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) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.PGroup
import... | Mathlib/GroupTheory/Torsion.lean | 367 | 368 | theorem not_isTorsionFree_iff : ¬IsTorsionFree G ↔ ∃ g : G, g ≠ 1 ∧ IsOfFinOrder g := by |
simp_rw [IsTorsionFree, Ne, not_forall, Classical.not_not, exists_prop]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 610 | 614 | theorem Gamma_mul_Gamma_add_half (s : ℝ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℝ) ^ (1 - 2 * s) * √π := by |
rw [← ofReal_inj]
simpa only [← Gamma_ofReal, ofReal_cpow zero_le_two, ofReal_mul, ofReal_add, ofReal_div,
ofReal_sub] using Complex.Gamma_mul_Gamma_add_half ↑s
|
/-
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.Group.Hom.Defs
import Mathlib.Algebra.Group.Subse... | Mathlib/Algebra/Group/Submonoid/Basic.lean | 331 | 332 | theorem coe_iInf {ι : Sort*} {S : ι → Submonoid M} : (↑(⨅ i, S i) : Set M) = ⋂ i, S i := by |
simp only [iInf, coe_sInf, Set.biInter_range]
|
/-
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.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.fil... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 499 | 502 | theorem hasBasis_toFilter (l : IntegrationParams) (I : Box ι) :
(l.toFilter I).HasBasis (fun r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) => ∀ c, l.RCond (r c))
fun r => { π | ∃ c, l.MemBaseSet I c (r c) π } := by |
simpa only [setOf_exists] using hasBasis_iSup (l.hasBasis_toFilterDistortion I)
|
/-
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 | 3,029 | 3,031 | theorem exists_mem_insert [DecidableEq α] (a : α) (s : Finset α) (p : α → Prop) :
(∃ x, x ∈ insert a s ∧ p x) ↔ p a ∨ ∃ x, x ∈ s ∧ p x := by |
simp only [mem_insert, or_and_right, exists_or, exists_eq_left]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-comm... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 284 | 287 | theorem setIntegral_compProd_univ_right (f : β × γ → E) {s : Set β} (hs : MeasurableSet s)
(hf : IntegrableOn f (s ×ˢ univ) ((κ ⊗ₖ η) a)) :
∫ z in s ×ˢ univ, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by |
simp_rw [setIntegral_compProd hs MeasurableSet.univ hf, Measure.restrict_univ]
|
/-
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.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 126 | 126 | theorem cons_bind : (a ::ₘ s).bind f = f a + s.bind f := by | simp [bind]
|
/-
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.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congr... | Mathlib/GroupTheory/Coprod/Basic.lean | 516 | 517 | theorem fst_toProd (x : M ∗ N) : (toProd x).1 = fst x := by |
rw [← fst_comp_toProd]; rfl
|
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.L... | Mathlib/Algebra/Order/Rearrangement.lean | 300 | 302 | theorem Antivary.sum_smul_eq_sum_smul_comp_perm_iff (hfg : Antivary f g) :
((∑ i, f i • g (σ i)) = ∑ i, f i • g i) ↔ Antivary f (g ∘ σ) := by |
simp [(hfg.antivaryOn _).sum_smul_eq_sum_smul_comp_perm_iff fun _ _ ↦ mem_univ _]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.MonCat.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import ... | Mathlib/Algebra/Category/MonCat/Colimits.lean | 188 | 191 | theorem cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j) :
(coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x := by |
rw [← cocone_naturality F f]
rfl
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,879 | 1,880 | theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by | simp_rw [image2_iUnion_left]
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Submonoid.Basic
/-!
# Submonoid of opposite monoids
For every monoid `M`, we construct an... | Mathlib/Algebra/Group/Submonoid/MulOpposite.lean | 136 | 139 | theorem op_closure (s : Set M) : (closure s).op = closure (MulOpposite.unop ⁻¹' s) := by |
simp_rw [closure, op_sInf, Set.preimage_setOf_eq, Submonoid.unop_coe]
congr with a
exact MulOpposite.unop_surjective.forall
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
#align_import category_theory.sites.closed from "leanprover-community/mathlib"@"4cfc30e317c... | Mathlib/CategoryTheory/Sites/Closed.lean | 300 | 305 | theorem topologyOfClosureOperator_close (c : ∀ X : C, ClosureOperator (Sieve X))
(pb : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), c Y (S.pullback f) = (c X S).pullback f) (X : C)
(S : Sieve X) : (topologyOfClosureOperator c pb).close S = c X S := by |
ext Y f
change c _ (Sieve.pullback f S) = ⊤ ↔ c _ S f
rw [pb, Sieve.pullback_eq_top_iff_mem]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
... | Mathlib/RingTheory/SimpleModule.lean | 91 | 94 | theorem isSimpleModule_iff_isCoatom : IsSimpleModule R (M ⧸ m) ↔ IsCoatom m := by |
rw [← Set.isSimpleOrder_Ici_iff_isCoatom]
apply OrderIso.isSimpleOrder_iff
exact Submodule.comapMkQRelIso m
|
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Init.Data.List.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.Con... | Mathlib/Data/Nat/Bits.lean | 393 | 395 | theorem bit0_mod_two : bit0 n % 2 = 0 := by |
rw [Nat.mod_two_of_bodd]
simp
|
/-
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.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264... | Mathlib/Algebra/MvPolynomial/Rename.lean | 324 | 327 | theorem coeff_rename_ne_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : (rename f φ).coeff d ≠ 0) : ∃ u : σ →₀ ℕ, u.mapDomain f = d ∧ φ.coeff u ≠ 0 := by |
contrapose! h
apply coeff_rename_eq_zero _ _ _ h
|
/-
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.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/... | Mathlib/Algebra/Lie/Solvable.lean | 116 | 124 | theorem derivedSeriesOfIdeal_add_le_add (J : LieIdeal R L) (k l : ℕ) :
D (k + l) (I + J) ≤ D k I + D l J := by |
let D₁ : LieIdeal R L →o LieIdeal R L :=
{ toFun := fun I => ⁅I, I⁆
monotone' := fun I J h => LieSubmodule.mono_lie I J I J h h }
have h₁ : ∀ I J : LieIdeal R L, D₁ (I ⊔ J) ≤ D₁ I ⊔ J := by
simp [D₁, LieSubmodule.lie_le_right, LieSubmodule.lie_le_left, le_sup_of_le_right]
rw [← D₁.iterate_sup_le_su... |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 474 | 477 | theorem monomial_pow (n : ℕ) (r : R) (k : ℕ) : monomial n r ^ k = monomial (n * k) (r ^ k) := by |
induction' k with k ih
· simp [pow_zero, monomial_zero_one]
· simp [pow_succ, ih, monomial_mul_monomial, Nat.succ_eq_add_one, mul_add, add_comm]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 73 | 74 | theorem mem_image₂_iff (hf : Injective2 f) : f a b ∈ image₂ f s t ↔ a ∈ s ∧ b ∈ t := by |
rw [← mem_coe, coe_image₂, mem_image2_iff hf, mem_coe, mem_coe]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_the... | Mathlib/FieldTheory/Minpoly/Field.lean | 259 | 266 | theorem coeff_zero_eq_zero (hx : IsIntegral A x) : coeff (minpoly A x) 0 = 0 ↔ x = 0 := by |
constructor
· intro h
have zero_root := zero_isRoot_of_coeff_zero_eq_zero h
rw [← root hx zero_root]
exact RingHom.map_zero _
· rintro rfl
simp
|
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 690 | 699 | theorem eq_of_bisim [Nonempty (M F)] (bisim : IsBisimulation R) : ∀ s₁ s₂, R s₁ s₂ → s₁ = s₂ := by |
inhabit M F
introv Hr; apply ext
introv
by_cases h : IsPath ps s₁ ∨ IsPath ps s₂
· have H := nth_of_bisim R bisim _ _ ps Hr h
exact H.left
· rw [not_or] at h
cases' h with h₀ h₁
simp only [iselect_eq_default, *, not_false_iff]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Compl... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 144 | 151 | theorem add (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g) :
IsBoundedLinearMap 𝕜 fun e => f e + g e :=
let ⟨hlf, Mf, _, hMf⟩ := hf
let ⟨hlg, Mg, _, hMg⟩ := hg
(hlf.mk' _ + hlg.mk' _).isLinear.with_bound (Mf + Mg) fun x =>
calc
‖f x + g x‖ ≤ Mf * ‖x‖ + Mg * ‖x‖ := norm_add_le_of_le (... | rw [add_mul]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 725 | 745 | theorem prodAssoc_prod [SFinite τ] :
map MeasurableEquiv.prodAssoc ((μ.prod ν).prod τ) = μ.prod (ν.prod τ) := by |
have : sum (fun (p : ℕ × ℕ × ℕ) ↦
(sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2)))
= sum (fun (p : (ℕ × ℕ) × ℕ) ↦
(sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))) := by
ext s hs
rw [sum_apply _ hs, sum_apply _ hs, ← (Equiv.prodAssoc _ _ ... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.MetricSpace.PseudoMetric
import Mathlib.Topology.UniformSpace.Equicontinuity
#align_import topology.metric_space.equicontinuity from "lea... | Mathlib/Topology/MetricSpace/Equicontinuity.lean | 103 | 114 | theorem uniformEquicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
(b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
(H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : UniformEquicontinuous F := by |
rw [Metric.uniformEquicontinuous_iff]
intro ε ε0
rcases tendsto_nhds_nhds.1 b_lim ε ε0 with ⟨δ, δ0, hδ⟩
refine ⟨δ, δ0, fun x y hxy i => ?_⟩
calc
dist (F i x) (F i y) ≤ b (dist x y) := H x y i
_ ≤ |b (dist x y)| := le_abs_self _
_ = dist (b (dist x y)) 0 := by simp [Real.dist_eq]
_ < ε := hδ (... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 604 | 607 | theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α := by |
nontriviality α; inhabit α
exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Scott Morrison
-/
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathli... | Mathlib/Data/List/Range.lean | 259 | 268 | theorem ranges_length (l : List ℕ) :
l.ranges.map length = l := by |
induction l with
| nil => simp only [ranges, map_nil]
| cons a l hl => -- (a :: l)
simp only [map, length_range, map_map, cons.injEq, true_and]
conv_rhs => rw [← hl]
apply map_congr
intro s _
simp only [Function.comp_apply, length_map]
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
#alig... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 219 | 236 | theorem frobenius_frobeniusRotation {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
frobenius (frobeniusRotation p ha₁ ha₂) * a₁ = frobeniusRotation p ha₁ ha₂ * a₂ := by |
ext n
cases' n with n
· simp only [WittVector.mul_coeff_zero, WittVector.coeff_frobenius_charP, frobeniusRotation,
frobeniusRotationCoeff, Nat.zero_eq]
apply solution_spec' _ ha₁
· simp only [nthRemainder_spec, WittVector.coeff_frobenius_charP, frobeniusRotationCoeff,
frobeniusRotation]
hav... |
/-
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 | 1,080 | 1,082 | theorem top_mul_of_pos {x : EReal} (h : 0 < x) : ⊤ * x = ⊤ := by |
rw [EReal.mul_comm]
exact mul_top_of_pos h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 556 | 558 | theorem mem_span_singleton_trans {x y z : M} (hxy : x ∈ R ∙ y) (hyz : y ∈ R ∙ z) : x ∈ R ∙ z := by |
rw [← SetLike.mem_coe, ← singleton_subset_iff] at *
exact Submodule.subset_span_trans hxy hyz
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,379 | 1,380 | theorem Real.dist_le_of_mem_Icc_01 {x y : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) (hy : y ∈ Icc (0 : ℝ) 1) :
dist x y ≤ 1 := by | simpa only [sub_zero] using Real.dist_le_of_mem_Icc hx hy
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 665 | 670 | theorem swap_mul_swap_same_mem_closure_three_cycles {a b c : α} (ab : a ≠ b) (ac : a ≠ c) :
swap a b * swap a c ∈ closure { σ : Perm α | IsThreeCycle σ } := by |
by_cases bc : b = c
· subst bc
simp [one_mem]
exact subset_closure (isThreeCycle_swap_mul_swap_same ab ac bc)
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 98 | 99 | theorem naturality' (η : ApplicativeTransformation F G) (x : t (F α)) :
η (sequence x) = sequence (@η _ <$> x) := by | simp [sequence, naturality, traverse_map]
|
/-
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 | 1,128 | 1,130 | theorem prime_mk {p : α} : Prime (Associates.mk p) ↔ Prime p := by |
rw [Prime, _root_.Prime, forall_associated]
simp only [forall_associated, mk_ne_zero, isUnit_mk, mk_mul_mk, mk_dvd_mk]
|
/-
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.Algebra.Order.Chebyshev
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Order.Partition.Equipartition
#align_... | Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | 198 | 205 | theorem hundred_lt_pow_initialBound_mul {ε : ℝ} (hε : 0 < ε) (l : ℕ) :
100 < ↑4 ^ initialBound ε l * ε ^ 5 := by |
rw [← rpow_natCast 4, ← div_lt_iff (pow_pos hε 5), lt_rpow_iff_log_lt _ zero_lt_four, ←
div_lt_iff, initialBound, Nat.cast_max, Nat.cast_max]
· push_cast
exact lt_max_of_lt_right (lt_max_of_lt_right <| Nat.lt_floor_add_one _)
· exact log_pos (by norm_num)
· exact div_pos (by norm_num) (pow_pos hε 5)
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
#align_import algebra.direct_sum.ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Algebra/DirectSum/Ring.lean | 283 | 288 | theorem ofPow {i} (a : A i) (n : ℕ) :
of _ i a ^ n = of _ (n • i) (GradedMonoid.GMonoid.gnpow _ a) := by |
induction' n with n n_ih
· exact of_eq_of_gradedMonoid_eq (pow_zero <| GradedMonoid.mk _ a).symm
· rw [pow_succ, n_ih, of_mul_of]
exact of_eq_of_gradedMonoid_eq (pow_succ (GradedMonoid.mk _ a) n).symm
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 387 | 391 | theorem Set.Finite.convexHull_eq {s : Set E} (hs : s.Finite) : convexHull R s =
{ x : E | ∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ hs.toFinset, w y = 1 ∧
hs.toFinset.centerMass w id = x } := by |
simpa only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, exists_prop] using
hs.toFinset.convexHull_eq
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68a... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 98 | 102 | theorem compress_self (u a : α) : compress u u a = a := by |
unfold compress
split_ifs with h
· exact h.1.symm.sup_sdiff_cancel_right
· rfl
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0... | Mathlib/Data/Finset/Option.lean | 165 | 166 | theorem insertNone_eraseNone [DecidableEq (Option α)] (s : Finset (Option α)) :
insertNone (eraseNone s) = insert none s := by | ext (_ | x) <;> simp
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.Spec... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 182 | 200 | theorem Gamma1_mem (N : ℕ) (A : SL(2, ℤ)) : A ∈ Gamma1 N ↔
((↑ₘA 0 0 : ℤ) : ZMod N) = 1 ∧ ((↑ₘA 1 1 : ℤ) : ZMod N) = 1 ∧ ((↑ₘA 1 0 : ℤ) : ZMod N) = 0 := by |
constructor
· intro ha
simp_rw [Gamma1, Subgroup.mem_map] at ha
obtain ⟨⟨x, hx⟩, hxx⟩ := ha
rw [Gamma1_to_Gamma0_mem] at hx
simp only [Subgroup.mem_top, true_and] at hxx
rw [← hxx]
convert hx
· intro ha
simp_rw [Gamma1, Subgroup.mem_map]
have hA : A ∈ Gamma0 N := by simp [ha.right... |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 282 | 286 | theorem adjoint_conj {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (S : F →L[𝕜] E) :
IsSelfAdjoint (ContinuousLinearMap.adjoint S ∘L T ∘L S) := by |
rw [isSelfAdjoint_iff'] at hT ⊢
simp only [hT, adjoint_comp, adjoint_adjoint]
exact ContinuousLinearMap.comp_assoc _ _ _
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import n... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 478 | 484 | theorem not_exists_int_prime_dvd_sub_of_prime_ne_two'
[hcycl : IsCyclotomicExtension {p} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑p) (hodd : p ≠ 2) :
¬(∃ n : ℤ, (p : 𝓞 K) ∣ (hζ.toInteger - n : 𝓞 K)) := by |
have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by simpa using hcycl
replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simpa using hζ
exact not_exists_int_prime_dvd_sub_of_prime_ne_two hζ hodd
|
/-
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 | 724 | 726 | theorem ofENat_lt {x y : ℕ∞} : ofENat x < ofENat y ↔ x < y := by |
classical
rw [← toWithTop_lt, toWithTop_ofENat, toWithTop_ofENat]
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
#align_import ring_theory.polynomial.cyclotomic.expand from "leanprover-comm... | Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 124 | 134 | theorem cyclotomic_mul_prime_eq_pow_of_not_dvd (R : Type*) {p n : ℕ} [hp : Fact (Nat.Prime p)]
[Ring R] [CharP R p] (hn : ¬p ∣ n) : cyclotomic (n * p) R = cyclotomic n R ^ (p - 1) := by |
letI : Algebra (ZMod p) R := ZMod.algebra _ _
suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1) by
rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),
this, Polynomial.map_pow]
apply mul_right_injective₀ (cyclotomic_ne_zero n <| ZMod p); dsim... |
/-
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.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.legendre_sym... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 80 | 81 | theorem quadraticCharFun_one : quadraticCharFun F 1 = 1 := by |
simp only [quadraticCharFun, one_ne_zero, isSquare_one, if_true, if_false, id]
|
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 59 | 60 | theorem boxProd_adj_left : (G □ H).Adj (a₁, b) (a₂, b) ↔ G.Adj a₁ a₂ := by |
simp only [boxProd_adj, and_true, SimpleGraph.irrefl, false_and, or_false]
|
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.G... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 52 | 54 | theorem lineMap_mono_left (ha : a ≤ a') (hr : r ≤ 1) : lineMap a b r ≤ lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_le_add_right (smul_le_smul_of_nonneg_left ha (sub_nonneg.2 hr)) _
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | Mathlib/CategoryTheory/Subobject/Basic.lean | 561 | 564 | theorem pullback_comp (f : X ⟶ Y) (g : Y ⟶ Z) (x : Subobject Z) :
(pullback (f ≫ g)).obj x = (pullback f).obj ((pullback g).obj x) := by |
induction' x using Quotient.inductionOn' with t
exact Quotient.sound ⟨(MonoOver.pullbackComp _ _).app t⟩
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 48 | 48 | theorem xgcd_zero_left {s t r' s' t'} : xgcdAux 0 s t r' s' t' = (r', s', t') := by | simp [xgcdAux]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.I... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 97 | 99 | theorem integral_comp_inv_smul (f : E → F) (R : ℝ) :
∫ x, f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ x, f x ∂μ := by |
rw [integral_comp_smul μ f R⁻¹, inv_pow, inv_inv]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56... | Mathlib/Order/Filter/Partial.lean | 236 | 240 | theorem pmap_res (l : Filter α) (s : Set α) (f : α → β) :
pmap (PFun.res f s) l = map f (l ⊓ 𝓟 s) := by |
ext t
simp only [PFun.core_res, mem_pmap, mem_map, mem_inf_principal, imp_iff_not_or]
rfl
|
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 383 | 414 | theorem LipschitzOnWith.extend_real {f : α → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → ℝ, LipschitzWith K g ∧ EqOn f g s := by |
/- An extension is given by `g y = Inf {f x + K * dist y x | x ∈ s}`. Taking `x = y`, one has
`g y ≤ f y` for `y ∈ s`, and the other inequality holds because `f` is `K`-Lipschitz, so that it
can not counterbalance the growth of `K * dist y x`. One readily checks from the formula that
the extended functio... |
/-
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 | 339 | 352 | theorem corec_eq (f : β → Option (α × β)) (b : β) :
destruct (corec f b) = omap (corec f) (f b) := by |
dsimp [corec, destruct, get]
-- Porting note: next two lines were `change`...`with`...
have h: Stream'.corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).1 := rfl
rw [h]
dsimp [Corec.f]
induction' h : f b with s; · rfl
cases' s with a b'; dsimp [Corec.f]
apply congr_arg fun b' => some (a, b')
apply... |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 530 | 534 | theorem one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : ℂ) :
(Gamma s)⁻¹ = s * (Gamma (s + 1))⁻¹ := by |
rcases ne_or_eq s 0 with (h | rfl)
· rw [Gamma_add_one s h, mul_inv, mul_inv_cancel_left₀ h]
· rw [zero_add, Gamma_zero, inv_zero, zero_mul]
|
/-
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.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768ef... | Mathlib/Data/Set/Pairwise/Lattice.lean | 147 | 153 | theorem Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion (h₀ : (s ∪ t).PairwiseDisjoint f)
(h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t := by |
rintro i hi
obtain ⟨a, hai⟩ := h₁ i hi
obtain ⟨j, hj, haj⟩ := mem_iUnion₂.1 (h <| mem_iUnion₂_of_mem hi hai)
rwa [h₀.eq (subset_union_left hi) (subset_union_right hj)
(not_disjoint_iff.2 ⟨a, hai, haj⟩)]
|
/-
Copyright (c) 2021 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.C... | Mathlib/Algebra/Category/ModuleCat/Subobject.lean | 111 | 120 | theorem cokernel_π_imageSubobject_ext {L M N : ModuleCat.{v} R} (f : L ⟶ M) [HasImage f]
(g : (imageSubobject f : ModuleCat.{v} R) ⟶ N) [HasCokernel g] {x y : N} (l : L)
(w : x = y + g (factorThruImageSubobject f l)) : cokernel.π g x = cokernel.π g y := by |
subst w
-- Porting note: The proof from here used to just be `simp`.
simp only [map_add, add_right_eq_self]
change ((cokernel.π g) ∘ (g) ∘ (factorThruImageSubobject f)) l = 0
rw [← coe_comp, ← coe_comp, Category.assoc]
simp only [cokernel.condition, comp_zero]
rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 270 | 271 | theorem set_support_apply_mem {p : Perm α} {a : α} :
p a ∈ { x | p x ≠ x } ↔ a ∈ { x | p x ≠ x } := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstance... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 131 | 132 | theorem nontrivial_iff_ne_bot : Nontrivial p ↔ p ≠ ⊥ := by |
rw [iff_not_comm, not_nontrivial_iff_subsingleton, subsingleton_iff_eq_bot]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"0... | Mathlib/Algebra/Squarefree/Basic.lean | 154 | 163 | theorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :
(∀ x : R, Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ x : R, ¬Irreducible x) ∨ Squarefree r := by |
refine ⟨fun h ↦ ?_, ?_⟩
· rcases eq_or_ne r 0 with (rfl | hr)
· exact .inl (by simpa using h)
· exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)
· rintro (⟨rfl, h⟩ | h)
· simpa using h
intro x hx t
exact hx.not_unit (h x t)
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 138 | 139 | theorem eq_of_perm_factors {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) (h : a.factors ~ b.factors) :
a = b := by | simpa [prod_factors ha, prod_factors hb] using List.Perm.prod_eq h
|
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 333 | 334 | theorem range_exp : range exp = Set.Ioi 0 := by |
rw [← coe_comp_expOrderIso, range_comp, expOrderIso.range_eq, image_univ, Subtype.range_coe]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 1,400 | 1,403 | theorem dist_def (f g : α →₁[μ] β) : dist f g = (∫⁻ a, edist (f a) (g a) ∂μ).toReal := by |
simp only [Lp.dist_def, snorm, one_ne_zero, snorm', Pi.sub_apply, one_toReal, ENNReal.rpow_one,
ne_eq, not_false_eq_true, div_self, ite_false]
simp [edist_eq_coe_nnnorm_sub]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_funct... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 154 | 157 | theorem one_of_mem_closedBall (hx : x ∈ closedBall c f.rIn) : f x = 1 := by |
apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn
simpa only [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg f.rIn_pos.le, ← div_eq_inv_mul,
div_le_one f.rIn_pos] using mem_closedBall_iff_norm.1 hx
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.qu... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 278 | 294 | theorem compact_open_induction_on {P : Opens X.carrier → Prop} (S : Opens X.carrier)
(hS : IsCompact S.1) (h₁ : P ⊥)
(h₂ : ∀ (S : Opens X.carrier) (_ : IsCompact S.1) (U : X.affineOpens), P S → P (S ⊔ U)) :
P S := by |
classical
obtain ⟨s, hs, hs'⟩ := (isCompact_open_iff_eq_finset_affine_union S.1).mp ⟨hS, S.2⟩
replace hs' : S = iSup fun i : s => (i : Opens X.carrier) := by ext1; simpa using hs'
subst hs'
apply @Set.Finite.induction_on _ _ _ hs
· convert h₁; rw [iSup_eq_bot]; rintro ⟨_, h⟩; exact h.elim
· intro x s _ h... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.qu... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 307 | 348 | theorem exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact (X : Scheme.{u})
{U : Opens X.carrier} (hU : IsCompact U.1) (x f : X.presheaf.obj (op U))
(H : x |_ X.basicOpen f = 0) : ∃ n : ℕ, f ^ n * x = 0 := by |
obtain ⟨s, hs, e⟩ := (isCompact_open_iff_eq_finset_affine_union U.1).mp ⟨hU, U.2⟩
replace e : U = iSup fun i : s => (i : Opens X.carrier) := by
ext1; simpa using e
have h₁ : ∀ i : s, i.1.1 ≤ U := by
intro i
change (i : Opens X.carrier) ≤ U
rw [e]
-- Porting note: `exact le_iSup _ _` no longer... |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Oliver Nash
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.D... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 144 | 146 | theorem univBall_symm_apply_center (c : P) (r : ℝ) : (univBall c r).symm c = 0 := by |
have : 0 ∈ (univBall c r).source := by simp
simpa only [univBall_apply_zero] using (univBall c r).left_inv this
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 320 | 324 | theorem equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔
∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U := by |
simp_rw [← equicontinuousWithinAt_univ, equicontinuousWithinAt_iff_pair (mem_univ x₀),
nhdsWithin_univ]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 629 | 631 | theorem coe_spanSingleton (x : P) : (spanSingleton S x : Submodule R P) = span R {x} := by |
rw [spanSingleton]
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, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Monotone functions on an order topology
This file contains lemmas about limits and contin... | Mathlib/Topology/Order/Monotone.lean | 282 | 292 | theorem Monotone.tendsto_nhdsWithin_Iio {α β : Type*} [LinearOrder α] [TopologicalSpace α]
[OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) := by |
rcases eq_empty_or_nonempty (Iio x) with (h | h); · simp [h]
refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩
· obtain ⟨z, zx, lz⟩ : ∃ a : α, a < x ∧ l < f a := by
simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using
exists_lt_of_lt_csSup (h.image _) hl
exact mem_of_supers... |
/-
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.RingHomProperties
import Mathlib.RingTheory.IntegralClosure
#align_import ring_theory.ring_hom.integral from "leanprover-community/mathlib"@"a7c0... | Mathlib/RingTheory/RingHom/Integral.lean | 35 | 41 | theorem isIntegral_stableUnderBaseChange : StableUnderBaseChange fun f => f.IsIntegral := by |
refine StableUnderBaseChange.mk _ isIntegral_respectsIso ?_
introv h x
refine TensorProduct.induction_on x ?_ ?_ ?_
· apply isIntegral_zero
· intro x y; exact IsIntegral.tmul x (h y)
· intro x y hx hy; exact IsIntegral.add hx hy
|
/-
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.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 69 | 74 | theorem card_support_mul_le : (p * q).support.card ≤ p.support.card * q.support.card := by |
calc (p * q).support.card
_ = (p.toFinsupp * q.toFinsupp).support.card := by rw [← support_toFinsupp, toFinsupp_mul]
_ ≤ (p.toFinsupp.support + q.toFinsupp.support).card :=
Finset.card_le_card (AddMonoidAlgebra.support_mul p.toFinsupp q.toFinsupp)
_ ≤ p.support.card * q.support.card := Finset.card_image... |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 517 | 519 | theorem addHaar_closedBall (x : E) {r : ℝ} (hr : 0 ≤ r) :
μ (closedBall x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) := by |
rw [addHaar_closedBall' μ x hr, addHaar_closed_unit_ball_eq_addHaar_unit_ball]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Wieser
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.p... | Mathlib/Data/Real/Pointwise.lean | 132 | 133 | theorem Real.iInf_mul_of_nonneg (ha : 0 ≤ r) (f : ι → ℝ) : (⨅ i, f i) * r = ⨅ i, f i * r := by |
simp only [Real.mul_iInf_of_nonneg ha, mul_comm]
|
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd29... | Mathlib/CategoryTheory/EqToHom.lean | 180 | 183 | theorem eqToHom_unop {X Y : Cᵒᵖ} (h : X = Y) :
(eqToHom h).unop = eqToHom (congr_arg unop h.symm) := by |
cases h
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, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 393 | 397 | theorem sumDiffSubset_symm_apply_of_not_mem {α} {s t : Set α} (h : s ⊆ t) [DecidablePred (· ∈ s)]
{x : t} (hx : x.1 ∉ s) : (Equiv.Set.sumDiffSubset h).symm x = Sum.inr ⟨x, ⟨x.2, hx⟩⟩ := by |
apply (Equiv.Set.sumDiffSubset h).injective
simp only [apply_symm_apply, sumDiffSubset_apply_inr]
exact Subtype.eq rfl
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.WellFound... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 512 | 516 | theorem one_eq_closure : (1 : AddSubmonoid R) = closure {1} := by |
rw [closure_singleton_eq, one_eq_mrange]
congr 1
ext
simp
|
/-
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 | 306 | 309 | theorem pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y ↔ IsRelPrime x y := by |
refine ⟨fun h ↦ ?_, IsRelPrime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm)
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 97 | 113 | theorem ae_eventually_measure_pos [SecondCountableTopology α] :
∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by |
set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
change μ s = 0
let f : α → Set (Set α) := fun _ => {a | μ a = 0}
have h : v.FineSubfamilyOn f s := by
intro x hx ε εpos
rw [hs] at hx
simp only [frequen... |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 1,311 | 1,317 | theorem frequently_lt_of_lt_limsup {α β} [ConditionallyCompleteLinearOrder β] {f : Filter α}
{u : α → β} {b : β}
(hu : f.IsCoboundedUnder (· ≤ ·) u := by | isBoundedDefault)
(h : b < limsup u f) : ∃ᶠ x in f, b < u x := by
contrapose! h
apply limsSup_le_of_le hu
simpa using h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Finite
#align_import data.finset.preimage from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
/-!... | Mathlib/Data/Finset/Preimage.lean | 126 | 129 | theorem sigma_preimage_mk_of_subset {β : α → Type*} [DecidableEq α] (s : Finset (Σa, β a))
{t : Finset α} (ht : s.image Sigma.fst ⊆ t) :
(t.sigma fun a => s.preimage (Sigma.mk a) sigma_mk_injective.injOn) = s := by |
rw [sigma_preimage_mk, filter_true_of_mem <| image_subset_iff.1 ht]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 224 | 224 | theorem e_isoExt_hom : F.e ≫ (isoExt hF hF').hom = F'.e := by | simp
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
/-!... | Mathlib/Order/Interval/Multiset.lean | 260 | 262 | theorem Ico_filter_le_of_le_left [DecidablePred (c ≤ ·)] (hca : c ≤ a) :
((Ico a b).filter fun x => c ≤ x) = Ico a b := by |
rw [Ico, ← Finset.filter_val, Finset.Ico_filter_le_of_le_left hca]
|
/-
Copyright (c) 2021 Ivan Sadofschi Costa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ivan Sadofschi Costa
-/
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# `co... | Mathlib/Data/Finsupp/Fin.lean | 68 | 73 | theorem cons_zero_zero : cons 0 (0 : Fin n →₀ M) = 0 := by |
ext a
by_cases c : a = 0
· simp [c]
· rw [← Fin.succ_pred a c, cons_succ]
simp
|
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
/-!
# Algebra instance on adic completion
In ... | Mathlib/RingTheory/AdicCompletion/Algebra.lean | 133 | 139 | theorem smul_mk {m n : ℕ} (hmn : m ≤ n) (r : AdicCauchySequence I R)
(x : AdicCauchySequence I M) :
r.val n • Submodule.Quotient.mk (p := (I ^ m • ⊤ : Submodule R M)) (x.val n) =
r.val m • Submodule.Quotient.mk (p := (I ^ m • ⊤ : Submodule R M)) (x.val m) := by |
rw [← Submodule.Quotient.mk_smul, ← Module.Quotient.mk_smul_mk,
AdicCauchySequence.mk_eq_mk hmn, Ideal.mk_eq_mk I hmn, Module.Quotient.mk_smul_mk,
Submodule.Quotient.mk_smul]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 600 | 602 | theorem padicValNat_prime_prime_pow {q : ℕ} [hp : Fact p.Prime] [hq : Fact q.Prime]
(n : ℕ) (neq : p ≠ q) : padicValNat p (q ^ n) = 0 := by |
rw [padicValNat.pow _ <| Nat.Prime.ne_zero hq.elim, padicValNat_primes neq, mul_zero]
|
/-
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 | 665 | 670 | theorem mul_inv {f : M →* N} (h : ∀ y : S, IsUnit (f y)) {x₁ x₂} {y₁ y₂ : S} :
f x₁ * (IsUnit.liftRight (f.restrict S) h y₁)⁻¹ =
f x₂ * (IsUnit.liftRight (f.restrict S) h y₂)⁻¹ ↔
f (x₁ * y₂) = f (x₂ * y₁) := by |
rw [mul_inv_right h, mul_assoc, mul_comm _ (f y₂), ← mul_assoc, mul_inv_left h, mul_comm x₂,
f.map_mul, f.map_mul]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin, Scott Morrison
-/
import Mathlib.Analysis.Normed.Group.SemiNormedGroupCat
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.CategoryTheory.L... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean | 394 | 399 | theorem explicitCokernelIso_hom_desc {X Y Z : SemiNormedGroupCat.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) :
(explicitCokernelIso f).hom ≫ cokernel.desc f g w = explicitCokernelDesc w := by |
ext1
simp [explicitCokernelDesc, explicitCokernelπ, explicitCokernelIso,
IsColimit.coconePointUniqueUpToIso]
|
/-
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.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 2,676 | 2,694 | theorem tr_respects_aux {q v T k} {S : ∀ k, List (Γ k)}
(hT : ∀ k, ListBlank.map (proj k) T = ListBlank.mk ((S k).map some).reverse) (o : StAct₂ k)
(IH : ∀ {v : σ} {S : ∀ k : K, List (Γ k)} {T : ListBlank (∀ k, Option (Γ k))},
(∀ k, ListBlank.map (proj k) T = ListBlank.mk ((S k).map some).reverse) →
... |
simp only [trNormal_run, step_run]
have hgo := tr_respects_aux₁ M o q v (hT k) _ le_rfl
obtain ⟨T', hT', hrun⟩ := tr_respects_aux₂ hT o
have := hgo.tail' rfl
rw [tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd,
stk_nth_val _ (hT k), List.get?_len_le (le_of_eq (List.length_rev... |
/-
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 | 1,089 | 1,098 | theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
(h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
setToL1 hT'' f = setToL1 hT f + setToL1 hT' f := by |
suffices setToL1 hT'' = setToL1 hT + setToL1 hT' by rw [this, ContinuousLinearMap.add_apply]
refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT'') _ _ _ _ ?_
ext1 f
suffices setToL1 hT f + setToL1 hT' f = setToL1SCLM α E μ hT'' f by rw [← this]; congr
rw [setToL1_eq_setToL1SCLM, setToL1_eq_setToL... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 387 | 390 | theorem filter_le_le_eq_Icc [DecidablePred fun j => a ≤ j ∧ j ≤ b] :
(univ.filter fun j => a ≤ j ∧ j ≤ b) = Icc a b := by |
ext
simp
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 64 | 77 | theorem arctan_tan {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re) (h₂ : z.re ≤ π / 2) :
arctan (tan z) = z := by |
have h := cos_ne_zero_of_arctan_bounds h₀ h₁ h₂
unfold arctan tan
-- multiply top and bottom by `cos z`
rw [← mul_div_mul_right (1 + _) _ h, add_mul, sub_mul, one_mul, ← mul_rotate, mul_div_cancel₀ _ h]
conv_lhs =>
enter [2, 1, 2]
rw [sub_eq_add_neg, ← neg_mul, ← sin_neg, ← cos_neg]
rw [← exp_mul_I... |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 81 | 86 | theorem charpoly_coeff_eq_prod_coeff_of_le {k : ℕ} (h : Fintype.card n - 1 ≤ k) :
M.charpoly.coeff k = (∏ i : n, (X - C (M i i))).coeff k := by |
apply eq_of_sub_eq_zero; rw [← coeff_sub]
apply Polynomial.coeff_eq_zero_of_degree_lt
apply lt_of_lt_of_le (charpoly_sub_diagonal_degree_lt M) ?_
rw [Nat.cast_le]; apply h
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 348 | 351 | theorem dvd_gcd_mul_iff_dvd_mul : x ∣ gcd x n * m ↔ x ∣ n * m := by |
refine ⟨(·.trans <| mul_dvd_mul_right (x.gcd_dvd_right n) m), fun ⟨y, hy⟩ ↦ ?_⟩
rw [← gcd_mul_right, hy, gcd_mul_left]
exact dvd_mul_right x (gcd m y)
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 433 | 443 | theorem norm_nonneg' (f : lp E p) : 0 ≤ ‖f‖ := by |
rcases p.trichotomy with (rfl | rfl | hp)
· simp [lp.norm_eq_card_dsupport f]
· cases' isEmpty_or_nonempty α with _i _i
· rw [lp.norm_eq_ciSup]
simp [Real.iSup_of_isEmpty]
inhabit α
exact (norm_nonneg (f default)).trans ((lp.isLUB_norm f).1 ⟨default, rfl⟩)
· rw [lp.norm_eq_tsum_rpow hp f]
... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Limits/Fubini.lean | 454 | 457 | theorem limitIsoLimitCurryCompLim_hom_π_π {j} {k} :
(limitIsoLimitCurryCompLim G).hom ≫ limit.π _ j ≫ limit.π _ k = limit.π _ (j, k) := by |
set_option tactic.skipAssignedInstances false in
simp [limitIsoLimitCurryCompLim, Trans.simple, HasLimit.isoOfNatIso, limitUncurryIsoLimitCompLim]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 710 | 715 | theorem toList_getLast? {i j} (w : NeWord M i j) : w.toList.getLast? = Option.some ⟨j, w.last⟩ := by |
rw [← Option.mem_def]
induction w
· rw [Option.mem_def]
rfl
· exact List.getLast?_append (by assumption)
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 519 | 523 | theorem equicontinuousWithinAt_iff_continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} :
EquicontinuousWithinAt F S x₀ ↔
ContinuousWithinAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) S x₀ := by |
rw [ContinuousWithinAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff]
rfl
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib... | Mathlib/Data/Sum/Interval.lean | 91 | 95 | theorem sumLift₂_nonempty :
(sumLift₂ f g a b).Nonempty ↔
(∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨
∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by |
simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da... | Mathlib/Algebra/Group/Ext.lean | 137 | 156 | theorem DivInvMonoid.ext {M : Type*} ⦃m₁ m₂ : DivInvMonoid M⦄
(h_mul : (letI := m₁; HMul.hMul : M → M → M) = (letI := m₂; HMul.hMul : M → M → M))
(h_inv : (letI := m₁; Inv.inv : M → M) = (letI := m₂; Inv.inv : M → M)) : m₁ = m₂ := by |
have h_mon := Monoid.ext h_mul
have h₁ : m₁.one = m₂.one := congr_arg (·.one) h_mon
let f : @MonoidHom M M m₁.toMulOneClass m₂.toMulOneClass :=
@MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)
(fun x y => congr_fun (congr_fun h_mul x) y)
have : m₁.npow = m₂.npow := congr_arg (·.npow) h_mon
have... |
/-
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.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
import Mathlib.MeasureTheory.Integral.CircleIntegral
i... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 496 | 502 | theorem circleIntegral_sub_inv_smul_of_differentiable_on_off_countable {R : ℝ} {c w : ℂ} {f : ℂ → E}
{s : Set ℂ} (hs : s.Countable) (hw : w ∈ ball c R) (hc : ContinuousOn f (closedBall c R))
(hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x) :
(∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w := by |
rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
hs hw hc hd, smul_inv_smul₀]
simp [Real.pi_ne_zero, I_ne_zero]
|
/-
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.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdB... | Mathlib/RingTheory/Finiteness.lean | 832 | 839 | theorem comp {g : B →+* C} {f : A →+* B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite := by |
letI := f.toAlgebra
letI := g.toAlgebra
letI := (g.comp f).toAlgebra
letI : IsScalarTower A B C := RestrictScalars.isScalarTower A B C
letI : Module.Finite A B := hf
letI : Module.Finite B C := hg
exact Module.Finite.trans B C
|
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