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) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 102 | 103 | theorem isBlock_empty : IsBlock G (⊥ : Set X) := by |
simp [IsBlock.def, Set.bot_eq_empty, Set.smul_set_empty]
|
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.D... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 115 | 117 | theorem vanishingIdeal_singleton (x : ProjectiveSpectrum 𝒜) :
vanishingIdeal ({x} : Set (ProjectiveSpectrum 𝒜)) = x.asHomogeneousIdeal := by |
simp [vanishingIdeal]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 329 | 332 | theorem IsMaximal.coprime_of_ne {M M' : Ideal α} (hM : M.IsMaximal) (hM' : M'.IsMaximal)
(hne : M ≠ M') : M ⊔ M' = ⊤ := by |
contrapose! hne with h
exact hM.eq_of_le hM'.ne_top (le_sup_left.trans_eq (hM'.eq_of_le h le_sup_right).symm)
|
/-
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 | 713 | 716 | theorem filterMap_mono (h : s ⊆ t) :
filterMap f s f_inj ⊆ filterMap f t f_inj := by |
rw [← val_le_iff] at h ⊢
exact Multiset.filterMap_le_filterMap f h
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 905 | 907 | theorem nontrivial_iff_exists_ne_one (H : Subgroup G) : Nontrivial H ↔ ∃ x ∈ H, x ≠ (1 : G) := by |
rw [Subtype.nontrivial_iff_exists_ne (fun x => x ∈ H) (1 : H)]
simp
|
/-
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
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,585 | 2,587 | theorem comap_surjective_eq_bot {f : Filter β} {m : α → β} (hm : Surjective m) :
comap m f = ⊥ ↔ f = ⊥ := by |
rw [comap_eq_bot_iff_compl_range, hm.range_eq, compl_univ, empty_mem_iff_bot]
|
/-
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 | 912 | 914 | theorem coprod.map_comp_id {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasBinaryCoproduct Z W]
[HasBinaryCoproduct Y W] [HasBinaryCoproduct X W] :
coprod.map (f ≫ g) (𝟙 W) = coprod.map f (𝟙 W) ≫ coprod.map g (𝟙 W) := by | simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
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 | 509 | 509 | theorem norm_zero : norm (0 : ℤ√d) = 0 := by | simp [norm]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 607 | 615 | theorem IsScalarTower.of_mclosure_eq_top {N α} [Monoid M] [MulAction M N] [SMul N α] [MulAction M α]
{s : Set M} (htop : Submonoid.closure s = ⊤)
(hs : ∀ x ∈ s, ∀ (y : N) (z : α), (x • y) • z = x • y • z) : IsScalarTower M N α := by |
refine ⟨fun x => Submonoid.induction_of_closure_eq_top_left htop x ?_ ?_⟩
· intro y z
rw [one_smul, one_smul]
· clear x
intro x hx x' hx' y z
rw [mul_smul, mul_smul, hs x hx, hx']
|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algeb... | Mathlib/Algebra/Module/PID.lean | 75 | 84 | theorem Submodule.isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (⊤ : Submodule R M).annihilator).toFinset =>
torsionBy R M
(IsPrincipal.generator (p : Ideal R) ^
(factors (⊤ : Submodule R M).annihilator).coun... |
convert isInternal_prime_power_torsion hM
ext p : 1
rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ← Ideal.span_singleton_pow,
Ideal.span_singleton_generator]
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib... | Mathlib/RingTheory/Polynomial/Vieta.lean | 75 | 77 | theorem prod_X_add_C_coeff' {σ} (s : Multiset σ) (r : σ → R) {k : ℕ} (h : k ≤ Multiset.card s) :
(s.map fun i => X + C (r i)).prod.coeff k = (s.map r).esymm (Multiset.card s - k) := by |
erw [← map_map (fun r => X + C r) r, prod_X_add_C_coeff] <;> rw [s.card_map r]; assumption
|
/-
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 | 1,254 | 1,257 | theorem val_eq_ite_valMinAbs {n : ℕ} [NeZero n] (a : ZMod n) :
(a.val : ℤ) = a.valMinAbs + if a.val ≤ n / 2 then 0 else n := by |
rw [valMinAbs_def_pos]
split_ifs <;> simp [add_zero, sub_add_cancel]
|
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/... | Mathlib/Analysis/SumIntegralComparisons.lean | 150 | 153 | theorem MonotoneOn.sum_le_integral (hf : MonotoneOn f (Icc x₀ (x₀ + a))) :
(∑ i ∈ Finset.range a, f (x₀ + i)) ≤ ∫ x in x₀..x₀ + a, f x := by |
rw [← neg_le_neg_iff, ← Finset.sum_neg_distrib, ← intervalIntegral.integral_neg]
exact hf.neg.integral_le_sum
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Batteries.Classes.Order
import Batteries.Control.ForInStep.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.Alias
/-!... | .lake/packages/batteries/Batteries/Data/RBMap/Basic.lean | 177 | 178 | theorem all_iff {t : RBNode α} : t.all p ↔ t.All (p ·) := by |
induction t <;> simp [*, all, All, and_assoc]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 309 | 315 | theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞)
(hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r := by |
refine le_antisymm (le_pre.2 fun s hs => ?_) (le_trim _)
rw [trim_eq_iInf]
refine iInf_le_of_le (closure s) <| iInf_le_of_le subset_closure <|
iInf_le_of_le measurableSet_closure ((pre_le ?_).trans_eq (hcl _))
rwa [diam_closure]
|
/-
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.Group.Even
import Mathlib.Algebra.Order.Group.Lattice
import Mathlib.Algebra.Order.Monoid.Unbu... | Mathlib/Algebra/Order/Group/Abs.lean | 429 | 430 | theorem abs_sub_le_iff : |a - b| ≤ c ↔ a - b ≤ c ∧ b - a ≤ c := by |
rw [abs_le, neg_le_sub_iff_le_add, sub_le_iff_le_add', and_comm, sub_le_iff_le_add']
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 403 | 417 | theorem digits_lt_base' {b m : ℕ} : ∀ {d}, d ∈ digits (b + 2) m → d < b + 2 := by |
apply Nat.strongInductionOn m
intro n IH d hd
cases' n with n
· rw [digits_zero] at hd
cases hd
-- base b+2 expansion of 0 has no digits
rw [digits_add_two_add_one] at hd
cases hd
· exact n.succ.mod_lt (by simp)
-- Porting note: Previous code (single line) contained linarith.
-- . exact IH _ (N... |
/-
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.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 595 | 596 | theorem iterate_prod_map (f : α → α) (g : β → β) (n : ℕ) :
(Prod.map f g)^[n] = Prod.map (f^[n]) (g^[n]) := by | induction n <;> simp [*, Prod.map_comp_map]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,330 | 3,334 | theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : G₁ →* G₃)
(hg : f.ker ≤ g.ker) (h : G₂ →* G₃) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ := by |
simp_rw [← hh]
exact ((f.liftOfRightInverse f_inv hf).apply_symm_apply _).symm
|
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.GroupTheory.PresentedGroup
import Mathlib.GroupTheory.Coxeter.Matrix
/-!
# Coxeter groups and Coxeter syst... | Mathlib/GroupTheory/Coxeter/Basic.lean | 110 | 119 | theorem reindex_relationsSet :
(M.reindex e).relationsSet =
FreeGroup.freeGroupCongr e '' M.relationsSet := let M' := M.reindex e; calc
Set.range (uncurry M'.relation)
_ = Set.range (uncurry M'.relation ∘ Prod.map e e) := by | simp [Set.range_comp]
_ = Set.range (FreeGroup.freeGroupCongr e ∘ uncurry M.relation) := by
apply congrArg Set.range
ext ⟨i, i'⟩
simp [relation, reindex_apply, M']
_ = _ := by simp [Set.range_comp, relationsSet]
|
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 352 | 361 | theorem ideal_eq_span_pow_irreducible {s : Ideal R} (hs : s ≠ ⊥) {ϖ : R} (hirr : Irreducible ϖ) :
∃ n : ℕ, s = Ideal.span {ϖ ^ n} := by |
have gen_ne_zero : generator s ≠ 0 := by
rw [Ne, ← eq_bot_iff_generator_eq_zero]
assumption
rcases associated_pow_irreducible gen_ne_zero hirr with ⟨n, u, hnu⟩
use n
have : span _ = _ := Ideal.span_singleton_generator s
rw [← this, ← hnu, span_singleton_eq_span_singleton]
use u
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.le... | Mathlib/Data/Int/Lemmas.lean | 55 | 57 | theorem natAbs_le_iff_sq_le {a b : ℤ} : a.natAbs ≤ b.natAbs ↔ a ^ 2 ≤ b ^ 2 := by |
rw [sq, sq]
exact natAbs_le_iff_mul_self_le
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 543 | 559 | theorem is_basis_iff_det {v : ι → M} :
LinearIndependent R v ∧ span R (Set.range v) = ⊤ ↔ IsUnit (e.det v) := by |
constructor
· rintro ⟨hli, hspan⟩
set v' := Basis.mk hli hspan.ge
rw [e.det_apply]
convert LinearEquiv.isUnit_det (LinearEquiv.refl R M) v' e using 2
ext i j
simp [v']
· intro h
rw [Basis.det_apply, Basis.toMatrix_eq_toMatrix_constr] at h
set v' := Basis.map e (LinearEquiv.ofIsUnitDet... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Ma... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 185 | 189 | theorem deriv_tsum (hu : Summable u) (hg : ∀ n, Differentiable 𝕜 (g n))
(hg' : ∀ n y, ‖deriv (g n) y‖ ≤ u n) (hg0 : Summable fun n => g n y₀) :
(deriv fun y => ∑' n, g n y) = fun y => ∑' n, deriv (g n) y := by |
ext1 x
exact deriv_tsum_apply hu hg hg' hg0 x
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.CategoryTheory.Category.Basic
import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Tactic.NthRewrite
imp... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 93 | 117 | theorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :
Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =
Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (𝟙 X) := by |
apply Quot.EqvGen_sound
induction' p with a b q f ih
· apply EqvGen.refl
· simp only [Quiver.Path.reverse]
fapply EqvGen.trans
-- Porting note: `Quiver.Path.*` and `Quiver.Hom.*` notation not working
· exact q ≫ Quiver.Path.reverse q
· apply EqvGen.symm
apply EqvGen.rel
have : Quoti... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 161 | 168 | theorem vars_C_mul (a : A) (ha : a ≠ 0) (φ : MvPolynomial σ A) :
(C a * φ : MvPolynomial σ A).vars = φ.vars := by |
ext1 i
simp only [mem_vars, exists_prop, mem_support_iff]
apply exists_congr
intro d
apply and_congr _ Iff.rfl
rw [coeff_C_mul, mul_ne_zero_iff, eq_true ha, true_and_iff]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import analysis.normed_space.complemented from "leanprover... | Mathlib/Analysis/NormedSpace/Complemented.lean | 139 | 143 | theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace 𝕜]
[FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented := by |
obtain ⟨q, hq⟩ : ∃ q, IsCompl p q := p.exists_isCompl
haveI : FiniteDimensional 𝕜 q := (p.quotientEquivOfIsCompl q hq).finiteDimensional
exact .of_isCompl_isClosed hq hp q.closed_of_finiteDimensional
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,678 | 1,680 | theorem norm_setToFun_le_mul_norm (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E)
(hC : 0 ≤ C) : ‖setToFun μ T hT f‖ ≤ C * ‖f‖ := by |
rw [L1.setToFun_eq_setToL1]; exact L1.norm_setToL1_le_mul_norm hT hC f
|
/-
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.Computability.Language
import Mathlib.Tactic.AdaptationNote
#align_import computability.regular_expressions from "leanprover-community/mathlib"@"369525b73f2... | Mathlib/Computability/RegularExpressions.lean | 227 | 228 | theorem one_rmatch_iff (x : List α) : rmatch 1 x ↔ x = [] := by |
induction x <;> simp [rmatch, matchEpsilon, *]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,534 | 1,536 | theorem cons_merge_cons (s : α → α → Bool) (a b l r) :
merge s (a::l) (b::r) = if s a b then a :: merge s l (b::r) else b :: merge s (a::l) r := by |
simp only [merge, merge.loop, cond]; split <;> (next hs => rw [hs, merge_loop]; rfl)
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 150 | 156 | theorem goldConj_irrational : Irrational ψ := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_sub 1
have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
convert this
norm_num
field_simp
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.Bin... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 189 | 190 | theorem pullbackZeroZeroIso_hom_snd (X Y : C) [HasBinaryProduct X Y] :
(pullbackZeroZeroIso X Y).hom ≫ prod.snd = pullback.snd := by | simp [← Iso.eq_inv_comp]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 251 | 259 | theorem comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.FiniteType) (hg : Surjective g) :
(g.comp f).FiniteType := by |
let _ : Algebra A B := f.toAlgebra
let _ : Algebra A C := (g.comp f).toAlgebra
exact Algebra.FiniteType.of_surjective hf
{ g with
toFun := g
commutes' := fun a => rfl }
hg
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.Gro... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 148 | 150 | theorem mul_cancel_left_mem_nonZeroDivisors {x y r : R'} (hr : r ∈ R'⁰) :
r * x = r * y ↔ x = y := by |
simp_rw [mul_comm r, mul_cancel_right_mem_nonZeroDivisors hr]
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
#align_import analysis.co... | Mathlib/Analysis/Convex/Jensen.lean | 296 | 303 | theorem ConvexOn.exists_ge_of_mem_convexHull (hf : ConvexOn 𝕜 (convexHull 𝕜 s) f) {x}
(hx : x ∈ convexHull 𝕜 s) : ∃ y ∈ s, f x ≤ f y := by |
rw [_root_.convexHull_eq] at hx
obtain ⟨α, t, w, p, hw₀, hw₁, hp, rfl⟩ := hx
rcases hf.exists_ge_of_centerMass hw₀ (hw₁.symm ▸ zero_lt_one) fun i hi =>
subset_convexHull 𝕜 s (hp i hi) with
⟨i, hit, Hi⟩
exact ⟨p i, hp i hit, Hi⟩
|
/-
Copyright (c) 2022 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.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 146 | 163 | theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
{φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ)
(h :
φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) :
φ = ψ := by |
rw [DFunLike.ext'_iff]
have : Dense (Set.range <| inclusion (le_topologicalClosure S)) := by
refine embedding_subtype_val.toInducing.dense_iff.2 fun x => ?_
convert show ↑x ∈ closure (S : Set A) from x.prop
rw [← Set.range_comp]
exact
Set.ext fun y =>
⟨by
rintro ⟨y, 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.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "l... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 102 | 105 | theorem _root_.IsClosed.measure_eq_univ_iff_eq [OpensMeasurableSpace X] [IsFiniteMeasure μ]
(hF : IsClosed F) :
μ F = μ univ ↔ F = univ := by |
rw [← ae_eq_univ_iff_measure_eq hF.measurableSet.nullMeasurableSet, hF.ae_eq_univ_iff_eq]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 695 | 716 | theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
(h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by |
choose! M hM using h'
suffices H : (s \ interior s).Countable by
have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
rw [this]
exact isOpen_interior.measurableSet.union H.measurableSet
have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
(mem_nhdsWithin_Ioi_if... |
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 129 | 136 | theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by |
by_cases hp : p = 0
· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
· rw [hq, mul_zero, trailingDegree_zero, add_top]
· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul ... |
/-
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 | 371 | 372 | theorem HasBasis.eventually_iff (hl : l.HasBasis p s) {q : α → Prop} :
(∀ᶠ x in l, q x) ↔ ∃ i, p i ∧ ∀ ⦃x⦄, x ∈ s i → q x := by | simpa using hl.mem_iff
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 804 | 804 | theorem mul_comm_div : a / b * c = a * (c / b) := by | simp
|
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheo... | Mathlib/Algebra/BigOperators/Fin.lean | 143 | 146 | theorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by |
rw [prod_univ_castSucc, prod_univ_four]
rfl
|
/-
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.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 504 | 508 | theorem tendsto_iff_forall_toWeakDualBCNN_tendsto {γ : Type*} {F : Filter γ}
{μs : γ → FiniteMeasure Ω} {μ : FiniteMeasure Ω} :
Tendsto μs F (𝓝 μ) ↔
∀ f : Ω →ᵇ ℝ≥0, Tendsto (fun i => (μs i).toWeakDualBCNN f) F (𝓝 (μ.toWeakDualBCNN f)) := by |
rw [tendsto_iff_weak_star_tendsto, tendsto_iff_forall_eval_tendsto_topDualPairing]; rfl
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 315 | 319 | theorem noncommProd_cons (s : Finset α) (a : α) (f : α → β)
(ha : a ∉ s) (comm) :
noncommProd (cons a s ha) f comm =
f a * noncommProd s f (comm.mono fun _ => Finset.mem_cons.2 ∘ .inr) := by |
simp_rw [noncommProd, Finset.cons_val, Multiset.map_cons, Multiset.noncommProd_cons]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,483 | 1,487 | theorem Products.max_eq_o_cons_tail' [Inhabited I] (l : Products I) (hl : l.val ≠ [])
(hlh : l.val.head! = term I ho) (hlc : List.Chain' (·>·) (term I ho :: l.Tail.val)) :
l = ⟨term I ho :: l.Tail.val, hlc⟩ := by |
simp_rw [← max_eq_o_cons_tail ho l hl hlh]
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, 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 | 461 | 463 | theorem image_comm {β'} [DecidableEq β'] [DecidableEq γ] {f : β → γ} {g : α → β} {f' : α → β'}
{g' : β' → γ} (h_comm : ∀ a, f (g a) = g' (f' a)) :
(s.image g).image f = (s.image f').image g' := by | simp_rw [image_image, comp, h_comm]
|
/-
Copyright (c) 2021 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Yaël Dillies
-/
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
/-!
... | Mathlib/Computability/EpsilonNFA.lean | 87 | 88 | theorem stepSet_empty (a : α) : M.stepSet ∅ a = ∅ := by |
simp_rw [stepSet, mem_empty_iff_false, iUnion_false, iUnion_empty]
|
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
/-!... | Mathlib/Combinatorics/Additive/ETransform.lean | 150 | 153 | theorem mulETransformRight.fst_mul_snd_subset :
(mulETransformRight e x).1 * (mulETransformRight e x).2 ⊆ x.1 * x.2 := by |
refine union_mul_inter_subset_union.trans (union_subset Subset.rfl ?_)
rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]
|
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb... | Mathlib/Data/Real/EReal.lean | 1,229 | 1,230 | theorem coe_abs (x : ℝ) : ((x : EReal).abs : EReal) = (|x| : ℝ) := by |
rw [abs_def, ← Real.coe_nnabs, ENNReal.ofReal_coe_nnreal]; rfl
|
/-
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.Algebra.Group.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#... | Mathlib/Data/Rat/Defs.lean | 186 | 202 | theorem lift_binop_eq (f : ℚ → ℚ → ℚ) (f₁ : ℤ → ℤ → ℤ → ℤ → ℤ) (f₂ : ℤ → ℤ → ℤ → ℤ → ℤ)
(fv :
∀ {n₁ d₁ h₁ c₁ n₂ d₂ h₂ c₂},
f ⟨n₁, d₁, h₁, c₁⟩ ⟨n₂, d₂, h₂, c₂⟩ = f₁ n₁ d₁ n₂ d₂ /. f₂ n₁ d₁ n₂ d₂)
(f0 : ∀ {n₁ d₁ n₂ d₂}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0) (a b c d : ℤ)
(b0 : b ≠ 0) (d0 : d ≠ 0... |
generalize ha : a /. b = x; cases' x with n₁ d₁ h₁ c₁; rw [mk'_eq_divInt] at ha
generalize hc : c /. d = x; cases' x with n₂ d₂ h₂ c₂; rw [mk'_eq_divInt] at hc
rw [fv]
have d₁0 := Int.ofNat_ne_zero.2 h₁
have d₂0 := Int.ofNat_ne_zero.2 h₂
exact (divInt_eq_iff (f0 d₁0 d₂0) (f0 b0 d0)).2
(H ((divInt_eq_if... |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
/-!
# The rank nullity theorem
In this file we provide the rank nullit... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 136 | 142 | theorem exists_linearIndependent_pair_of_one_lt_rank [StrongRankCondition R]
[NoZeroSMulDivisors R M] (h : 1 < Module.rank R M) {x : M} (hx : x ≠ 0) :
∃ y, LinearIndependent R ![x, y] := by |
obtain ⟨y, hy⟩ := exists_linearIndependent_snoc_of_lt_rank (linearIndependent_unique ![x] hx) h
have : Fin.snoc ![x] y = ![x, y] := Iff.mp List.ofFn_inj rfl
rw [this] at hy
exact ⟨y, hy⟩
|
/-
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.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 1,119 | 1,124 | theorem choose_subtype_eq {α : Type*} (p : α → Prop) [Fintype { a : α // p a }] [DecidableEq α]
(x : { a : α // p a })
(h : ∃! a : { a // p a }, (a : α) = x :=
⟨x, rfl, fun y hy => by simpa [Subtype.ext_iff] using hy⟩) :
Fintype.choose (fun y : { a : α // p a } => (y : α) = x) h = x := by |
rw [Subtype.ext_iff, Fintype.choose_spec (fun y : { a : α // p a } => (y : α) = x) _]
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#alig... | Mathlib/LinearAlgebra/Dimension/Free.lean | 193 | 196 | theorem Module.finite_of_finrank_pos (h : 0 < finrank R M) :
Module.Finite R M := by |
contrapose h
simp [finrank_of_not_finite h]
|
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespace List
variable ... | Mathlib/Data/List/ReduceOption.lean | 80 | 85 | theorem reduceOption_concat (l : List (Option α)) (x : Option α) :
(l.concat x).reduceOption = l.reduceOption ++ x.toList := by |
induction' l with hd tl hl generalizing x
· cases x <;> simp [Option.toList]
· simp only [concat_eq_append, reduceOption_append] at hl
cases hd <;> simp [hl, reduceOption_append]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Batteries.Classes.SatisfiesM
/-!
# Results about monadic operations on `Array`, in terms of `SatisfiesM`.
The pure versions of these theorems... | .lake/packages/batteries/Batteries/Data/Array/Monadic.lean | 18 | 30 | theorem SatisfiesM_foldlM [Monad m] [LawfulMonad m]
{as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → m β}
(hf : ∀ i : Fin as.size, ∀ b, motive i.1 b → SatisfiesM (motive (i.1 + 1)) (f b as[i])) :
SatisfiesM (motive as.size) (as.foldlM f init) := by |
let rec go {i j b} (h₁ : j ≤ as.size) (h₂ : as.size ≤ i + j) (H : motive j b) :
SatisfiesM (motive as.size) (foldlM.loop f as as.size (Nat.le_refl _) i j b) := by
unfold foldlM.loop; split
· next hj =>
split
· cases Nat.not_le_of_gt (by simp [hj]) h₂
· exact (hf ⟨j, hj⟩ b H).bind fun _ ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,165 | 1,167 | theorem eval₂_eta (p : MvPolynomial σ R) : eval₂ C X p = p := by |
apply MvPolynomial.induction_on p <;>
simp (config := { contextual := true }) [eval₂_add, eval₂_mul]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neumann bou... | Mathlib/Analysis/Convex/GaugeRescale.lean | 58 | 61 | theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by |
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 366 | 368 | theorem ofReal_pow {p : ℝ} (hp : 0 ≤ p) (n : ℕ) :
ENNReal.ofReal (p ^ n) = ENNReal.ofReal p ^ n := by |
rw [ofReal_eq_coe_nnreal hp, ← coe_pow, ← ofReal_coe_nnreal, NNReal.coe_pow, NNReal.coe_mk]
|
/-
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 | 1,336 | 1,337 | theorem disjoint_insert_left : Disjoint (insert a s) t ↔ a ∉ t ∧ Disjoint s t := by |
simp only [disjoint_left, mem_insert, or_imp, forall_and, forall_eq]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 403 | 404 | theorem filter_apply_neg [∀ i, Zero (β i)] {p : ι → Prop} [DecidablePred p] (f : Π₀ i, β i) {i : ι}
(h : ¬p i) : f.filter p i = 0 := by | simp only [filter_apply, if_neg h]
|
/-
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 | 682 | 691 | theorem zero_right (X : C) : IsPushout (0 : X ⟶ 0) (𝟙 X) (0 : (0 : C) ⟶ 0) (0 : X ⟶ 0) :=
{ w := by | simp
isColimit' :=
⟨{ desc := fun s => 0
fac := fun s => by
have c :=
@PushoutCocone.coequalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _)
(by simp [eq_iff_true_of_subsingleton]) (by simpa using PushoutCocone.condition s)
dsimp at c
simpa usin... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 712 | 713 | theorem fderiv_const_sub (c : F) : fderiv 𝕜 (fun y => c - f y) x = -fderiv 𝕜 f x := by |
simp only [← fderivWithin_univ, fderivWithin_const_sub uniqueDiffWithinAt_univ]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 278 | 286 | theorem irreducible_units_mul (a : αˣ) (b : α) : Irreducible (↑a * b) ↔ Irreducible b := by |
simp only [irreducible_iff, Units.isUnit_units_mul, and_congr_right_iff]
refine fun _ => ⟨fun h A B HAB => ?_, fun h A B HAB => ?_⟩
· rw [← a.isUnit_units_mul]
apply h
rw [mul_assoc, ← HAB]
· rw [← a⁻¹.isUnit_units_mul]
apply h
rw [mul_assoc, ← HAB, Units.inv_mul_cancel_left]
|
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
... | Mathlib/Dynamics/OmegaLimit.lean | 204 | 207 | theorem omegaLimit_eq_iInter_inter {v : Set τ} (hv : v ∈ f) :
ω f ϕ s = ⋂ u : ↥f.sets, closure (image2 ϕ (u ∩ v) s) := by |
rw [omegaLimit_eq_biInter_inter _ _ _ hv]
apply biInter_eq_iInter
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 99 | 103 | theorem choose_two_right (n : ℕ) : choose n 2 = n * (n - 1) / 2 := by |
induction' n with n ih
· simp
· rw [triangle_succ n, choose, ih]
simp [Nat.add_comm]
|
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib... | Mathlib/NumberTheory/Multiplicity.lean | 155 | 158 | theorem pow_sub_pow_of_prime {p : R} (hp : Prime p) {x y : R} (hxy : p ∣ x - y) (hx : ¬p ∣ x)
{n : ℕ} (hn : ¬p ∣ n) : multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) := by |
rw [← geom_sum₂_mul, multiplicity.mul hp, multiplicity_eq_zero.2 (not_dvd_geom_sum₂ hp hxy hx hn),
zero_add]
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 87 | 88 | theorem star_mul_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (star x * x) := by |
simp only [IsSelfAdjoint, star_mul, star_star]
|
/-
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 | 871 | 879 | theorem val_coe_unit_coprime {n : ℕ} (u : (ZMod n)ˣ) : Nat.Coprime (u : ZMod n).val n := by |
cases' n with n
· rcases Int.units_eq_one_or u with (rfl | rfl) <;> simp
apply Nat.coprime_of_mul_modEq_one ((u⁻¹ : Units (ZMod (n + 1))) : ZMod (n + 1)).val
have := Units.ext_iff.1 (mul_right_inv u)
rw [Units.val_one] at this
rw [← eq_iff_modEq_nat, Nat.cast_one, ← this]; clear this
rw [← natCast_zmod_v... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87... | Mathlib/Algebra/Group/Indicator.lean | 239 | 242 | theorem comp_mulIndicator (h : M → β) (f : α → M) {s : Set α} {x : α} [DecidablePred (· ∈ s)] :
h (s.mulIndicator f x) = s.piecewise (h ∘ f) (const α (h 1)) x := by |
letI := Classical.decPred (· ∈ s)
convert s.apply_piecewise f (const α 1) (fun _ => h) (x := x) using 2
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 150 | 152 | theorem map_succ (a : Fin (n + 1) → ℕ) :
map d a = a 0 + (∑ x : Fin n, a x.succ * d ^ (x : ℕ)) * d := by |
simp [map, Fin.sum_univ_succ, _root_.pow_succ, ← mul_assoc, ← sum_mul]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 226 | 228 | theorem IntegrableOn.add_measure (hμ : IntegrableOn f s μ) (hν : IntegrableOn f s ν) :
IntegrableOn f s (μ + ν) := by |
delta IntegrableOn; rw [Measure.restrict_add]; exact hμ.integrable.add_measure hν
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,430 | 1,430 | theorem lf_neg_iff {x y : PGame} : y ⧏ -x ↔ x ⧏ -y := by | rw [← neg_neg x, neg_lf_neg_iff, neg_neg]
|
/-
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 | 339 | 340 | theorem range_basisSingleton (ι : Type*) [Unique ι] (h : finrank K V = 1) (v : V) (hv : v ≠ 0) :
Set.range (basisSingleton ι h v hv) = {v} := by | rw [Set.range_unique, basisSingleton_apply]
|
/-
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.CliffordAlgebra.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.clifford_algeb... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 91 | 122 | theorem GradedAlgebra.lift_ι_eq (i' : ZMod 2) (x' : evenOdd Q i') :
-- Porting note: added a second `by apply`
lift Q ⟨by apply GradedAlgebra.ι Q, by apply GradedAlgebra.ι_sq_scalar Q⟩ x' =
DirectSum.of (fun i => evenOdd Q i) i' x' := by |
cases' x' with x' hx'
dsimp only [Subtype.coe_mk, DirectSum.lof_eq_of]
induction hx' using Submodule.iSup_induction' with
| mem i x hx =>
obtain ⟨i, rfl⟩ := i
-- Porting note: `dsimp only [Subtype.coe_mk] at hx` doesn't work, use `change` instead
change x ∈ LinearMap.range (ι Q) ^ i at hx
induc... |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 171 | 172 | theorem of'_modOf (g : G) : of' k G g %ᵒᶠ g = 0 := by |
simpa only [one_mul] using mul_of'_modOf (1 : k[G]) g
|
/-
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.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 45 | 47 | theorem range_arcsin : range arcsin = Icc (-(π / 2)) (π / 2) := by |
rw [arcsin, range_comp Subtype.val]
simp [Icc]
|
/-
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.Data.Rat.Encodable
import Mathlib.Data.Real.EReal
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Topology.Order.MonotoneContinuity
#al... | Mathlib/Topology/Instances/EReal.lean | 203 | 208 | theorem continuousAt_add_bot_coe (a : ℝ) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, a) := by |
simp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]
refine fun r ↦ ((gt_mem_nhds (bot_lt_coe (r - (a + 1)))).prod_nhds
(gt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| lt_add_one _)).mono fun _ h ↦ ?_
simpa only [← coe_add, sub_add_cancel] using add_lt_add h.1 h.2
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Topology.GDelta
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.Instances.EReal
import Mathlib.Analysis.Normed.G... | Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 327 | 330 | theorem measurable_of_isOpen {f : δ → γ} (hf : ∀ s, IsOpen s → MeasurableSet (f ⁻¹' s)) :
Measurable f := by |
rw [‹BorelSpace γ›.measurable_eq]
exact measurable_generateFrom hf
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,309 | 1,311 | theorem tendsto_pi_nhds {f : Y → ∀ i, π i} {g : ∀ i, π i} {u : Filter Y} :
Tendsto f u (𝓝 g) ↔ ∀ x, Tendsto (fun i => f i x) u (𝓝 (g x)) := by |
rw [nhds_pi, Filter.tendsto_pi]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 166 | 167 | theorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by |
rw [← Stream'.eta (map f (a::s)), map_eq]; rfl
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 147 | 149 | theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x := by |
rw [rpow_logb_eq_abs b_pos b_ne_one hx.ne']
exact abs_of_pos hx
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 125 | 127 | theorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by |
simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
exists_and_left, and_assoc, and_comm, and_left_comm]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 386 | 391 | theorem IsOrtho.comap (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 F} (h : U ⟂ V) :
U.comap f ⟂ V.comap f := by |
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_comap, ← f.inner_map_map]
intro u hu v hv
exact h _ hu _ hv
|
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
/-... | Mathlib/Data/Real/Pi/Wallis.lean | 101 | 114 | theorem tendsto_W_nhds_pi_div_two : Tendsto W atTop (𝓝 <| π / 2) := by |
refine tendsto_of_tendsto_of_tendsto_of_le_of_le ?_ tendsto_const_nhds le_W W_le
have : 𝓝 (π / 2) = 𝓝 ((1 - 0) * (π / 2)) := by rw [sub_zero, one_mul]
rw [this]
refine Tendsto.mul ?_ tendsto_const_nhds
have h : ∀ n : ℕ, ((2 : ℝ) * n + 1) / (2 * n + 2) = 1 - 1 / (2 * n + 2) := by
intro n
rw [sub_div... |
/-
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 | 449 | 453 | theorem inCoordinates_tangent_bundle_core_model_space (x₀ x : H) (y₀ y : H') (ϕ : E →L[𝕜] E') :
inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x y₀ y ϕ = ϕ := by |
erw [VectorBundleCore.inCoordinates_eq] <;> try trivial
simp_rw [tangentBundleCore_indexAt, tangentBundleCore_coordChange_model_space,
ContinuousLinearMap.id_comp, ContinuousLinearMap.comp_id]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 342 | 350 | theorem ae_differentiableWithinAt_of_mem_pi
{ι : Type*} [Fintype ι] {f : E → ι → ℝ} {s : Set E}
(hf : LipschitzOnWith C f s) : ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x := by |
have A : ∀ i : ι, LipschitzWith 1 (fun x : ι → ℝ ↦ x i) := fun i => LipschitzWith.eval i
have : ∀ i : ι, ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ (fun x : E ↦ f x i) s x := fun i ↦ by
apply ae_differentiableWithinAt_of_mem_of_real
exact LipschitzWith.comp_lipschitzOnWith (A i) hf
filter_upwards [ae_all_... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multis... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 225 | 228 | theorem prod_induction (p : α → Prop) (s : Multiset α) (p_mul : ∀ a b, p a → p b → p (a * b))
(p_one : p 1) (p_s : ∀ a ∈ s, p a) : p s.prod := by |
rw [prod_eq_foldr]
exact foldr_induction (· * ·) (fun x y z => by simp [mul_left_comm]) 1 p s p_mul p_one p_s
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 461 | 468 | theorem locallyConnectedSpace [i : LocallyConnectedSpace Y] (h : X ≃ₜ Y) :
LocallyConnectedSpace X := by |
have : ∀ x, (𝓝 x).HasBasis (fun s ↦ IsOpen s ∧ h x ∈ s ∧ IsConnected s)
(h.symm '' ·) := fun x ↦ by
rw [← h.symm_map_nhds_eq]
exact (i.1 _).map _
refine locallyConnectedSpace_of_connected_bases _ _ this fun _ _ hs ↦ ?_
exact hs.2.2.2.image _ h.symm.continuous.continuousOn
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 61 | 73 | theorem div_le_iff (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b :=
⟨fun h =>
calc
a = a / b * b := by | rw [div_mul_cancel₀ _ (ne_of_lt hb).symm]
_ ≤ c * b := mul_le_mul_of_nonneg_right h hb.le
,
fun h =>
calc
a / b = a * (1 / b) := div_eq_mul_one_div a b
_ ≤ c * b * (1 / b) := mul_le_mul_of_nonneg_right h (one_div_pos.2 hb).le
_ = c * b / b := (div_eq_mul_one_div (c * b) b).symm
... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 290 | 290 | theorem cof_ord_le (c : Cardinal) : c.ord.cof ≤ c := by | simpa using cof_le_card c.ord
|
/-
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.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071a... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 109 | 117 | theorem jacobiSymNat.even_even (a b : ℕ) (hb₀ : Nat.beq (b / 2) 0 = false) (ha : a % 2 = 0)
(hb₁ : b % 2 = 0) : jacobiSymNat a b = 0 := by |
refine jacobiSym.eq_zero_iff.mpr
⟨ne_of_gt ((Nat.pos_of_ne_zero (Nat.ne_of_beq_eq_false hb₀)).trans_le (Nat.div_le_self b 2)),
fun hf => ?_⟩
have h : 2 ∣ a.gcd b := Nat.dvd_gcd (Nat.dvd_of_mod_eq_zero ha) (Nat.dvd_of_mod_eq_zero hb₁)
change 2 ∣ (a : ℤ).gcd b at h
rw [hf, ← even_iff_two_dvd] at h
ex... |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
impor... | Mathlib/Order/SupIndep.lean | 328 | 332 | theorem SetIndependent.disjoint_sSup {x : α} {y : Set α} (hx : x ∈ s) (hy : y ⊆ s) (hxy : x ∉ y) :
Disjoint x (sSup y) := by |
have := (hs.mono <| insert_subset_iff.mpr ⟨hx, hy⟩) (mem_insert x _)
rw [insert_diff_of_mem _ (mem_singleton _), diff_singleton_eq_self hxy] at this
exact this
|
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functorial constructions
Lemmas on the interaction... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 39 | 42 | theorem tprod_pi_single [DecidableEq β] (b : β) (a : α) : ∏' b', Pi.mulSingle b a b' = a := by |
rw [tprod_eq_mulSingle b]
· simp
· intro b' hb'; simp [hb']
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanpro... | Mathlib/GroupTheory/Schreier.lean | 135 | 143 | theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
Group.rank H ≤ H.index * Group.rank G := by |
haveI := H.fg_of_index_ne_zero
obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G
obtain ⟨T, hT₀, hT⟩ := exists_finset_card_le_mul H hS
calc
Group.rank H ≤ T.card := Group.rank_le H hT
_ ≤ H.index * S.card := hT₀
_ = H.index * Group.rank G := congr_arg (H.index * ·) hS₀
|
/-
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 | 1,388 | 1,390 | theorem ker_eq_bot : f.ker = ⊥ ↔ Function.Injective f := by |
rw [← LieSubmodule.coe_toSubmodule_eq_iff, ker_coeSubmodule, LieSubmodule.bot_coeSubmodule,
LinearMap.ker_eq_bot, coe_toLinearMap]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fi... | Mathlib/Data/Fin/Tuple/Sort.lean | 164 | 170 | theorem eq_sort_iff' : σ = sort f ↔ StrictMono (σ.trans <| graphEquiv₁ f) := by |
constructor <;> intro h
· rw [h, sort, Equiv.trans_assoc, Equiv.symm_trans_self]
exact (graphEquiv₂ f).strictMono
· have := Subsingleton.elim (graphEquiv₂ f) (h.orderIsoOfSurjective _ <| Equiv.surjective _)
ext1 x
exact (graphEquiv₁ f).apply_eq_iff_eq_symm_apply.1 (DFunLike.congr_fun this x).symm
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 86 | 92 | theorem aeval_verschiebung_poly' (x : 𝕎 R) (n : ℕ) :
aeval x.coeff (verschiebungPoly n) = (verschiebungFun x).coeff n := by |
cases' n with n
· simp only [verschiebungPoly, Nat.zero_eq, ge_iff_le, tsub_eq_zero_of_le, ite_true, map_zero,
verschiebungFun_coeff_zero]
· rw [verschiebungPoly, verschiebungFun_coeff_succ, if_neg n.succ_ne_zero, aeval_X,
add_tsub_cancel_right]
|
/-
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 | 259 | 294 | theorem Memℒp.snorm_indicator_norm_ge_le (hf : Memℒp f p μ) (hmeas : StronglyMeasurable f) {ε : ℝ}
(hε : 0 < ε) : ∃ M : ℝ, snorm ({ x | M ≤ ‖f x‖₊ }.indicator f) p μ ≤ ENNReal.ofReal ε := by |
by_cases hp_ne_zero : p = 0
· refine ⟨1, hp_ne_zero.symm ▸ ?_⟩
simp [snorm_exponent_zero]
by_cases hp_ne_top : p = ∞
· subst hp_ne_top
obtain ⟨M, hM⟩ := hf.snormEssSup_indicator_norm_ge_eq_zero hmeas
refine ⟨M, ?_⟩
simp only [snorm_exponent_top, hM, zero_le]
obtain ⟨M, hM', hM⟩ := Memℒp.integ... |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 640 | 643 | theorem mul_inv_left {f : M →* N} (h : ∀ y : S, IsUnit (f y)) (y : S) (w z : N) :
w * (IsUnit.liftRight (f.restrict S) h y)⁻¹ = z ↔ w = f y * z := by |
rw [mul_comm]
exact Units.inv_mul_eq_iff_eq_mul (IsUnit.liftRight (f.restrict S) h y)
|
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.GroupTheory.PresentedGroup
import Mathlib.GroupTheory.Coxeter.Matrix
/-!
# Coxeter groups and Coxeter syst... | Mathlib/GroupTheory/Coxeter/Basic.lean | 392 | 399 | theorem alternatingWord_succ' (i i' : B) (m : ℕ) :
alternatingWord i i' (m + 1) = (if Even m then i' else i) :: alternatingWord i i' m := by |
induction' m with m ih generalizing i i'
· simp [alternatingWord]
· rw [alternatingWord]
nth_rw 1 [ih i' i]
rw [alternatingWord]
simp [Nat.even_add_one]
|
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