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) 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.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 812 | 814 | theorem some_zero [Zero M] : (0 : Option α →₀ M).some = 0 := by |
ext
simp
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprove... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 470 | 493 | theorem tendsto_normalize_testAgainstNN_of_tendsto {γ : Type*} {F : Filter γ}
{μs : γ → FiniteMeasure Ω} (μs_lim : Tendsto μs F (𝓝 μ)) (nonzero : μ ≠ 0) (f : Ω →ᵇ ℝ≥0) :
Tendsto (fun i => (μs i).normalize.toFiniteMeasure.testAgainstNN f) F
(𝓝 (μ.normalize.toFiniteMeasure.testAgainstNN f)) := by |
have lim_mass := μs_lim.mass
have aux : {(0 : ℝ≥0)}ᶜ ∈ 𝓝 μ.mass :=
isOpen_compl_singleton.mem_nhds (μ.mass_nonzero_iff.mpr nonzero)
have eventually_nonzero : ∀ᶠ i in F, μs i ≠ 0 := by
simp_rw [← mass_nonzero_iff]
exact lim_mass aux
have eve : ∀ᶠ i in F,
(μs i).normalize.toFiniteMeasure.testA... |
/-
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.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 362 | 366 | theorem convexBodySum_mem {x : K} :
mixedEmbedding K x ∈ (convexBodySum K B) ↔
∑ w : InfinitePlace K, (mult w) * w.val x ≤ B := by |
simp_rw [Set.mem_setOf_eq, convexBodySumFun, normAtPlace_apply]
rfl
|
/-
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 | 832 | 845 | theorem integral_hasFDerivWithinAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b)
{s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb]
(hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb)
(ha : Tendsto f (la ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (lb ⊓ ... |
rw [HasFDerivWithinAt, nhdsWithin_prod_eq]
have :=
integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae hf hmeas_a hmeas_b ha hb
(tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[s] a)) tendsto_fst
(tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[t] b)) tendst... |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 201 | 202 | theorem rightAdjointMate_id {X : C} [HasRightDual X] : (𝟙 X)ᘁ = 𝟙 (Xᘁ) := by |
simp [rightAdjointMate]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
/-!
# Type... | Mathlib/ModelTheory/Types.lean | 129 | 132 | theorem setOf_mem_eq_univ_iff (φ : L[[α]].Sentence) :
{ p : T.CompleteType α | φ ∈ p } = Set.univ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ φ := by |
rw [models_iff_not_satisfiable, ← compl_empty_iff, compl_setOf_mem, ← setOf_subset_eq_empty_iff]
simp
|
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 255 | 256 | theorem nim_equiv_iff_eq {o₁ o₂ : Ordinal} : (nim o₁ ≈ nim o₂) ↔ o₁ = o₂ := by |
rw [Impartial.equiv_iff_add_equiv_zero, nim_add_equiv_zero_iff]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 448 | 450 | theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by |
rw [← cosh_sq_sub_sinh_sq x]
ring
|
/-
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, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.VectorBundle.Basic
import Mathlib.Analysis.Convex.Normed
#align_import geometry.manifold.vector_bundle.tangent ... | Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean | 389 | 391 | theorem tangentBundle_model_space_coe_chartAt (p : TangentBundle I H) :
⇑(chartAt (ModelProd H E) p) = TotalSpace.toProd H E := by |
rw [← PartialHomeomorph.coe_coe, tangentBundle_model_space_chartAt]; rfl
|
/-
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.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 287 | 293 | theorem isOpen_setOf_nat_le_rank (n : ℕ) :
IsOpen { f : E →L[𝕜] F | ↑n ≤ (f : E →ₗ[𝕜] F).rank } := by |
simp only [LinearMap.le_rank_iff_exists_linearIndependent_finset, setOf_exists, ← exists_prop]
refine isOpen_biUnion fun t _ => ?_
have : Continuous fun f : E →L[𝕜] F => fun x : (t : Set E) => f x :=
continuous_pi fun x => (ContinuousLinearMap.apply 𝕜 F (x : E)).continuous
exact isOpen_setOf_linearIndepe... |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 393 | 397 | theorem Nontrivial.infsep_lt_iff {d} (hs : s.Nontrivial) :
s.infsep < d ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ dist x y < d := by |
rw [← not_iff_not]
push_neg
exact hs.le_infsep_iff
|
/-
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 | 1,361 | 1,362 | theorem lift_le_nat_iff {a : Cardinal.{u}} {n : ℕ} : lift.{v} a ≤ n ↔ a ≤ n := by |
rw [← lift_natCast.{v,u}, lift_le]
|
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Uthaiwat, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.I... | Mathlib/RingTheory/Polynomial/Nilpotent.lean | 174 | 182 | theorem not_isUnit_of_natDegree_pos_of_isReduced [IsReduced R] (p : R[X])
(hpl : 0 < p.natDegree) : ¬ IsUnit p := by |
simp only [ne_eq, isNilpotent_iff_eq_zero, not_and, not_forall, exists_prop,
Polynomial.isUnit_iff_coeff_isUnit_isNilpotent]
intro _
refine ⟨p.natDegree, hpl.ne', ?_⟩
contrapose! hpl
simp only [coeff_natDegree, leadingCoeff_eq_zero] at hpl
simp [hpl]
|
/-
Copyright (c) 2023 Kim Liesinger. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Liesinger
-/
import Mathlib.Algebra.Group.Defs
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two `List α`,
with a customizable ... | Mathlib/Data/List/EditDistance/Defs.lean | 241 | 245 | theorem suffixLevenshtein_cons₁_fst (x : α) (xs ys) :
(suffixLevenshtein C (x :: xs) ys).1 =
levenshtein C (x :: xs) ys ::
(suffixLevenshtein C xs ys).1 := by |
simp [suffixLevenshtein_cons₁]
|
/-
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.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from ... | Mathlib/Algebra/Order/Invertible.lean | 35 | 35 | theorem invOf_lt_zero [Invertible a] : ⅟ a < 0 ↔ a < 0 := by | simp only [← not_le, invOf_nonneg]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 282 | 283 | theorem EuclideanSpace.inner_single_right (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι) :
⟪v, EuclideanSpace.single i (a : 𝕜)⟫ = a * conj (v i) := by | simp [apply_ite conj, mul_comm]
|
/-
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.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.Unifo... | Mathlib/Topology/Instances/Real.lean | 244 | 248 | theorem tendsto_zmultiplesHom_cofinite {a : ℝ} (ha : a ≠ 0) :
Tendsto (zmultiplesHom ℝ a) cofinite (cocompact ℝ) := by |
apply (zmultiplesHom ℝ a).tendsto_coe_cofinite_of_discrete <| smul_left_injective ℤ ha
rw [AddSubgroup.range_zmultiplesHom]
infer_instance
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 683 | 689 | theorem modEq_mersenne (n k : ℕ) : k ≡ k / 2 ^ n + k % 2 ^ n [MOD 2 ^ n - 1] :=
-- See https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/help.20finding.20a.20lemma/near/177698446
calc
k = 2 ^ n * (k / 2 ^ n) + k % 2 ^ n := (Nat.div_add_mod k (2 ^ n)).symm
_ ≡ 1 * (k / 2 ^ n) + k % 2 ^... | rw [one_mul]
|
/-
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.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Categor... | Mathlib/AlgebraicGeometry/Properties.lean | 288 | 299 | theorem isIntegralOfOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[IsIntegral Y] [Nonempty X.carrier] : IsIntegral X := by |
constructor; · infer_instance
intro U hU
have : U = (Opens.map f.1.base).obj (H.base_open.isOpenMap.functor.obj U) := by
ext1; exact (Set.preimage_image_eq _ H.base_open.inj).symm
rw [this]
have : IsDomain (Y.presheaf.obj (op (H.base_open.isOpenMap.functor.obj U))) := by
apply (config := { allowSynth... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 1,539 | 1,542 | theorem mk_quaternionAlgebra : #(ℍ[R,c₁,c₂]) = #R ^ 4 := by |
rw [mk_congr (QuaternionAlgebra.equivProd c₁ c₂)]
simp only [mk_prod, lift_id]
ring
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_impor... | Mathlib/Data/Int/AbsoluteValue.lean | 28 | 29 | theorem AbsoluteValue.map_units_int (abv : AbsoluteValue ℤ S) (x : ℤˣ) : abv x = 1 := by |
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "lea... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 835 | 851 | theorem uniformIntegrable_of [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ i, AEStronglyMeasurable (f i) μ)
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, snorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UniformIntegrable f p μ := by |
set g : ι → α → β := fun i => (hf i).choose
have hgmeas : ∀ i, StronglyMeasurable (g i) := fun i => (Exists.choose_spec <| hf i).1
have hgeq : ∀ i, g i =ᵐ[μ] f i := fun i => (Exists.choose_spec <| hf i).2.symm
refine (uniformIntegrable_of' hp hp' hgmeas fun ε hε => ?_).ae_eq hgeq
obtain ⟨C, hC⟩ := h ε hε
r... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_im... | Mathlib/RingTheory/Coprime/Lemmas.lean | 245 | 248 | theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 196 | 203 | theorem affineIndependent_iff_le_finrank_vectorSpan [Fintype ι] (p : ι → P) {n : ℕ}
(hc : Fintype.card ι = n + 1) :
AffineIndependent k p ↔ n ≤ finrank k (vectorSpan k (Set.range p)) := by |
rw [affineIndependent_iff_finrank_vectorSpan_eq k p hc]
constructor
· rintro rfl
rfl
· exact fun hle => le_antisymm (finrank_vectorSpan_range_le k p hc) hle
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 92 | 95 | theorem HasDerivAt.smul (hc : HasDerivAt c c' x) (hf : HasDerivAt f f' x) :
HasDerivAt (fun y => c y • f y) (c x • f' + c' • f x) x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.smul hf
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 222 | 224 | theorem NonUnitalSubsemiring.toSubsemiring_toNonUnitalSubsemiring (S : NonUnitalSubsemiring R)
(h1 : (1 : R) ∈ S) : (NonUnitalSubsemiring.toSubsemiring S h1).toNonUnitalSubsemiring = S := by |
cases S; rfl
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 518 | 522 | theorem _root_.Continuous.path_trans {f : Y → Path x y} {g : Y → Path y z} :
Continuous f → Continuous g → Continuous fun t => (f t).trans (g t) := by |
intro hf hg
apply continuous_uncurry_iff.mp
exact trans_continuous_family _ (continuous_uncurry_iff.mpr hf) _ (continuous_uncurry_iff.mpr hg)
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 3,541 | 3,548 | theorem getLast_reverse {l : List α} (hl : l.reverse ≠ [])
(hl' : 0 < l.length := (by
contrapose! hl
simpa [length_eq_zero] using hl)) :
l.reverse.getLast hl = l.get ⟨0, hl'⟩ := by |
rw [getLast_eq_get, get_reverse']
· simp
· simpa using hl'
|
/-
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.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 818 | 824 | theorem Dense.diff_finset [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
(t : Finset X) : Dense (s \ t) := by |
induction t using Finset.induction_on with
| empty => simpa using hs
| insert _ ih =>
rw [Finset.coe_insert, ← union_singleton, ← diff_diff]
exact ih.diff_singleton _
|
/-
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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 115 | 116 | theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) := by |
rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 814 | 825 | theorem Submodule.orthogonal_orthogonal [HasOrthogonalProjection K] : Kᗮᗮ = K := by |
ext v
constructor
· obtain ⟨y, hy, z, hz, rfl⟩ := K.exists_add_mem_mem_orthogonal v
intro hv
have hz' : z = 0 := by
have hyz : ⟪z, y⟫ = 0 := by simp [hz y hy, inner_eq_zero_symm]
simpa [inner_add_right, hyz] using hv z hz
simp [hy, hz']
· intro hv w hw
rw [inner_eq_zero_symm]
ex... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 968 | 970 | theorem tan_pi_div_three : tan (π / 3) = sqrt 3 := by |
rw [tan_eq_sin_div_cos, sin_pi_div_three, cos_pi_div_three]
ring
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 225 | 228 | theorem aleph'_limit {o : Ordinal} (ho : o.IsLimit) : aleph' o = ⨆ a : Iio o, aleph' a := by |
refine le_antisymm ?_ (ciSup_le' fun i => aleph'_le.2 (le_of_lt i.2))
rw [aleph'_le_of_limit ho]
exact fun a ha => le_ciSup (bddAbove_of_small _) (⟨a, ha⟩ : Iio o)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.subobject.factor_thru from ... | Mathlib/CategoryTheory/Subobject/FactorThru.lean | 132 | 135 | theorem factorThru_mk_self (f : X ⟶ Y) [Mono f] :
(mk f).factorThru f (mk_factors_self f) = (underlyingIso f).inv := by |
ext
simp
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedb... | Mathlib/LinearAlgebra/Projection.lean | 233 | 234 | theorem ofIsCompl_left_apply (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} (u : p) :
ofIsCompl h φ ψ (u : E) = φ u := by | simp [ofIsCompl]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 511 | 511 | theorem cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x := by | simp [cos_add]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 609 | 610 | theorem uncurry_apply (f : M →ₗ[R] N →ₗ[R] P) (m : M) (n : N) :
uncurry R M N P f (m ⊗ₜ n) = f m n := by | rw [uncurry, LinearMap.flip_apply, lift.tmul]; rfl
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 119 | 130 | theorem pow_eq_one_of_norm_eq_one {x : K} (hxi : IsIntegral ℤ x) (hx : ∀ φ : K →+* A, ‖φ x‖ = 1) :
∃ (n : ℕ) (_ : 0 < n), x ^ n = 1 := by |
obtain ⟨a, -, b, -, habne, h⟩ :=
@Set.Infinite.exists_ne_map_eq_of_mapsTo _ _ _ _ (x ^ · : ℕ → K) Set.infinite_univ
(by exact fun a _ => ⟨hxi.pow a, fun φ => by simp [hx φ]⟩) (finite_of_norm_le K A (1 : ℝ))
wlog hlt : b < a
· exact this K A hxi hx b a habne.symm h.symm (habne.lt_or_lt.resolve_right hlt... |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 25 | 26 | theorem min_def (a b : α) : min a b = if a ≤ b then a else b := by |
rw [LinearOrder.min_def a]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 94 | 96 | theorem iUnion_left_iff [Countable ι] {s : ι → Set α} :
AEDisjoint μ (⋃ i, s i) t ↔ ∀ i, AEDisjoint μ (s i) t := by |
simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 148 | 152 | theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
[(ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N) :
generator (ϕ.submoduleImage N) ∣ ϕ ⟨x, hNO hx⟩ := by |
rw [← mem_iff_generator_dvd, LinearMap.mem_submoduleImage_of_le hNO]
exact ⟨x, hx, rfl⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 648 | 654 | theorem toWithTop_le {x y : PartENat} [hx : Decidable x.Dom] [hy : Decidable y.Dom] :
toWithTop x ≤ toWithTop y ↔ x ≤ y := by |
induction y using PartENat.casesOn generalizing hy
· simp
induction x using PartENat.casesOn generalizing hx
· simp
· simp -- Porting note: this takes too long.
|
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 116 | 119 | theorem toDualContinuousMultilinearMap_le_projectiveSeminorm (x : ⨂[𝕜] i, E i) :
‖toDualContinuousMultilinearMap F x‖ ≤ projectiveSeminorm x := by |
simp only [toDualContinuousMultilinearMap, LinearMap.coe_mk, AddHom.coe_mk]
apply LinearMap.mkContinuous_norm_le _ (apply_nonneg _ _)
|
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury G. Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepar... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 298 | 304 | theorem measurableSingletonClass_of_countablySeparated
[MeasurableSpace α] [CountablySeparated α] :
MeasurableSingletonClass α := by |
rcases measurable_injection_nat_bool_of_countablySeparated α with ⟨f, fmeas, finj⟩
refine ⟨fun x ↦ ?_⟩
rw [← finj.preimage_image {x}, image_singleton]
exact fmeas $ MeasurableSet.singleton _
|
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 334 | 334 | theorem re_T_smul : (T • z).re = z.re + 1 := by | simpa using re_T_zpow_smul z 1
|
/-
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.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 196 | 201 | theorem le_iff_exists_sup : a ≤ b ↔ ∃ c, b = a ⊔ c := by |
constructor
· intro h
exact ⟨b, (sup_eq_right.mpr h).symm⟩
· rintro ⟨c, rfl : _ = _ ⊔ _⟩
exact le_sup_left
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Support
#align_import algebra.monoid_algebra.degree from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 281 | 292 | theorem supDegree_prod_le {R A B : Type*} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B]
[SemilatticeSup B] [OrderBot B]
[CovariantClass B B (· + ·) (· ≤ ·)] [CovariantClass B B (Function.swap (· + ·)) (· ≤ ·)]
{D : A → B} (hzero : D 0 = 0) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
{ι} {s : Fins... |
refine s.induction ?_ ?_
· rw [Finset.prod_empty, Finset.sum_empty, one_def, supDegree_single]
split_ifs; exacts [bot_le, hzero.le]
· intro i s his ih
rw [Finset.prod_insert his, Finset.sum_insert his]
exact (supDegree_mul_le hadd).trans (add_le_add_left ih _)
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,293 | 1,303 | theorem mk_bounded_set_le (α : Type u) (c : Cardinal) :
#{ t : Set α // #t ≤ c } ≤ max #α ℵ₀ ^ c := by |
trans #{ t : Set (Sum (ULift.{u} ℕ) α) // #t ≤ c }
· refine ⟨Embedding.subtypeMap ?_ ?_⟩
· apply Embedding.image
use Sum.inr
apply Sum.inr.inj
intro s hs
exact mk_image_le.trans hs
apply (mk_bounded_set_le_of_infinite (Sum (ULift.{u} ℕ) α) c).trans
rw [max_comm, ← add_eq_max] <;> rfl
|
/-
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.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 912 | 913 | theorem ae_map_iff {p : β → Prop} {μ : Measure α} : (∀ᵐ x ∂μ.map f, p x) ↔ ∀ᵐ x ∂μ, p (f x) := by |
simp only [ae_iff, hf.map_apply, preimage_setOf_eq]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
/-!
## `norm_num` plugin for `^`.
-/
set_option autoImplicit true
namespac... | Mathlib/Tactic/NormNum/Pow.lean | 201 | 204 | theorem isNat_zpow_neg {α : Type*} [DivisionSemiring α] {a : α} {b : ℤ} {nb ne : ℕ}
(pb : IsInt b (Int.negOfNat nb)) (pe' : IsNat (a ^ nb)⁻¹ ne) :
IsNat (a ^ b) ne := by |
rwa [pb.out, Int.cast_negOfNat, zpow_neg, zpow_natCast]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 1,653 | 1,658 | theorem mem_sUnion {x y : Class.{u}} : y ∈ ⋃₀ x ↔ ∃ z, z ∈ x ∧ y ∈ z := by |
constructor
· rintro ⟨w, rfl, z, hzx, hwz⟩
exact ⟨z, hzx, coe_mem.2 hwz⟩
· rintro ⟨w, hwx, z, rfl, hwz⟩
exact ⟨z, rfl, w, hwx, hwz⟩
|
/-
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 | 2,458 | 2,459 | theorem toFinset_sum_count_eq (s : Multiset α) : ∑ a in s.toFinset, s.count a = card s := by |
simpa using (Finset.sum_multiset_map_count s (fun _ => (1 : ℕ))).symm
|
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.TensorProduct.Basic
/-! # Right-exactness properties of tensor product
## Modules
* `LinearMa... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 381 | 387 | theorem rTensor_exact : Exact (rTensor Q f) (rTensor Q g) := by |
rw [exact_iff, ← Submodule.ker_mkQ (p := range (rTensor Q f)),
← rTensor.inverse_comp_rTensor Q hfg hg]
apply symm
apply ker_comp_of_ker_eq_bot
rw [ker_eq_bot]
exact (rTensor.equiv Q hfg hg).symm.injective
|
/-
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.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.S... | Mathlib/AlgebraicGeometry/Scheme.lean | 399 | 406 | theorem Scheme.Spec_map_presheaf_map_eqToHom {X : Scheme} {U V : Opens X} (h : U = V) (W) :
(Scheme.Spec.map (X.presheaf.map (eqToHom h).op).op).val.c.app W =
eqToHom (by cases h; induction W using Opposite.rec'; dsimp; simp) := by |
have : Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 _ := by
rw [X.presheaf.map_id, op_id, Scheme.Spec.map_id]
cases h
refine (Scheme.congr_app this _).trans ?_
simp [eqToHom_map]
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce1... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 32 | 49 | theorem Basis.ext_multilinear_fin {f g : MultilinearMap R M M₂} {ι₁ : Fin n → Type*}
(e : ∀ i, Basis (ι₁ i) R (M i))
(h : ∀ v : ∀ i, ι₁ i, (f fun i => e i (v i)) = g fun i => e i (v i)) : f = g := by |
induction' n with m hm
· ext x
convert h finZeroElim
· apply Function.LeftInverse.injective uncurry_curryLeft
refine Basis.ext (e 0) ?_
intro i
apply hm (Fin.tail e)
intro j
convert h (Fin.cons i j)
iterate 2
rw [curryLeft_apply]
congr 1 with x
refine Fin.cases rfl (... |
/-
Copyright (c) 2018 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Module.Pi
#align_import data.holor from "leanprover-community/mathlib"@"509de852e1de5... | Mathlib/Data/Holor.lean | 156 | 157 | theorem mul_assoc [Semigroup α] (x : Holor α ds₁) (y : Holor α ds₂) (z : Holor α ds₃) :
HEq (mul (mul x y) z) (mul x (mul y z)) := by | simp [cast_heq, mul_assoc0, assocLeft]
|
/-
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.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 280 | 280 | theorem intCast_re (n : ℤ) : (n : ℤ√d).re = n := by | cases n <;> rfl
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprove... | Mathlib/Analysis/Complex/ReImTopology.lean | 94 | 95 | theorem interior_setOf_re_le (a : ℝ) : interior { z : ℂ | z.re ≤ a } = { z | z.re < a } := by |
simpa only [interior_Iic] using interior_preimage_re (Iic a)
|
/-
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.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 212 | 214 | theorem pairwise_disjoint_Ioc_zpow :
Pairwise (Disjoint on fun n : ℤ => Ioc (b ^ n) (b ^ (n + 1))) := by |
simpa only [one_mul] using pairwise_disjoint_Ioc_mul_zpow 1 b
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 176 | 180 | theorem fourier_norm [Fact (0 < T)] (n : ℤ) : ‖@fourier T n‖ = 1 := by |
rw [ContinuousMap.norm_eq_iSup_norm]
have : ∀ x : AddCircle T, ‖fourier n x‖ = 1 := fun x => abs_coe_circle _
simp_rw [this]
exact @ciSup_const _ _ _ Zero.instNonempty _
|
/-
Copyright (c) 2019 Michael Howes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Howes, Newell Jensen
-/
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.presented_group from "leanprover-communit... | Mathlib/GroupTheory/PresentedGroup.lean | 53 | 58 | theorem closure_range_of (rels : Set (FreeGroup α)) :
Subgroup.closure (Set.range (PresentedGroup.of : α → PresentedGroup rels)) = ⊤ := by |
have : (PresentedGroup.of : α → PresentedGroup rels) = QuotientGroup.mk' _ ∘ FreeGroup.of := rfl
rw [this, Set.range_comp, ← MonoidHom.map_closure (QuotientGroup.mk' _),
FreeGroup.closure_range_of, ← MonoidHom.range_eq_map]
exact MonoidHom.range_top_of_surjective _ (QuotientGroup.mk'_surjective _)
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 558 | 562 | theorem eq_of_le_of_finrank_le {S₁ S₂ : Submodule K V} [FiniteDimensional K S₂] (hle : S₁ ≤ S₂)
(hd : finrank K S₂ ≤ finrank K S₁) : S₁ = S₂ := by |
rw [← LinearEquiv.finrank_eq (Submodule.comapSubtypeEquivOfLe hle)] at hd
exact le_antisymm hle (Submodule.comap_subtype_eq_top.1
(eq_top_of_finrank_eq (le_antisymm (comap (Submodule.subtype S₂) S₁).finrank_le hd)))
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 399 | 409 | theorem absNorm_eq_zero_iff {I : Ideal S} : Ideal.absNorm I = 0 ↔ I = ⊥ := by |
constructor
· intro hI
rw [← le_bot_iff]
intros x hx
rw [mem_bot, ← Algebra.norm_eq_zero_iff (R := ℤ), ← Int.natAbs_eq_zero,
← Ideal.absNorm_span_singleton, ← zero_dvd_iff, ← hI]
apply Ideal.absNorm_dvd_absNorm_of_le
rwa [Ideal.span_singleton_le_iff_mem]
· rintro rfl
exact absNorm_b... |
/-
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 | 586 | 591 | theorem prime_of_degree_eq_one (hp1 : degree p = 1) : Prime p := by |
classical
have : Prime (normalize p) :=
Monic.prime_of_degree_eq_one (hp1 ▸ degree_normalize)
(monic_normalize fun hp0 => absurd hp1 (hp0.symm ▸ by simp [degree_zero]))
exact (normalize_associated _).prime this
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 655 | 656 | theorem mul_eq_left {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) (hb' : b ≠ 0) : a * b = a := by |
rw [mul_eq_max_of_aleph0_le_left ha hb', max_eq_left hb]
|
/-
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 | 499 | 502 | theorem ones_embedding (i : Fin (ones n).length) (h : 0 < (ones n).blocksFun i) :
(ones n).embedding i ⟨0, h⟩ = ⟨i, lt_of_lt_of_le i.2 (ones n).length_le⟩ := by |
ext
simpa using i.2.le
|
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 687 | 688 | theorem mul_pos_of_neg_of_neg {a b : α} (ha : a < 0) (hb : b < 0) : 0 < a * b := by |
simpa only [zero_mul] using mul_lt_mul_of_neg_right ha hb
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 821 | 824 | theorem HasBasis.forall_mem_mem (h : HasBasis l p s) {x : α} :
(∀ t ∈ l, x ∈ t) ↔ ∀ i, p i → x ∈ s i := by |
simp only [h.mem_iff, exists_imp, and_imp]
exact ⟨fun h i hi => h (s i) i hi Subset.rfl, fun h t i hi ht => ht (h i hi)⟩
|
/-
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.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathl... | Mathlib/Order/JordanHolder.lean | 448 | 459 | theorem eq_of_head_eq_head_of_last_eq_last_of_length_eq_zero {s₁ s₂ : CompositionSeries X}
(hb : s₁.head = s₂.head) (ht : s₁.last = s₂.last) (hs₁0 : s₁.length = 0) : s₁ = s₂ := by |
have : ∀ x, x ∈ s₁ ↔ x = s₁.last := fun x =>
⟨fun hx => subsingleton_of_length_eq_zero hs₁0 hx s₁.last_mem, fun hx => hx.symm ▸ s₁.last_mem⟩
have : ∀ x, x ∈ s₂ ↔ x = s₂.last := fun x =>
⟨fun hx =>
subsingleton_of_length_eq_zero
(length_eq_zero_of_head_eq_head_of_last_eq_last_of_length_eq_zer... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f1... | Mathlib/Data/List/NodupEquivFin.lean | 211 | 232 | theorem duplicate_iff_exists_distinct_get {l : List α} {x : α} :
l.Duplicate x ↔
∃ (n m : Fin l.length) (_ : n < m),
x = l.get n ∧ x = l.get m := by |
classical
rw [duplicate_iff_two_le_count, le_count_iff_replicate_sublist,
sublist_iff_exists_fin_orderEmbedding_get_eq]
constructor
· rintro ⟨f, hf⟩
refine ⟨f ⟨0, by simp⟩, f ⟨1, by simp⟩, f.lt_iff_lt.2 (Nat.zero_lt_one), ?_⟩
rw [← hf, ← hf]; simp
· rintro ⟨n, m, hnm, h, h'⟩
r... |
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 177 | 180 | theorem alternatingFaceMapComplex_obj_d (X : SimplicialObject C) (n : ℕ) :
((alternatingFaceMapComplex C).obj X).d (n + 1) n = AlternatingFaceMapComplex.objD X n := by |
dsimp only [alternatingFaceMapComplex, AlternatingFaceMapComplex.obj]
apply ChainComplex.of_d
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import linear_algebra.affine_space.midpoint_zero from "leanprover-... | Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean | 29 | 31 | theorem lineMap_one_half {R : Type*} {V P : Type*} [DivisionRing R] [CharZero R] [AddCommGroup V]
[Module R V] [AddTorsor V P] (a b : P) : lineMap a b (1 / 2 : R) = midpoint R a b := by |
rw [one_div, lineMap_inv_two]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 1,252 | 1,254 | theorem sup'_inf_distrib_right (f : ι → α) (a : α) :
s.sup' hs f ⊓ a = s.sup' hs fun i => f i ⊓ a := by |
rw [inf_comm, sup'_inf_distrib_left]; simp_rw [inf_comm]
|
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | Mathlib/Data/List/Sym.lean | 40 | 43 | theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs).sym2 ↔ z = s(x, x) ∨ (∃ y, y ∈ xs ∧ z = s(x, y)) ∨ z ∈ xs.sym2 := by |
simp only [List.sym2, map_cons, cons_append, mem_cons, mem_append, mem_map]
simp only [eq_comm]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import... | Mathlib/Algebra/Algebra/Operations.lean | 385 | 388 | theorem mem_mul_span_singleton {x y : A} : x ∈ P * span R {y} ↔ ∃ z ∈ P, z * y = x := by |
-- Porting note: need both `*` and `Mul.mul`
simp_rw [(· * ·), Mul.mul, map₂_span_singleton_eq_map_flip]
rfl
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 1,117 | 1,125 | theorem OrderIso.strictConcaveOn_symm (f : α ≃o β) (hf : StrictConvexOn 𝕜 univ f) :
StrictConcaveOn 𝕜 univ f.symm := by |
refine ⟨convex_univ, fun x _ y _ hxy a b ha hb hab => ?_⟩
obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩
obtain ⟨y', hy''⟩ := f.surjective.exists.mp ⟨y, rfl⟩
have hxy' : x' ≠ y' := by rw [← f.injective.ne_iff, ← hx'', ← hy'']; exact hxy
simp only [hx'', hy'', OrderIso.symm_apply_apply, gt_iff_lt]
rw ... |
/-
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 | 402 | 406 | theorem finprod_eq_finset_prod_of_mulSupport_subset (f : α → M) {s : Finset α}
(h : mulSupport f ⊆ (s : Set α)) : ∏ᶠ i, f i = ∏ i ∈ s, f i :=
haveI h' : (s.finite_toSet.subset h).toFinset ⊆ s := by |
simpa [← Finset.coe_subset, Set.coe_toFinset]
finprod_eq_prod_of_mulSupport_toFinset_subset _ _ h'
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Nat.SuccPred
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Algebra.Order.Ring.WithTop
#align_i... | Mathlib/Data/ENat/Basic.lean | 217 | 221 | theorem toNat_sub {n : ℕ∞} (hn : n ≠ ⊤) (m : ℕ∞) : toNat (m - n) = toNat m - toNat n := by |
lift n to ℕ using hn
induction m
· rw [top_sub_coe, toNat_top, zero_tsub]
· rw [← coe_sub, toNat_coe, toNat_coe, toNat_coe]
|
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Computability.Language
import Mathlib.Tactic.NormNum
#align_import computability.DFA from "leanprover-community/mathlib"@"3... | Mathlib/Computability/DFA.lean | 151 | 166 | theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
(hlen : Fintype.card σ ≤ List.length x) :
∃ a b c,
x = a ++ b ++ c ∧
a.length + b.length ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts := by |
obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl
use a, b, c, hx, hlen, hnil
intro y hy
rw [Language.mem_mul] at hy
rcases hy with ⟨ab, hab, c', hc', rfl⟩
rw [Language.mem_mul] at hab
rcases hab with ⟨a', ha', b', hb', rfl⟩
rw [Set.mem_singleton_iff] at ha' h... |
/-
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.Data.Nat.ModEq
import Mathlib.Tactic.Abel
import Mathlib.Tactic.GCongr.Core
#align_import data.int.modeq from "leanprover-community/mathlib"@"47a1a73351de... | Mathlib/Data/Int/ModEq.lean | 286 | 287 | theorem modEq_sub_fac {a b n : ℤ} (c : ℤ) (ha : a ≡ b [ZMOD n]) : a - n * c ≡ b [ZMOD n] := by |
convert Int.modEq_add_fac (-c) ha using 1; rw [mul_neg, sub_eq_add_neg]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Anal... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 762 | 765 | theorem self_le_sinh_iff : x ≤ sinh x ↔ 0 ≤ x :=
calc
x ≤ sinh x ↔ sinh 0 - 0 ≤ sinh x - x := by | simp
_ ↔ 0 ≤ x := sinh_sub_id_strictMono.le_iff_le
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 78 | 82 | theorem measurable_map (f : α → β) (hf : Measurable f) :
Measurable fun μ : Measure α => map f μ := by |
refine measurable_of_measurable_coe _ fun s hs => ?_
simp_rw [map_apply hf hs]
exact measurable_coe (hf hs)
|
/-
Copyright (c) 2020 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ruben Van de Velde
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import anal... | Mathlib/Analysis/NormedSpace/Extend.lean | 88 | 90 | theorem extendTo𝕜'_apply_re (fr : F →ₗ[ℝ] ℝ) (x : F) : re (fr.extendTo𝕜' x : 𝕜) = fr x := by |
simp only [extendTo𝕜'_apply, map_sub, zero_mul, mul_zero, sub_zero,
rclike_simps]
|
/-
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.QuadraticForm.IsometryEquiv
/-!
# Linear equivalences of tensor products as isometrie... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 37 | 46 | theorem tmul_comp_tensorMap
{Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}
{Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}
(f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :
(Q₂.tmul Q₄).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Q₁.tmul Q₃ := by |
have h₁ : Q₁ = Q₂.comp f.toLinearMap := QuadraticForm.ext fun x => (f.map_app x).symm
have h₃ : Q₃ = Q₄.comp g.toLinearMap := QuadraticForm.ext fun x => (g.map_app x).symm
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₁ m₃ m₁' m₃'
simp [-associated_apply, h₁, h₃, associated_tmul]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 756 | 764 | theorem hom_left_act_hom' :
(P.tensorBimod (regular S)).actLeft ≫ hom P = (R.X ◁ hom P) ≫ P.actLeft := by |
dsimp; dsimp [hom, TensorBimod.actLeft, regular]
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_
dsimp
slice_lhs 1 4 => rw [id_tensor_π_preserves_coequalizer_inv_colimMap_desc]
slice_lhs 2 3 => rw [middle_assoc]
slice_rhs 1 2 => rw [← MonoidalCategory.whiskerLeft_comp, coequalizer.π_desc]... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 559 | 565 | theorem finite_factors (I : FractionalIdeal R⁰ K) :
∀ᶠ v : HeightOneSpectrum R in Filter.cofinite, count K v I = 0 := by |
by_cases hI : I = 0
· simp only [hI, count_zero, Filter.eventually_cofinite, not_true_eq_false, setOf_false,
finite_empty]
· convert finite_factors' hI (choose_spec (choose_spec (exists_eq_spanSingleton_mul I))).2
rw [count_ne_zero K _ hI]
|
/-
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.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/m... | Mathlib/Topology/Order/LeftRightLim.lean | 186 | 188 | theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x) := by |
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within f (hf.tendsto_leftLim x)
filter_upwards [@self_mem_nhdsWithin _ _ x (Iio x)] with y hy using hf.le_leftLim hy
|
/-
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.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... | Mathlib/RingTheory/LocalProperties.lean | 200 | 209 | theorem RingHom.PropertyIsLocal.ofLocalizationSpan (hP : RingHom.PropertyIsLocal @P) :
RingHom.OfLocalizationSpan @P := by |
introv R hs hs'
apply_fun Ideal.map f at hs
rw [Ideal.map_span, Ideal.map_top] at hs
apply hP.OfLocalizationSpanTarget _ _ hs
rintro ⟨_, r, hr, rfl⟩
convert hP.StableUnderComposition
_ _ (hP.HoldsForLocalizationAway (Localization.Away r) r) (hs' ⟨r, hr⟩) using 1
exact (IsLocalization.map_comp _).symm... |
/-
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.RBMap.Basic
import Batteries.Tactic.SeqFocus
/-!
# Lemmas for Red-black trees
The main theorem in this file is `WF_def`, which shows that the ... | .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 218 | 220 | theorem balance1_eq {l : RBNode α} {v : α} {r : RBNode α}
(hl : l.Balanced c n) : balance1 l v r = node black l v r := by |
unfold balance1; split <;> first | rfl | nomatch hl
|
/-
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.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | Mathlib/Analysis/MeanInequalitiesPow.lean | 196 | 200 | theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
(a ^ p + b ^ p) ^ (1 / p) ≤ a + b := by |
rw [← @NNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]
rw [one_div_one_div]
exact add_rpow_le_rpow_add _ _ hp1
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 247 | 261 | theorem unit_mem_mul_iff_mem_swap_mul {a b : A} {r : Rˣ} : ↑r ∈ σ (a * b) ↔ ↑r ∈ σ (b * a) := by |
have h₁ : ∀ x y : A, IsUnit (1 - x * y) → IsUnit (1 - y * x) := by
refine fun x y h => ⟨⟨1 - y * x, 1 + y * h.unit.inv * x, ?_, ?_⟩, rfl⟩
· calc
(1 - y * x) * (1 + y * (IsUnit.unit h).inv * x) =
1 - y * x + y * ((1 - x * y) * h.unit.inv) * x := by noncomm_ring
_ = 1 := by simp onl... |
/-
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 | 335 | 335 | theorem ceil_zero : ⌈(0 : α)⌉₊ = 0 := by | rw [← Nat.cast_zero, ceil_natCast]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology... | Mathlib/Topology/TietzeExtension.lean | 73 | 77 | theorem ContinuousMap.exists_extension (f : C(X₁, Y)) :
∃ (g : C(X, Y)), g.comp ⟨e, he.continuous⟩ = f := by |
let e' : X₁ ≃ₜ Set.range e := Homeomorph.ofEmbedding _ he.toEmbedding
obtain ⟨g, hg⟩ := (f.comp e'.symm).exists_restrict_eq he.isClosed_range
exact ⟨g, by ext x; simpa using congr($(hg) ⟨e' x, x, rfl⟩)⟩
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Extensions of Groups
This file defines the HNN extension of a group `G`, `HNN... | Mathlib/GroupTheory/HNNExtension.lean | 225 | 227 | theorem ext {w w' : NormalWord d}
(h1 : w.head = w'.head) (h2 : w.toList = w'.toList): w = w' := by |
rcases w with ⟨⟨⟩, _⟩; cases w'; simp_all
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 175 | 184 | theorem sin_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖ := by |
by_cases h0 : x = 0 ∧ y = 0; · simp [h0]
rw [not_and_or] at h0
rw [sin_angle_add_of_inner_eq_zero h h0, div_mul_cancel₀]
rw [← mul_self_ne_zero, norm_add_sq_eq_norm_sq_add_norm_sq_real h]
refine (ne_of_lt ?_).symm
rcases h0 with (h0 | h0)
· exact Left.add_pos_of_pos_of_nonneg (mul_self_pos.2 (norm_ne_zer... |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "lean... | Mathlib/Algebra/CharZero/Lemmas.lean | 166 | 168 | theorem one_eq_bit1 {a : R} : 1 = bit1 a ↔ a = 0 := by |
rw [eq_comm]
exact bit1_eq_one
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 969 | 970 | theorem disjoint_iff_ne : Disjoint s t ↔ ∀ a ∈ s, ∀ b ∈ t, a ≠ b := by |
simp only [disjoint_left, imp_not_comm, forall_eq']
|
/-
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.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Categor... | Mathlib/AlgebraicGeometry/Properties.lean | 260 | 279 | theorem isIntegralOfIsIrreducibleIsReduced [IsReduced X] [H : IrreducibleSpace X.carrier] :
IsIntegral X := by |
constructor; · infer_instance
intro U hU
haveI := (@LocallyRingedSpace.component_nontrivial X.toLocallyRingedSpace U hU).1
have : NoZeroDivisors
(X.toLocallyRingedSpace.toSheafedSpace.toPresheafedSpace.presheaf.obj (op U)) := by
refine ⟨fun {a b} e => ?_⟩
simp_rw [← basicOpen_eq_bot_iff, ← Opens.... |
/-
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 | 245 | 246 | theorem head_eq_destruct (s : Seq α) : head.{u} s = Prod.fst.{u} <$> destruct.{u} s := by |
unfold destruct head; cases get? s 0 <;> rfl
|
/-
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 | 81 | 81 | theorem one_add (n : PosNum) : 1 + n = succ n := by | cases n <;> rfl
|
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