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) 2020 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb... | Mathlib/Analysis/Convex/Hull.lean | 214 | 217 | theorem affineSpan_convexHull (s : Set E) : affineSpan 𝕜 (convexHull 𝕜 s) = affineSpan 𝕜 s := by |
refine le_antisymm ?_ (affineSpan_mono 𝕜 (subset_convexHull 𝕜 s))
rw [affineSpan_le]
exact convexHull_subset_affineSpan s
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathl... | Mathlib/Order/JordanHolder.lean | 109 | 113 | theorem isMaximal_of_eq_inf (x b : X) {a y : X} (ha : x ⊓ y = a) (hxy : x ≠ y) (hxb : IsMaximal x b)
(hyb : IsMaximal y b) : IsMaximal a y := by |
have hb : x ⊔ y = b := sup_eq_of_isMaximal hxb hyb hxy
substs a b
exact isMaximal_inf_right_of_isMaximal_sup hxb hyb
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,442 | 2,449 | theorem pullbackAssoc_hom_snd_fst [HasPullback ((pullback.snd : Z₁ ⟶ X₂) ≫ f₃) f₄]
[HasPullback f₁ ((pullback.fst : Z₂ ⟶ X₂) ≫ f₂)] : (pullbackAssoc f₁ f₂ f₃ f₄).hom ≫
pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
trans l₂ ≫ pullback.fst
· rw [← Category.assoc]
congr 1
exact IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.right
· exact pullback.lift_fst _ _ _
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 659 | 659 | theorem any_iff (s : String) (p : Char → Bool) : any s p ↔ ∃ c ∈ s.1, p c := by | simp [any_eq]
|
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.Bicategory.Functor.Oplax
#align_import category_theory.bicategory.natural_transformation from "leanprover-community/mathlib"@"4ff75f5b8502275... | Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean | 118 | 124 | theorem whiskerLeft_naturality_comp (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c) :
f ◁ θ.naturality (g ≫ h) ≫ f ◁ θ.app a ◁ H.mapComp g h =
f ◁ G.mapComp g h ▷ θ.app c ≫
f ◁ (α_ _ _ _).hom ≫
f ◁ G.map g ◁ θ.naturality h ≫
f ◁ (α_ _ _ _).inv ≫ f ◁ θ.naturality g ▷ H.map h ≫ f ◁ (α_ ... |
simp_rw [← whiskerLeft_comp, naturality_comp]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of options
... | Mathlib/Data/Option/NAry.lean | 124 | 127 | theorem map₂_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
map₂ f (map₂ g a b) c = map₂ f' a (map₂ g' b c) := by |
cases a <;> cases b <;> cases c <;> simp [h_assoc]
|
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
#align_import computability.reduce from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
/-!
#... | Mathlib/Computability/Reduce.lean | 352 | 353 | theorem manyOneEquiv_toNat (p : Set α) (q : Set β) :
ManyOneEquiv (toNat p) (toNat q) ↔ ManyOneEquiv p q := by | simp [ManyOneEquiv]
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 390 | 391 | theorem leftUnitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
η ≫ (λ_ g).inv = (λ_ f).inv ≫ 𝟙 a ◁ η := 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, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f70... | Mathlib/Data/PFun.lean | 375 | 387 | theorem fixInduction'_fwd {C : α → Sort*} {f : α →. Sum β α} {b : β} {a a' : α} (h : b ∈ f.fix a)
(h' : b ∈ f.fix a') (fa : Sum.inr a' ∈ f a)
(hbase : ∀ a_final : α, Sum.inl b ∈ f a_final → C a_final)
(hind : ∀ a₀ a₁ : α, b ∈ f.fix a₁ → Sum.inr a₁ ∈ f a₀ → C a₁ → C a₀) :
@fixInduction' _ _ C _ _ _ h hba... |
unfold fixInduction'
rw [fixInduction_spec]
-- Porting note: the explicit motive required because `simp` behaves differently
refine Eq.rec (motive := fun x e =>
Sum.casesOn (motive := fun y => (f a).get (dom_of_mem_fix h) = y → C a) x ?_ ?_
(Eq.trans (Part.get_eq_of_mem fa (dom_of_mem_fix h)) e) = ... |
/-
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 | 275 | 278 | theorem isFiniteMeasure_withDensity_ofReal {f : α → ℝ} (hfi : HasFiniteIntegral f μ) :
IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal <| f x) := by |
refine isFiniteMeasure_withDensity ((lintegral_mono fun x => ?_).trans_lt hfi).ne
exact Real.ofReal_le_ennnorm (f x)
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.monotone_convergence from "leanprover-community/mathlib"@"4c19a16e4b705bf... | Mathlib/Topology/Order/MonotoneConvergence.lean | 147 | 148 | theorem tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <| range f) :
Tendsto f atTop (𝓝 (⨅ i, f i)) := by | convert tendsto_atBot_ciSup h_anti.dual hbdd.dual using 1
|
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44... | Mathlib/Topology/StoneCech.lean | 58 | 62 | theorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by |
rw [← isOpen_compl_iff]
convert ultrafilter_isOpen_basic sᶜ using 1
ext u
exact Ultrafilter.compl_mem_iff_not_mem.symm
|
/-
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.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,502 | 2,502 | theorem count_pos {a : α} {s : Multiset α} : 0 < count a s ↔ a ∈ s := by | simp [count, countP_pos]
|
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 144 | 145 | theorem catalan_two : catalan 2 = 2 := by |
norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 946 | 970 | theorem sigma_pi_composition_eq_iff
(u v : Σ c : Composition n, ∀ i : Fin c.length, Composition (c.blocksFun i)) :
u = v ↔ (ofFn fun i => (u.2 i).blocks) = ofFn fun i => (v.2 i).blocks := by |
refine ⟨fun H => by rw [H], fun H => ?_⟩
rcases u with ⟨a, b⟩
rcases v with ⟨a', b'⟩
dsimp at H
have h : a = a' := by
ext1
have :
map List.sum (ofFn fun i : Fin (Composition.length a) => (b i).blocks) =
map List.sum (ofFn fun i : Fin (Composition.length a') => (b' i).blocks) := by
... |
/-
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 | 770 | 773 | theorem inv_antitoneOn_Icc_right (ha : 0 < a) :
AntitoneOn (fun x:α ↦ x⁻¹) (Set.Icc a b) := by |
convert sub_inv_antitoneOn_Icc_right ha
exact (sub_zero _).symm
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 255 | 264 | theorem ker_mulVecLin_transpose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aᵀ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, ← mulVec_mulVec]
constructor
· intro h
replace h := congr_arg (dotProduct x) h
rwa [dotProduct_mulVec, dotProduct_zero, vecMul_transpose, dotProduct_self_eq_zero] at h
· intro h
rw [h, mulVec_zero]
|
/-
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.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 203 | 212 | theorem content_eq_gcd_leadingCoeff_content_eraseLead (p : R[X]) :
p.content = GCDMonoid.gcd p.leadingCoeff (eraseLead p).content := by |
by_cases h : p = 0
· simp [h]
rw [← leadingCoeff_eq_zero, leadingCoeff, ← Ne, ← mem_support_iff] at h
rw [content, ← Finset.insert_erase h, Finset.gcd_insert, leadingCoeff, content,
eraseLead_support]
refine congr rfl (Finset.gcd_congr rfl fun i hi => ?_)
rw [Finset.mem_erase] at hi
rw [eraseLead_coe... |
/-
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 | 332 | 337 | theorem IsSemisimpleRing.ideal_eq_span_idempotent [IsSemisimpleRing R] (I : Ideal R) :
∃ e : R, IsIdempotentElem e ∧ I = .span {e} := by |
obtain ⟨J, h⟩ := exists_isCompl I
obtain ⟨f, idem, rfl⟩ := I.isIdempotentElemEquiv.symm (I.isComplEquivProj ⟨J, h⟩)
exact ⟨f 1, LinearMap.isIdempotentElem_apply_one_iff.mpr idem, by
erw [LinearMap.range_eq_map, ← Ideal.span_one, LinearMap.map_span, Set.image_singleton]; rfl⟩
|
/-
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 | 108 | 109 | theorem xgcdAux_val (x y) : xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y) := by |
rw [xgcd, ← xgcdAux_fst x y 1 0 0 1]
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 376 | 380 | theorem smul_le_smul {p q : Seminorm 𝕜 E} {a b : ℝ≥0} (hpq : p ≤ q) (hab : a ≤ b) :
a • p ≤ b • q := by |
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (apply_nonneg p x) (NNReal.coe_nonneg b)
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 842 | 846 | theorem integral_comp_sub_mul (hc : c ≠ 0) (d) :
(∫ x in a..b, f (d - c * x)) = c⁻¹ • ∫ x in d - c * b..d - c * a, f x := by |
simp only [sub_eq_add_neg, neg_mul_eq_neg_mul]
rw [integral_comp_add_mul f (neg_ne_zero.mpr hc) d, integral_symm]
simp only [inv_neg, smul_neg, neg_neg, neg_smul]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import ... | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 121 | 129 | theorem squashSeq_nth_of_lt {m : ℕ} (m_lt_n : m < n) : (squashSeq s n).get? m = s.get? m := by |
cases s_succ_nth_eq : s.get? (n + 1) with
| none => rw [squashSeq_eq_self_of_terminated s_succ_nth_eq]
| some =>
obtain ⟨gp_n, s_nth_eq⟩ : ∃ gp_n, s.get? n = some gp_n :=
s.ge_stable n.le_succ s_succ_nth_eq
obtain ⟨gp_m, s_mth_eq⟩ : ∃ gp_m, s.get? m = some gp_m :=
s.ge_stable (le_of_lt m_lt_n... |
/-
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.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.m... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 526 | 536 | theorem is_left_invariant_chaar {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
chaar K₀ (K.map _ <| continuous_mul_left g) = chaar K₀ K := by |
let eval : (Compacts G → ℝ) → ℝ := fun f => f (K.map _ <| continuous_mul_left g) - f K
have : Continuous eval := (continuous_apply (K.map _ _)).sub (continuous_apply K)
rw [← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
unfold clPrehaar; rw [IsClose... |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 118 | 169 | theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p) := by |
classical
by_cases hn : n = 0
· simp_all
have hpos := Nat.pos_of_ne_zero hn
by_contra hdiff
set P := minpoly ℤ μ
set Q := minpoly ℤ (μ ^ p)
have Pmonic : P.Monic := minpoly.monic (h.isIntegral hpos)
have Qmonic : Q.Monic := minpoly.monic ((h.pow_of_prime hprime.1 hdiv).isIntegral hpos)
have Pirr : ... |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import... | Mathlib/Order/Filter/Cocardinal.lean | 61 | 68 | theorem hasBasis_cocardinal : HasBasis (cocardinal α hreg) {s : Set α | #s < c} compl :=
⟨fun s =>
⟨fun h => ⟨sᶜ, h, (compl_compl s).subset⟩, fun ⟨_t, htf, hts⟩ => by
have : #↑sᶜ < c := by |
apply lt_of_le_of_lt _ htf
rw [compl_subset_comm] at hts
apply Cardinal.mk_le_mk_of_subset hts
simp_all only [mem_cocardinal] ⟩⟩
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Batteries.Data.Nat.Lemmas
import Batteries.WF
import Mathlib.Init.Data.Nat.Basic
import Mathlib.Util.AssertExists
#align_import init.... | Mathlib/Init/Data/Nat/Lemmas.lean | 541 | 545 | theorem cond_decide_mod_two (x : ℕ) [d : Decidable (x % 2 = 1)] :
cond (@decide (x % 2 = 1) d) 1 0 = x % 2 := by |
by_cases h : x % 2 = 1
· simp! [*]
· cases mod_two_eq_zero_or_one x <;> simp! [*, Nat.zero_ne_one]
|
/-
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 | 272 | 275 | theorem ptendsto_of_ptendsto' {f : α →. β} {l₁ : Filter α} {l₂ : Filter β} :
PTendsto' f l₁ l₂ → PTendsto f l₁ l₂ := by |
rw [ptendsto_def, ptendsto'_def]
exact fun h s sl₂ => mem_of_superset (h s sl₂) (PFun.preimage_subset_core _ _)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | Mathlib/Topology/Algebra/Monoid.lean | 438 | 442 | theorem exists_open_nhds_one_split {s : Set M} (hs : s ∈ 𝓝 (1 : M)) :
∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ ∀ v ∈ V, ∀ w ∈ V, v * w ∈ s := by |
have : (fun a : M × M => a.1 * a.2) ⁻¹' s ∈ 𝓝 ((1, 1) : M × M) :=
tendsto_mul (by simpa only [one_mul] using hs)
simpa only [prod_subset_iff] using exists_nhds_square this
|
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_th... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 469 | 472 | theorem IsNormal.apply_lt_nfp {f} (H : IsNormal f) {a b} : f b < nfp f a ↔ b < nfp f a := by |
unfold nfp
rw [← @apply_lt_nfpFamily_iff Unit (fun _ => f) _ (fun _ => H) a b]
exact ⟨fun h _ => h, fun h => h Unit.unit⟩
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.PowerBasis
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import ring_theory.adjoin.power_basis from "le... | Mathlib/RingTheory/Adjoin/PowerBasis.lean | 173 | 183 | theorem toMatrix_isIntegral {B B' : PowerBasis K S} {P : R[X]} (h : aeval B.gen P = B'.gen)
(hB : IsIntegral R B.gen) (hmin : minpoly K B.gen = (minpoly R B.gen).map (algebraMap R K)) :
∀ i j, IsIntegral R (B.basis.toMatrix B'.basis i j) := by |
intro i j
rw [B.basis.toMatrix_apply, B'.coe_basis]
refine repr_pow_isIntegral hB (fun i => ?_) hmin _ _
rw [← h, aeval_eq_sum_range, map_sum, Finset.sum_apply']
refine IsIntegral.sum _ fun n _ => ?_
rw [Algebra.smul_def, IsScalarTower.algebraMap_apply R K S, ← Algebra.smul_def,
LinearEquiv.map_smul, a... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 112 | 114 | theorem sub_mem_pNilradical_iff_pow_expChar_pow_eq {R : Type*} [CommRing R] {p : ℕ} [ExpChar R p]
{x y : R} : x - y ∈ pNilradical R p ↔ ∃ n : ℕ, x ^ p ^ n = y ^ p ^ n := by |
simp_rw [mem_pNilradical, sub_pow_expChar_pow, sub_eq_zero]
|
/-
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 | 100 | 122 | theorem quadraticCharFun_mul (a b : F) :
quadraticCharFun F (a * b) = quadraticCharFun F a * quadraticCharFun F b := by |
by_cases ha : a = 0
· rw [ha, zero_mul, quadraticCharFun_zero, zero_mul]
-- now `a ≠ 0`
by_cases hb : b = 0
· rw [hb, mul_zero, quadraticCharFun_zero, mul_zero]
-- now `a ≠ 0` and `b ≠ 0`
have hab := mul_ne_zero ha hb
by_cases hF : ringChar F = 2
·-- case `ringChar F = 2`
rw [quadraticCharFun_eq_... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 634 | 637 | theorem lintegral_of_isEmpty {α} [MeasurableSpace α] [IsEmpty α] (μ : Measure α) (f : α → ℝ≥0∞) :
∫⁻ x, f x ∂μ = 0 := by |
have : Subsingleton (Measure α) := inferInstance
convert lintegral_zero_measure f
|
/-
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.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.Cat... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 660 | 727 | theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
{c₁ : Cofan fun i : Fin n ↦ f i.succ} {c₂ : BinaryCofan (f 0) c₁.pt}
(t₁ : IsVanKampenColimit c₁) (t₂ : IsVanKampenColimit c₂)
[∀ {Z} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i]
[HasFiniteCoproducts C] :
IsVanKampenColimit (extendCofan ... |
intro F c α i e hα
refine ⟨?_, isUniversalColimit_extendCofan f t₁.isUniversal t₂.isUniversal c α i e hα⟩
intro ⟨Hc⟩ ⟨j⟩
have t₂' := (@t₂ (pair (F.obj ⟨0⟩) (∐ fun (j : Fin n) ↦ F.obj ⟨j.succ⟩))
(BinaryCofan.mk (P := c.pt) (c.ι.app _) (Sigma.desc fun b ↦ c.ι.app _))
(mapPair (α.app _) (Sigma.desc fun b ... |
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 125 | 126 | theorem card [NeZero n] : Fintype.card (DihedralGroup n) = 2 * n := by |
rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison, Jakob von Raumer, Joël Riou
-/
import Mathlib.CategoryTheory.Preadditive.ProjectiveResolution
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathl... | Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean | 268 | 279 | theorem exact_d_f {X Y : C} (f : X ⟶ Y) :
(ShortComplex.mk (d f) f (by simp)).Exact := by |
let α : ShortComplex.mk (d f) f (by simp) ⟶ ShortComplex.mk (kernel.ι f) f (by simp) :=
{ τ₁ := Projective.π _
τ₂ := 𝟙 _
τ₃ := 𝟙 _ }
have : Epi α.τ₁ := by dsimp; infer_instance
have : IsIso α.τ₂ := by dsimp; infer_instance
have : Mono α.τ₃ := by dsimp; infer_instance
rw [ShortComplex.exact_... |
/-
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.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 569 | 574 | theorem le_induced_generateFrom {α β} [t : TopologicalSpace α] {b : Set (Set β)} {f : α → β}
(h : ∀ a : Set β, a ∈ b → IsOpen (f ⁻¹' a)) : t ≤ induced f (generateFrom b) := by |
rw [induced_generateFrom_eq]
apply le_generateFrom
simp only [mem_image, and_imp, forall_apply_eq_imp_iff₂, exists_imp]
exact h
|
/-
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 | 210 | 212 | theorem lift_mrange_le {N} [Monoid N] (f : ∀ i, M i →* N) {s : Submonoid N} :
MonoidHom.mrange (lift f) ≤ s ↔ ∀ i, MonoidHom.mrange (f i) ≤ s := by |
simp [mrange_eq_iSup]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac6... | Mathlib/NumberTheory/RamificationInertia.lean | 626 | 641 | theorem rank_pow_quot [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime] (hP0 : P ≠ ⊥)
(i : ℕ) (hi : i ≤ e) :
Module.rank (R ⧸ p) (Ideal.map (Ideal.Quotient.mk (P ^ e)) (P ^ i)) =
(e - i) • Module.rank (R ⧸ p) (S ⧸ P) := by |
-- Porting note: Lean cannot figure out what to prove by itself
let Q : ℕ → Prop :=
fun i => Module.rank (R ⧸ p) { x // x ∈ map (Quotient.mk (P ^ e)) (P ^ i) }
= (e - i) • Module.rank (R ⧸ p) (S ⧸ P)
refine Nat.decreasingInduction' (P := Q) (fun j lt_e _le_j ih => ?_) hi ?_
· dsimp only [Q]
rw [ran... |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Cont... | Mathlib/Data/Real/Cardinality.lean | 93 | 96 | theorem summable_cantor_function (f : ℕ → Bool) (h1 : 0 ≤ c) (h2 : c < 1) :
Summable (cantorFunctionAux c f) := by |
apply (summable_geometric_of_lt_one h1 h2).summable_of_eq_zero_or_self
intro n; cases h : f n <;> simp [h]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 575 | 580 | theorem exists_le_of_tendsto_atTop [SemilatticeSup α] [Preorder β] {u : α → β}
(h : Tendsto u atTop atTop) (a : α) (b : β) : ∃ a' ≥ a, b ≤ u a' := by |
have : Nonempty α := ⟨a⟩
have : ∀ᶠ x in atTop, a ≤ x ∧ b ≤ u x :=
(eventually_ge_atTop a).and (h.eventually <| eventually_ge_atTop b)
exact this.exists
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 255 | 257 | theorem inter_product [DecidableEq α] [DecidableEq β] : (s ∩ s') ×ˢ t = s ×ˢ t ∩ s' ×ˢ t := by |
ext ⟨x, y⟩
simp only [← and_and_right, mem_inter, mem_product]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 731 | 732 | theorem div_le_one' : a / b ≤ 1 ↔ a ≤ b := by |
rw [← mul_le_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
|
/-
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 | 353 | 358 | theorem bot_lt_ker_of_det_eq_zero {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] {f : M →ₗ[𝕜] M}
(hf : LinearMap.det f = 0) : ⊥ < LinearMap.ker f := by |
have : FiniteDimensional 𝕜 M := by simp [f.finiteDimensional_of_det_ne_one, hf]
contrapose hf
simp only [bot_lt_iff_ne_bot, Classical.not_not, ← isUnit_iff_ker_eq_bot] at hf
exact isUnit_iff_ne_zero.1 (f.isUnit_det hf)
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 312 | 323 | theorem nnnorm_eq_sup_normAtPlace (x : E K) :
‖x‖₊ = univ.sup fun w ↦ ⟨normAtPlace w x, normAtPlace_nonneg w x⟩ := by |
rw [show (univ : Finset (InfinitePlace K)) = (univ.image
(fun w : {w : InfinitePlace K // IsReal w} ↦ w.1)) ∪
(univ.image (fun w : {w : InfinitePlace K // IsComplex w} ↦ w.1))
by ext; simp [isReal_or_isComplex], sup_union, univ.sup_image, univ.sup_image, sup_eq_max,
Prod.nnnorm_def', Pi.nnnorm_def, P... |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 488 | 491 | theorem norm_iteratedFDeriv_le_seminorm (f : 𝓢(E, F)) (n : ℕ) (x₀ : E) :
‖iteratedFDeriv ℝ n f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 n) f := by |
have := SchwartzMap.le_seminorm 𝕜 0 n f x₀
rwa [pow_zero, one_mul] at this
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomi... | Mathlib/Algebra/Polynomial/Taylor.lean | 137 | 142 | theorem eq_zero_of_hasseDeriv_eq_zero {R} [CommRing R] (f : R[X]) (r : R)
(h : ∀ k, (hasseDeriv k f).eval r = 0) : f = 0 := by |
apply taylor_injective r
rw [LinearMap.map_zero]
ext k
simp only [taylor_coeff, h, coeff_zero]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 46 | 50 | theorem oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arccos (‖y‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two h
|
/-
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 | 448 | 451 | theorem X_sub_C_pow_dvd_iff {p : R[X]} {t : R} {n : ℕ} :
(X - C t) ^ n ∣ p ↔ X ^ n ∣ p.comp (X + C t) := by |
convert (map_dvd_iff <| algEquivAevalXAddC t).symm using 2
simp [C_eq_algebraMap]
|
/-
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.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 210 | 212 | theorem mul_inv (t : TransvectionStruct n R) : t.toMatrix * t.inv.toMatrix = 1 := by |
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
|
/-
Copyright (c) 2022 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.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-com... | Mathlib/Algebra/Lie/Normalizer.lean | 138 | 141 | theorem mem_normalizer_iff (x : L) : x ∈ H.normalizer ↔ ∀ y : L, y ∈ H → ⁅x, y⁆ ∈ H := by |
rw [mem_normalizer_iff']
refine forall₂_congr fun y hy => ?_
rw [← lie_skew, neg_mem_iff (G := L)]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 137 | 145 | theorem mdifferentiableWithinAt_iff {f : M → M'} {s : Set M} {x : M} :
MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧
DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f)
((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s) ((extChartAt I x) x) := by |
rw [mdifferentiableWithinAt_iff']
refine and_congr Iff.rfl (exists_congr fun f' => ?_)
rw [inter_comm]
simp only [HasFDerivWithinAt, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq]
|
/-
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 | 215 | 218 | theorem noncommProd_eq_prod {α : Type*} [CommMonoid α] (s : Multiset α) :
(noncommProd s fun _ _ _ _ _ => Commute.all _ _) = prod s := by |
induction s using Quotient.inductionOn
simp
|
/-
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, Anne Baanen
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Finsupp
#align_import algebra.big_operators.associated from "leanp... | Mathlib/Algebra/BigOperators/Associated.lean | 159 | 168 | theorem exists_mem_multiset_le_of_prime {s : Multiset (Associates α)} {p : Associates α}
(hp : Prime p) : p ≤ s.prod → ∃ a ∈ s, p ≤ a :=
Multiset.induction_on s (fun ⟨d, Eq⟩ => (hp.ne_one (mul_eq_one_iff.1 Eq.symm).1).elim)
fun a s ih h =>
have : p ≤ a * s.prod := by | simpa using h
match Prime.le_or_le hp this with
| Or.inl h => ⟨a, Multiset.mem_cons_self a s, h⟩
| Or.inr h =>
let ⟨a, has, h⟩ := ih h
⟨a, Multiset.mem_cons_of_mem has, h⟩
|
/-
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 | 152 | 155 | theorem div_le_of_nonneg_of_le_mul (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c := by |
rcases eq_or_lt_of_le hb with (rfl | hb')
· simp only [div_zero, hc]
· rwa [div_le_iff hb']
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 411 | 416 | theorem finset_sup_le_sum (p : ι → Seminorm 𝕜 E) (s : Finset ι) : s.sup p ≤ ∑ i ∈ s, p i := by |
classical
refine Finset.sup_le_iff.mpr ?_
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
|
/-
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.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | Mathlib/Topology/Connected/LocallyConnected.lean | 135 | 142 | theorem OpenEmbedding.locallyConnectedSpace [LocallyConnectedSpace α] [TopologicalSpace β]
{f : β → α} (h : OpenEmbedding f) : LocallyConnectedSpace β := by |
refine locallyConnectedSpace_of_connected_bases (fun _ s ↦ f ⁻¹' s)
(fun x s ↦ (IsOpen s ∧ f x ∈ s ∧ IsConnected s) ∧ s ⊆ range f) (fun x ↦ ?_)
(fun x s hxs ↦ hxs.1.2.2.isPreconnected.preimage_of_isOpenMap h.inj h.isOpenMap hxs.2)
rw [h.nhds_eq_comap]
exact LocallyConnectedSpace.open_connected_basis (f x... |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 220 | 226 | theorem smeval_assoc_X_pow (m n : ℕ) :
p.smeval x * x ^ m * x ^ n = p.smeval x * (x ^ m * x ^ n) := by |
induction p using Polynomial.induction_on' with
| h_add p q ph qh =>
simp only [smeval_add, ph, qh, add_mul]
| h_monomial n a =>
rw [smeval_monomial, smul_mul_assoc, smul_mul_assoc, npow_mul_assoc, ← smul_mul_assoc]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 522 | 527 | theorem IsHermitian.fromBlocks₂₂ [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜)
{D : Matrix n n 𝕜} (hD : D.IsHermitian) :
(Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (A - B * D⁻¹ * Bᴴ).IsHermitian := by |
rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert IsHermitian.fromBlocks₁₁ _ _ hD <;> simp
|
/-
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 | 1,095 | 1,097 | theorem continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by |
simp [ContinuousOn, continuousWithinAt_iff']
|
/-
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 | 1,410 | 1,410 | theorem neg_le_iff {x y : PGame} : -y ≤ x ↔ -x ≤ y := by | rw [← neg_neg x, neg_le_neg_iff, neg_neg]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 769 | 769 | theorem zneg_toZNum (n : Num) : -n.toZNum = n.toZNumNeg := by | cases n <;> rfl
|
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 99 | 109 | theorem mem_essImage_of_unit_isSplitMono [Reflective i] {A : C}
[IsSplitMono ((reflectorAdjunction i).unit.app A)] : A ∈ i.essImage := by |
let η : 𝟭 C ⟶ reflector i ⋙ i := (reflectorAdjunction i).unit
haveI : IsIso (η.app (i.obj ((reflector i).obj A))) :=
Functor.essImage.unit_isIso ((i.obj_mem_essImage _))
have : Epi (η.app A) := by
refine @epi_of_epi _ _ _ _ _ (retraction (η.app A)) (η.app A) ?_
rw [show retraction _ ≫ η.app A = _ fr... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 189 | 192 | theorem lipschitzExtensionConstant_pos (E' : Type*) [NormedAddCommGroup E'] [NormedSpace ℝ E']
[FiniteDimensional ℝ E'] : 0 < lipschitzExtensionConstant E' := by |
rw [lipschitzExtensionConstant]
exact zero_lt_one.trans_le (le_max_right _ _)
|
/-
Copyright (c) 2019 Zhouhang Zhou. 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.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 987 | 995 | theorem HasFiniteIntegral.tendsto_setIntegral_nhds_zero {ι} {f : α → G}
(hf : HasFiniteIntegral f μ) {l : Filter ι} {s : ι → Set α} (hs : Tendsto (μ ∘ s) l (𝓝 0)) :
Tendsto (fun i => ∫ x in s i, f x ∂μ) l (𝓝 0) := by |
rw [tendsto_zero_iff_norm_tendsto_zero]
simp_rw [← coe_nnnorm, ← NNReal.coe_zero, NNReal.tendsto_coe, ← ENNReal.tendsto_coe,
ENNReal.coe_zero]
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(tendsto_set_lintegral_zero (ne_of_lt hf) hs) (fun i => zero_le _)
fun i => ennnorm_integra... |
/-
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.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
#align_import algebra.order.hom.basic from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf... | Mathlib/Algebra/Order/Hom/Basic.lean | 248 | 248 | theorem map_div_rev : f (x / y) = f (y / x) := by | rw [← inv_div, map_inv_eq_map]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series... | Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 117 | 128 | theorem ofPowerSeries_C (r : R) : ofPowerSeries Γ R (PowerSeries.C R r) = HahnSeries.C r := by |
ext n
simp only [ofPowerSeries_apply, C, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, ne_eq,
single_coeff]
split_ifs with hn
· subst hn
convert @embDomain_coeff ℕ R _ _ Γ _ _ _ 0 <;> simp
· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff... |
/-
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.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "lean... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 394 | 397 | theorem measure_condCDF_Iic (ρ : Measure (α × ℝ)) (a : α) (x : ℝ) :
(condCDF ρ a).measure (Iic x) = ENNReal.ofReal (condCDF ρ a x) := by |
rw [← sub_zero (condCDF ρ a x)]
exact (condCDF ρ a).measure_Iic (tendsto_condCDF_atBot ρ a) _
|
/-
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.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 158 | 161 | theorem pullbackDiagonalMapIso_hom_snd :
(pullbackDiagonalMapIso f i i₁ i₂).hom ≫ pullback.snd = pullback.snd ≫ pullback.snd := by |
delta pullbackDiagonalMapIso
simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Ta... | Mathlib/RingTheory/Valuation/Basic.lean | 303 | 305 | theorem map_sub_le {x y g} (hx : v x ≤ g) (hy : v y ≤ g) : v (x - y) ≤ g := by |
rw [sub_eq_add_neg]
exact v.map_add_le hx (le_trans (le_of_eq (v.map_neg y)) hy)
|
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variable {α : Type*}
namespace List
... | Mathlib/Data/List/Iterate.lean | 21 | 22 | theorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by |
induction n generalizing a <;> simp [*]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprove... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 305 | 311 | theorem tendsto_iff_forall_lintegral_tendsto {γ : Type*} {F : Filter γ}
{μs : γ → ProbabilityMeasure Ω} {μ : ProbabilityMeasure Ω} :
Tendsto μs F (𝓝 μ) ↔
∀ f : Ω →ᵇ ℝ≥0,
Tendsto (fun i => ∫⁻ ω, f ω ∂(μs i : Measure Ω)) F (𝓝 (∫⁻ ω, f ω ∂(μ : Measure Ω))) := by |
rw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]
exact FiniteMeasure.tendsto_iff_forall_lintegral_tendsto
|
/-
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.Nonneg.Ring
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Int.Lemmas
#align_import data.rat.nnrat fr... | Mathlib/Data/NNRat/Defs.lean | 319 | 320 | theorem toNNRat_lt_iff_lt_coe {p : ℚ≥0} (hq : 0 ≤ q) : toNNRat q < p ↔ q < ↑p := by |
rw [← coe_lt_coe, Rat.coe_toNNRat q hq]
|
/-
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.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
/-!
# Cycles of a li... | Mathlib/Data/List/Cycle.lean | 940 | 941 | theorem chain_singleton (r : α → α → Prop) (a : α) : Chain r [a] ↔ r a a := by |
rw [chain_coe_cons, nil_append, List.chain_singleton]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.Pi
im... | Mathlib/Algebra/Module/LinearMap/Basic.lean | 170 | 171 | theorem map_smul_inv {σ' : S →+* R} [RingHomInvPair σ σ'] (c : S) (x : M) :
c • f x = f (σ' c • x) := by | simp [map_smulₛₗ _]
|
/-
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.Geometry.RingedSpace.PresheafedSpace
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Topology.Sheaves.Limits
import Mathlib.Categor... | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | 70 | 79 | theorem map_comp_c_app (F : J ⥤ PresheafedSpace.{_, _, v} C) {j₁ j₂ j₃}
(f : j₁ ⟶ j₂) (g : j₂ ⟶ j₃) (U) :
(F.map (f ≫ g)).c.app (op U) =
(F.map g).c.app (op U) ≫
(pushforwardMap (F.map g).base (F.map f).c).app (op U) ≫
(Pushforward.comp (F.obj j₁).presheaf (F.map f).base (F.map g).base).... |
cases U
simp [PresheafedSpace.congr_app (F.map_comp f g)]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.c... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 299 | 301 | theorem HasDerivAtFilter.sub (hf : HasDerivAtFilter f f' x L) (hg : HasDerivAtFilter g g' x L) :
HasDerivAtFilter (fun x => f x - g x) (f' - g') x L := by |
simpa only [sub_eq_add_neg] using hf.add hg.neg
|
/-
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.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 338 | 340 | theorem comap_bracket_incl_of_le {I₁ I₂ : LieIdeal R L} (h₁ : I₁ ≤ I) (h₂ : I₂ ≤ I) :
⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I₁, I₂⁆ := by |
rw [comap_bracket_incl]; rw [← inf_eq_right] at h₁ h₂; rw [h₁, h₂]
|
/-
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.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-communi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 140 | 145 | theorem CNF_lt_snd {b o : Ordinal.{u}} {x : Ordinal × Ordinal} : x ∈ CNF b o → 0 < x.2 := by |
refine CNFRec b (by simp) (fun o ho IH ↦ ?_) o
rw [CNF_ne_zero ho]
rintro (h | ⟨_, h⟩)
· exact div_opow_log_pos b ho
· exact IH h
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 331 | 336 | theorem separable_map {S} [CommRing S] [Nontrivial S] (f : F →+* S) {p : F[X]} :
(p.map f).Separable ↔ p.Separable := by |
refine ⟨fun H ↦ ?_, fun H ↦ H.map⟩
obtain ⟨m, hm⟩ := Ideal.exists_maximal S
have := Separable.map H (f := Ideal.Quotient.mk m)
rwa [map_map, separable_def, derivative_map, isCoprime_map] at this
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 71 | 81 | theorem intDegree_mul {x y : RatFunc K} (hx : x ≠ 0) (hy : y ≠ 0) :
intDegree (x * y) = intDegree x + intDegree y := by |
simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]
norm_cast
rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←
Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))
(mul_ne_zero x.denom_ne_zero y.denom_ne_zero),
← Polynomial.... |
/-
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 | 561 | 563 | theorem integrable_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → β} {μ : ι → Measure α}
{s : Finset ι} : Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i) := by |
induction s using Finset.induction_on <;> simp [*]
|
/-
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.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 95 | 98 | theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
(S† ∘L T ∘L S).IsPositive := by |
convert hT.conj_adjoint (S†)
rw [adjoint_adjoint]
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 918 | 922 | theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
p.closedBall x r = ∅ := by |
ext
rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false_iff, not_le]
exact hr.trans_le (apply_nonneg _ _)
|
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.R... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 341 | 351 | theorem diagonalHomEquiv_apply (f : Rep.ofMulAction k G (Fin (n + 1) → G) ⟶ A) (x : Fin n → G) :
diagonalHomEquiv n A f x = f.hom (Finsupp.single (Fin.partialProd x) 1) := by |
/- Porting note (#11039): broken proof was
unfold diagonalHomEquiv
simpa only [LinearEquiv.trans_apply, Rep.leftRegularHomEquiv_apply,
MonoidalClosed.linearHomEquivComm_hom, Finsupp.llift_symm_apply, TensorProduct.curry_apply,
Linear.homCongr_apply, Iso.refl_hom, Iso.trans_inv, Action.comp_hom, ModuleCat.c... |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sebastien Gouezel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Ba... | Mathlib/Topology/VectorBundle/Basic.lean | 521 | 530 | theorem symm_apply_eq_mk_continuousLinearEquivAt_symm (e : Trivialization F (π F E)) [e.IsLinear R]
(b : B) (hb : b ∈ e.baseSet) (z : F) :
e.toPartialHomeomorph.symm ⟨b, z⟩ = ⟨b, (e.continuousLinearEquivAt R b hb).symm z⟩ := by |
have h : (b, z) ∈ e.target := by
rw [e.target_eq]
exact ⟨hb, mem_univ _⟩
apply e.injOn (e.map_target h)
· simpa only [e.source_eq, mem_preimage]
· simp_rw [e.right_inv h, coe_coe, e.apply_eq_prod_continuousLinearEquivAt R b hb,
ContinuousLinearEquiv.apply_symm_apply]
|
/-
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.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 50 | 51 | theorem birkhoffAverage_one (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 1 x = g x := by | simp [birkhoffAverage]
|
/-
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.Data.Set.Function
import Mathlib.Logic.Function.Iterate
import Mathlib.GroupTheory.Perm.Basic
#align_import dynamics.fixed_points.basic from "leanpr... | Mathlib/Dynamics/FixedPoints/Basic.lean | 97 | 100 | theorem preimage_iterate {s : Set α} (h : IsFixedPt (Set.preimage f) s) (n : ℕ) :
IsFixedPt (Set.preimage f^[n]) s := by |
rw [Set.preimage_iterate_eq]
exact h.iterate n
|
/-
Copyright (c) 2021 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Basic
#align_import group_theory.group_action.conj_act ... | Mathlib/GroupTheory/GroupAction/ConjAct.lean | 300 | 302 | theorem fixedPoints_eq_center : fixedPoints (ConjAct G) G = center G := by |
ext x
simp [mem_center_iff, smul_def, mul_inv_eq_iff_eq_mul]
|
/-
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.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 809 | 810 | theorem LinearMap.toMatrixAlgEquiv_id : LinearMap.toMatrixAlgEquiv v₁ id = 1 := by |
simp_rw [LinearMap.toMatrixAlgEquiv, AlgEquiv.ofLinearEquiv_apply, LinearMap.toMatrix_id]
|
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.fiber_bundle.constructions from "leanprove... | Mathlib/Topology/FiberBundle/Constructions.lean | 322 | 326 | theorem Pullback.continuous_totalSpaceMk [∀ x, TopologicalSpace (E x)] [FiberBundle F E]
{f : B' → B} {x : B'} : Continuous (@TotalSpace.mk _ F (f *ᵖ E) x) := by |
simp only [continuous_iff_le_induced, Pullback.TotalSpace.topologicalSpace, induced_compose,
induced_inf, Function.comp, induced_const, top_inf_eq, pullbackTopology_def]
exact le_of_eq (FiberBundle.totalSpaceMk_inducing F E (f x)).induced
|
/-
Copyright (c) 2024 Lean FRO LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.Catego... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 260 | 263 | theorem tensorObj_comul' (A B : Comon_ C) :
(A ⊗ B).comul =
(A.comul ⊗ B.comul) ≫ (tensor_μ Cᵒᵖ (op A.X, op B.X) (op A.X, op B.X)).unop := by |
rfl
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
/-!
# The... | Mathlib/CategoryTheory/Comma/Arrow.lean | 171 | 174 | theorem iso_w {f g : Arrow T} (e : f ≅ g) : g.hom = e.inv.left ≫ f.hom ≫ e.hom.right := by |
have eq := Arrow.hom.congr_right e.inv_hom_id
rw [Arrow.comp_right, Arrow.id_right] at eq
erw [Arrow.w_assoc, eq, Category.comp_id]
|
/-
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 | 1,461 | 1,461 | theorem neg_equiv_zero_iff {x : PGame} : (-x ≈ 0) ↔ (x ≈ 0) := by | rw [neg_equiv_iff, neg_zero]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 838 | 842 | theorem centroid_univ (s : Finset P) : univ.centroid k ((↑) : s → P) = s.centroid k id := by |
rw [centroid, centroid, ← s.attach_affineCombination_coe]
congr
ext
simp
|
/-
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 | 244 | 248 | theorem boxProd_degree (x : α × β)
[Fintype (G.neighborSet x.1)] [Fintype (H.neighborSet x.2)] [Fintype ((G □ H).neighborSet x)] :
(G □ H).degree x = G.degree x.1 + H.degree x.2 := by |
rw [degree, degree, degree, boxProd_neighborFinset, Finset.card_disjUnion]
simp_rw [Finset.card_product, Finset.card_singleton, mul_one, one_mul]
|
/-
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.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
/-!
# Cycles of a li... | Mathlib/Data/List/Cycle.lean | 619 | 629 | theorem nontrivial_coe_nodup_iff {l : List α} (hl : l.Nodup) :
Nontrivial (l : Cycle α) ↔ 2 ≤ l.length := by |
rw [Nontrivial]
rcases l with (_ | ⟨hd, _ | ⟨hd', tl⟩⟩)
· simp
· simp
· simp only [mem_cons, exists_prop, mem_coe_iff, List.length, Ne, Nat.succ_le_succ_iff,
Nat.zero_le, iff_true_iff]
refine ⟨hd, hd', ?_, by simp⟩
simp only [not_or, mem_cons, nodup_cons] at hl
exact hl.left.left
|
/-
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.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 176 | 184 | theorem convexOn_Gamma : ConvexOn ℝ (Ioi 0) Gamma := by |
refine
((convexOn_exp.subset (subset_univ _) ?_).comp convexOn_log_Gamma
(exp_monotone.monotoneOn _)).congr
fun x hx => exp_log (Gamma_pos_of_pos hx)
rw [convex_iff_isPreconnected]
refine isPreconnected_Ioi.image _ fun x hx => ContinuousAt.continuousWithinAt ?_
refine (differentiableAt_Gamm... |
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 1,941 | 1,943 | theorem insert_diff_of_mem (s) (h : a ∈ t) : insert a s \ t = s \ t := by |
ext
constructor <;> simp (config := { contextual := true }) [or_imp, h]
|
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