff_injective_aeval.1 hx) MvPolynomial.C).2
(MvPolynomial.C_injective _ _)
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 71 | 85 | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by |
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, Bin... |
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