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 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 277 | 278 | theorem toIcoDiv_sub_zsmul (a b : α) (m : ℤ) : toIcoDiv hp a (b - m • p) = toIcoDiv hp a b - m := by |
rw [sub_eq_add_neg, ← neg_smul, toIcoDiv_add_zsmul, sub_eq_add_neg]
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_sp... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 116 | 126 | theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) := by |
simp_rw [exp, FormalMultilinearSeries.sum]
by_cases h : Summable (fun (n : ℕ) => (expSeries 𝕜 R n) fun x_1 ↦ fst x)
· refine (hasSum_expSeries_of_smul_comm 𝕜 x hx ?_).tsum_eq
exact h.hasSum
· rw [tsum_eq_zero_of_not_summable h, zero_smul, inr_zero, inl_zero, zero_add,
tsum_eq_zero_of_not_summable]
... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb8... | Mathlib/Data/Finset/Card.lean | 664 | 666 | theorem card_eq_one : s.card = 1 ↔ ∃ a, s = {a} := by |
cases s
simp only [Multiset.card_eq_one, Finset.card, ← val_inj, singleton_val]
|
/-
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.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 320 | 323 | theorem log_le_sub_one_of_pos {x : ℝ} (hx : 0 < x) : log x ≤ x - 1 := by |
rw [le_sub_iff_add_le]
convert add_one_le_exp (log x)
rw [exp_log hx]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.subobject.factor_thru from ... | Mathlib/CategoryTheory/Subobject/FactorThru.lean | 201 | 206 | theorem factorThru_add_sub_factorThru_right {X Y : C} {P : Subobject Y} (f g : X ⟶ Y)
(w : P.Factors (f + g)) (wg : P.Factors g) :
P.factorThru (f + g) w - P.factorThru g wg =
P.factorThru f (factors_left_of_factors_add f g w wg) := by |
ext
simp
|
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.Cyclotomic.Rat
/-!
# Fermat Last Theorem in the case `n = 3`
The g... | Mathlib/NumberTheory/FLT/Three.lean | 94 | 114 | theorem fermatLastTheoremThree_of_three_dvd_only_c
(H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :
FermatLastTheoremFor 3 := by |
rw [fermatLastTheoremFor_iff_int]
refine fermatLastTheoremWith_of_fermatLastTheoremWith_coprime (fun a b c ha hb hc Hgcd hF ↦?_)
by_cases h1 : 3 ∣ a * b * c
swap
· exact fermatLastTheoremThree_case_1 h1 hF
rw [(prime_three).dvd_mul, (prime_three).dvd_mul] at h1
rw [← sub_eq_zero, sub_eq_add_neg, ← (show ... |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir, Yury Kudryashov
-/
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanpro... | Mathlib/Order/Filter/FilterProduct.lean | 171 | 172 | theorem const_abs [LinearOrderedAddCommGroup β] (x : β) : (↑|x| : β*) = |↑x| := by |
rw [abs_def, map_const]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 451 | 454 | theorem hasFDerivAt_apply (i : ι) (f : ∀ i, F' i) :
HasFDerivAt (𝕜:=𝕜) (fun f : ∀ i, F' i => f i) (proj i) f := by |
apply HasStrictFDerivAt.hasFDerivAt
apply hasStrictFDerivAt_apply
|
/-
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 | 549 | 582 | theorem mem_smul_iff {i j : ι} {m₁ : M i} {m₂ : M j} {w : Word M} :
⟨_, m₁⟩ ∈ (of m₂ • w).toList ↔
(¬i = j ∧ ⟨i, m₁⟩ ∈ w.toList)
∨ (m₁ ≠ 1 ∧ ∃ (hij : i = j),(⟨i, m₁⟩ ∈ w.toList.tail) ∨
(∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = hij ▸ (m₂ * m')) ∨
(w.fstIdx ≠ some j ∧ m₁ = hij ▸ m₂)) := by |
rw [of_smul_def, mem_rcons_iff, mem_equivPair_tail_iff, equivPair_head, or_assoc]
by_cases hij : i = j
· subst i
simp only [not_true, ne_eq, false_and, exists_prop, true_and, false_or]
by_cases hw : ⟨j, m₁⟩ ∈ w.toList.tail
· simp [hw, show m₁ ≠ 1 from w.ne_one _ (List.mem_of_mem_tail hw)]
· simp ... |
/-
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 | 420 | 421 | theorem imageMonoIsoSource_inv_ι [Mono f] : (imageMonoIsoSource f).inv ≫ image.ι f = f := by |
simp [imageMonoIsoSource]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,001 | 1,005 | theorem Dense.biUnion_uniformity_ball {s : Set α} {U : Set (α × α)} (hs : Dense s) (hU : U ∈ 𝓤 α) :
⋃ x ∈ s, ball x U = univ := by |
refine iUnion₂_eq_univ_iff.2 fun y => ?_
rcases hs.inter_nhds_nonempty (mem_nhds_right y hU) with ⟨x, hxs, hxy : (x, y) ∈ U⟩
exact ⟨x, hxs, hxy⟩
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 925 | 930 | theorem multiplicative_factorization' {β : Type*} [CommMonoid β] (f : ℕ → β)
(h_mult : ∀ x y : ℕ, Coprime x y → f (x * y) = f x * f y) (hf0 : f 0 = 1) (hf1 : f 1 = 1) :
f n = n.factorization.prod fun p k => f (p ^ k) := by |
obtain rfl | hn := eq_or_ne n 0
· simpa
· exact multiplicative_factorization _ h_mult hf1 hn
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 854 | 861 | theorem induction_on [Semiring k] [Monoid G] {p : MonoidAlgebra k G → Prop} (f : MonoidAlgebra k G)
(hM : ∀ g, p (of k G g)) (hadd : ∀ f g : MonoidAlgebra k G, p f → p g → p (f + g))
(hsmul : ∀ (r : k) (f), p f → p (r • f)) : p f := by |
refine Finsupp.induction_linear f ?_ (fun f g hf hg => hadd f g hf hg) fun g r => ?_
· simpa using hsmul 0 (of k G 1) (hM 1)
· convert hsmul r (of k G g) (hM g)
-- Porting note: Was `simp only`.
rw [of_apply, smul_single', mul_one]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,093 | 2,095 | theorem transpose_smul {R : Type*} [SMul R α] (c : R) (M : Matrix m n α) : (c • M)ᵀ = c • Mᵀ := by |
ext
rfl
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Eric Wieser
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.Tacti... | Mathlib/Algebra/Group/Pi/Basic.lean | 505 | 509 | theorem extend_div [Div γ] (f : α → β) (g₁ g₂ : α → γ) (e₁ e₂ : β → γ) :
Function.extend f (g₁ / g₂) (e₁ / e₂) = Function.extend f g₁ e₁ / Function.extend f g₂ e₂ := by |
classical
funext x
simp [Function.extend_def, apply_dite₂]
|
/-
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.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 120 | 135 | theorem mul_polyOfInterest_aux1 (n : ℕ) :
∑ i ∈ range (n + 1), (p : 𝕄) ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n := by |
simp only [wittPolyProd]
convert wittStructureInt_prop p (X (0 : Fin 2) * X 1) n using 1
· simp only [wittPolynomial, wittMul]
rw [AlgHom.map_sum]
congr 1 with i
congr 1
have hsupp : (Finsupp.single i (p ^ (n - i))).support = {i} := by
rw [Finsupp.support_eq_singleton]
simp only [and_... |
/-
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 | 318 | 319 | theorem inv_mul_lt_one_iff_lt : a * b⁻¹ < 1 ↔ a < b := by |
rw [← mul_lt_mul_iff_right b, inv_mul_cancel_right, one_mul]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 161 | 165 | theorem geom_sum_mul_add [Semiring α] (x : α) (n : ℕ) :
(∑ i ∈ range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n := by |
have := (Commute.one_right x).geom_sum₂_mul_add n
rw [one_pow, geom_sum₂_with_one] at this
exact this
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 231 | 231 | theorem div_ball_one : s / ball 1 δ = thickening δ s := by | simp [div_eq_mul_inv, mul_ball_one]
|
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Mathlib/Algebra/Ring/Ext.lean | 497 | 499 | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!... | Mathlib/InformationTheory/Hamming.lean | 61 | 67 | theorem hammingDist_triangle (x y z : ∀ i, β i) :
hammingDist x z ≤ hammingDist x y + hammingDist y z := by |
classical
unfold hammingDist
refine le_trans (card_mono ?_) (card_union_le _ _)
rw [← filter_or]
exact monotone_filter_right _ fun i h ↦ (h.ne_or_ne _).imp_right Ne.symm
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 611 | 613 | theorem Base.insert_dep (hB : M.Base B) (h : e ∈ M.E \ B) : M.Dep (insert e B) := by |
rw [← not_indep_iff (insert_subset h.1 hB.subset_ground)]
exact h.2 ∘ (fun hi ↦ insert_eq_self.mp (hB.eq_of_subset_indep hi (subset_insert e B)).symm)
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 3,188 | 3,190 | theorem map_diff [DecidableEq β] {f : α → β} (finj : Injective f) {l₁ l₂ : List α} :
map f (l₁.diff l₂) = (map f l₁).diff (map f l₂) := by |
simp only [diff_eq_foldl, foldl_map, map_foldl_erase finj]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 138 | 141 | theorem preimage_emetric_closedBall (h : Isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' EMetric.closedBall (f x) r = EMetric.closedBall x r := by |
ext y
simp [h.edist_eq]
|
/-
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.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-comm... | Mathlib/Analysis/Convex/Star.lean | 199 | 201 | theorem starConvex_singleton (x : E) : StarConvex 𝕜 x {x} := by |
rintro y (rfl : y = x) a b _ _ hab
exact Convex.combo_self hab _
|
/-
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.Algebra.CharZero.Lemmas
import Mathlib.Order.Interval.Finset.Basic
#align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd... | Mathlib/Data/Int/Interval.lean | 189 | 213 | theorem image_Ico_emod (n a : ℤ) (h : 0 ≤ a) : (Ico n (n + a)).image (· % a) = Ico 0 a := by |
obtain rfl | ha := eq_or_lt_of_le h
· simp
ext i
simp only [mem_image, mem_range, mem_Ico]
constructor
· rintro ⟨i, _, rfl⟩
exact ⟨emod_nonneg i ha.ne', emod_lt_of_pos i ha⟩
intro hia
have hn := Int.emod_add_ediv n a
obtain hi | hi := lt_or_le i (n % a)
· refine ⟨i + a * (n / a + 1), ⟨?_, ?_⟩, ... |
/-
Copyright (c) 2024 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
/-!
# Continuous affine equivalences
In this file, we define c... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 84 | 87 | theorem coe_injective : Function.Injective ((↑) : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) := by |
intro e e' H
cases e
congr
|
/-
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 | 248 | 251 | theorem iSup₂_lintegral_le {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
⨆ (i) (j), ∫⁻ a, f i j a ∂μ ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ := by |
convert (monotone_lintegral μ).le_map_iSup₂ f with a
simp only [iSup_apply]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.GradedObject
#align_import algebra.hom... | Mathlib/Algebra/Homology/Homology.lean | 276 | 280 | theorem boundariesToCycles'_naturality (i : ι) :
boundariesMap f i ≫ C₂.boundariesToCycles' i =
C₁.boundariesToCycles' i ≫ cycles'Map f i := by |
ext
simp
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import al... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 73 | 85 | theorem σ_comp_πSummand_id_eq_zero {n : ℕ} (i : Fin (n + 1)) :
X.σ i ≫ s.πSummand (IndexSet.id (op [n + 1])) = 0 := by |
apply s.hom_ext'
intro A
dsimp only [SimplicialObject.σ]
rw [comp_zero, s.cofan_inj_epi_naturality_assoc A (SimplexCategory.σ i).op,
cofan_inj_πSummand_eq_zero]
rw [ne_comm]
change ¬(A.epiComp (SimplexCategory.σ i).op).EqId
rw [IndexSet.eqId_iff_len_eq]
have h := SimplexCategory.len_le_of_epi (infe... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.ConcreteCategory.Bun... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean | 111 | 114 | theorem isZero_of_subsingleton (V : SemiNormedGroupCat) [Subsingleton V] : Limits.IsZero V := by |
refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩
· ext x; have : x = 0 := Subsingleton.elim _ _; simp only [this, map_zero]
· ext; apply Subsingleton.elim
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
/--
A `Heap` is the nodes of the pairing heap.
Each node have two pointers: `child` going to the first c... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 119 | 121 | theorem Heap.size_merge_node (le) (a₁ : α) (c₁ s₁ : Heap α) (a₂ : α) (c₂ s₂ : Heap α) :
(merge le (.node a₁ c₁ s₁) (.node a₂ c₂ s₂)).size = c₁.size + c₂.size + 2 := by |
unfold merge; dsimp; split <;> simp_arith [size]
|
/-
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 | 36 | 72 | theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Nat.Prime p) (hdiv : ¬p ∣ n)
(R : Type*) [CommRing R] :
expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· simp
haveI := NeZero.of_pos hnpos
suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, map_cyclotomic_int,
map_cyclotomic]
refine eq_of_monic_of_dvd_of_natDe... |
/-
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 | 622 | 625 | theorem haarMeasure_apply {K₀ : PositiveCompacts G} {s : Set G} (hs : MeasurableSet s) :
haarMeasure K₀ s = (haarContent K₀).outerMeasure s / (haarContent K₀).measure K₀ := by |
change ((haarContent K₀).measure K₀)⁻¹ * (haarContent K₀).measure s = _
simp only [hs, div_eq_mul_inv, mul_comm, Content.measure_apply]
|
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Probability/ConditionalProbability.lean | 207 | 209 | theorem cond_eq_inv_mul_cond_mul [IsFiniteMeasure μ]
(hms : MeasurableSet s) (hmt : MeasurableSet t) : μ[t|s] = (μ s)⁻¹ * μ[s|t] * μ t := by |
rw [mul_assoc, cond_mul_eq_inter μ hmt s, Set.inter_comm, cond_apply _ hms]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 208 | 237 | theorem tendsto_approxBounded_of_norm_le {β} {f : α → β} [NormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} (hf : StronglyMeasurable[m] f) {c : ℝ} {x : α} (hfx : ‖f x‖ ≤ c) :
Tendsto (fun n => hf.approxBounded c n x) atTop (𝓝 (f x)) := by |
have h_tendsto := hf.tendsto_approx x
simp only [StronglyMeasurable.approxBounded, SimpleFunc.coe_map, Function.comp_apply]
by_cases hfx0 : ‖f x‖ = 0
· rw [norm_eq_zero] at hfx0
rw [hfx0] at h_tendsto ⊢
have h_tendsto_norm : Tendsto (fun n => ‖hf.approx n x‖) atTop (𝓝 0) := by
convert h_tendsto.... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 535 | 539 | theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
(hx : x ∉ s) (hy : y ∈ s) : firstDiff x y ≤ longestPrefix x s := by |
rw [longestPrefix, le_tsub_iff_right]
· exact firstDiff_lt_shortestPrefixDiff hs hx hy
· exact shortestPrefixDiff_pos hs ⟨y, hy⟩ hx
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 594 | 595 | theorem natCast_zmod_eq_zero_iff_dvd (a b : ℕ) : (a : ZMod b) = 0 ↔ b ∣ a := by |
rw [← Nat.cast_zero, ZMod.natCast_eq_natCast_iff, Nat.modEq_zero_iff_dvd]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymp... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 42 | 46 | theorem pow_div_pow_eventuallyEq_atTop {p q : ℕ} :
(fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atTop] fun x => x ^ ((p : ℤ) - q) := by |
apply (eventually_gt_atTop (0 : 𝕜)).mono fun x hx => _
intro x hx
simp [zpow_sub₀ hx.ne']
|
/-
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.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 159 | 160 | theorem edgeDensity_empty_right (s : Finset α) : edgeDensity r s ∅ = 0 := by |
rw [edgeDensity, Finset.card_empty, Nat.cast_zero, mul_zero, div_zero]
|
/-
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.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community... | Mathlib/LinearAlgebra/StdBasis.lean | 333 | 353 | theorem stdBasis_eq_stdBasisMatrix (i : m) (j : n) [DecidableEq m] [DecidableEq n] :
stdBasis R m n (i, j) = stdBasisMatrix i j (1 : R) := by |
-- Porting note: `simp` fails to apply `Pi.basis_apply`
ext a b
by_cases hi : i = a <;> by_cases hj : j = b
· simp [stdBasis, hi, hj, Basis.coe_reindex, comp_apply, Equiv.sigmaEquivProd_symm_apply,
StdBasisMatrix.apply_same]
erw [Pi.basis_apply]
simp
· simp only [stdBasis, hi, Basis.coe_reindex... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 608 | 611 | theorem norm_ofSubsingleton_id_le [Subsingleton ι] (i : ι) :
‖ofSubsingleton 𝕜 G G i (.id _ _)‖ ≤ 1 := by |
rw [norm_ofSubsingleton]
apply ContinuousLinearMap.norm_id_le
|
/-
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.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2d... | Mathlib/Algebra/Lie/Abelian.lean | 91 | 96 | theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A := by |
have h₁ : Std.Commutative (α := A) (· * ·) ↔ ∀ a b : A, a * b = b * a :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
have h₂ : IsLieAbelian A ↔ ∀ a b : A, ⁅a, b⁆ = 0 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
simp only [h₁, h₂, LieRing.of_associative_ring_bracket, sub_eq_zero]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 424 | 430 | theorem card_support_eq_one : f.support.card = 1 ↔
∃ (k : ℕ) (x : R) (hx : x ≠ 0), f = C x * X ^ k := by |
refine ⟨fun h => ?_, ?_⟩
· obtain ⟨k, x, _, hx, rfl⟩ := card_support_eq.mp h
exact ⟨k 0, x 0, hx 0, Fin.sum_univ_one _⟩
· rintro ⟨k, x, hx, rfl⟩
rw [support_C_mul_X_pow k hx, card_singleton]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#a... | Mathlib/Data/Real/Archimedean.lean | 110 | 115 | theorem exists_isGLB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddBelow S) : ∃ x, IsGLB S x := by |
have hne' : (-S).Nonempty := Set.nonempty_neg.mpr hne
have hbdd' : BddAbove (-S) := bddAbove_neg.mpr hbdd
use -Classical.choose (Real.exists_isLUB hne' hbdd')
rw [← isLUB_neg]
exact Classical.choose_spec (Real.exists_isLUB hne' hbdd')
|
/-
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.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe8353... | Mathlib/Data/List/OfFn.lean | 168 | 177 | theorem ofFn_mul {m n} (f : Fin (m * n) → α) :
List.ofFn f = List.join (List.ofFn fun i : Fin m => List.ofFn fun j : Fin n => f ⟨i * n + j,
calc
↑i * n + j < (i + 1) * n :=
(Nat.add_lt_add_left j.prop _).trans_eq (by rw [Nat.add_mul, Nat.one_mul])
_ ≤ _ := Nat.mul_le_mul_right _ i.prop⟩) := ... |
induction' m with m IH
· simp [ofFn_zero, Nat.zero_mul, ofFn_zero, join]
· simp_rw [ofFn_succ', succ_mul, join_concat, ofFn_add, IH]
rfl
|
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Data... | Mathlib/Data/Complex/Module.lean | 471 | 473 | theorem realPart_apply_coe (a : A) : (ℜ a : A) = (2 : ℝ)⁻¹ • (a + star a) := by |
unfold realPart
simp only [selfAdjointPart_apply_coe, invOf_eq_inv]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#al... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 324 | 334 | theorem IsClosed.vadd_right_of_isCompact {s : Set V} {t : Set P} (hs : IsClosed s)
(ht : IsCompact t) : IsClosed (s +ᵥ t) := by |
-- This result is still true for any `AddTorsor` where `-ᵥ` is continuous,
-- but we don't yet have a nice way to state it.
refine IsSeqClosed.isClosed (fun u p husv hup ↦ ?_)
choose! a ha v hv hav using husv
rcases ht.isSeqCompact hv with ⟨q, hqt, φ, φ_mono, hφq⟩
refine ⟨p -ᵥ q, hs.mem_of_tendsto ((hup.co... |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 169 | 183 | theorem cylinder_eq_empty_iff [h_nonempty : Nonempty (∀ i, α i)] (s : Finset ι)
(S : Set (∀ i : s, α i)) :
cylinder s S = ∅ ↔ S = ∅ := by |
refine ⟨fun h ↦ ?_, fun h ↦ by (rw [h]; exact cylinder_empty _)⟩
by_contra hS
rw [← Ne, ← nonempty_iff_ne_empty] at hS
let f := hS.some
have hf : f ∈ S := hS.choose_spec
classical
let f' : ∀ i, α i := fun i ↦ if hi : i ∈ s then f ⟨i, hi⟩ else h_nonempty.some i
have hf' : f' ∈ cylinder s S := by
rw ... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 203 | 204 | theorem mul_T_assoc (f : R[T;T⁻¹]) (m n : ℤ) : f * T m * T n = f * T (m + n) := by |
simp [← T_add, mul_assoc]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb537... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 232 | 243 | theorem galActionHom_injective [Fact (p.Splits (algebraMap F E))] :
Function.Injective (galActionHom p E) := by |
rw [injective_iff_map_eq_one]
intro ϕ hϕ
ext (x hx)
have key := Equiv.Perm.ext_iff.mp hϕ (rootsEquivRoots p E ⟨x, hx⟩)
change
rootsEquivRoots p E (ϕ • (rootsEquivRoots p E).symm (rootsEquivRoots p E ⟨x, hx⟩)) =
rootsEquivRoots p E ⟨x, hx⟩
at key
rw [Equiv.symm_apply_apply] at key
exact Subt... |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Bilin... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 106 | 112 | theorem integralClosure_le_span_dualBasis [IsSeparable K L] {ι : Type*} [Fintype ι] [DecidableEq ι]
(b : Basis ι K L) (hb_int : ∀ i, IsIntegral A (b i)) [IsIntegrallyClosed A] :
Subalgebra.toSubmodule (integralClosure A L) ≤
Submodule.span A (Set.range <| (traceForm K L).dualBasis (traceForm_nondegenerate K... |
refine le_trans ?_ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)
intro x hx
exact ⟨⟨x, hx⟩, rfl⟩
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 299 | 308 | theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :
succ (log b x) = sInf { o : Ordinal | x < b ^ o } := by |
let t := sInf { o : Ordinal | x < b ^ o }
have : x < (b^t) := csInf_mem (log_nonempty hb)
rcases zero_or_succ_or_limit t with (h | h | h)
· refine ((one_le_iff_ne_zero.2 hx).not_lt ?_).elim
simpa only [h, opow_zero] using this
· rw [show log b x = pred t from log_def hb x, succ_pred_iff_is_succ.2 h]
· ... |
/-
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 | 144 | 144 | theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by | simp only [mem_Icc, and_true_iff, le_rfl]
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,123 | 1,124 | theorem le_pred_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ 0) : j ≤ pred i hi ↔ succ j ≤ i := by |
rw [← succ_le_succ_iff, succ_pred]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 739 | 740 | theorem inf_sup_right (a b c : α) : (a ⊔ b) ⊓ c = a ⊓ c ⊔ b ⊓ c := by |
simp only [inf_sup_left, inf_comm _ c, eq_self_iff_true]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 365 | 366 | theorem span_pair_comm {x y : α} : (span {x, y} : Ideal α) = span {y, x} := by |
simp only [span_insert, sup_comm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"... | Mathlib/Data/Sum/Basic.lean | 218 | 222 | theorem exists_of_isLeft_left (h₁ : LiftRel r s x y) (h₂ : x.isLeft) :
∃ a c, r a c ∧ x = inl a ∧ y = inl c := by |
rcases isLeft_iff.mp h₂ with ⟨_, rfl⟩
simp only [liftRel_iff, false_and, and_false, exists_false, or_false] at h₁
exact h₁
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subri... | Mathlib/LinearAlgebra/Ray.lean | 708 | 714 | theorem exists_pos_left_iff_sameRay_and_ne_zero (hx : x ≠ 0) :
(∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y ∧ y ≠ 0 := by |
constructor
· rintro ⟨r, hr, rfl⟩
simp [hx, hr.le, hr.ne']
· rintro ⟨hxy, hy⟩
exact (exists_pos_left_iff_sameRay hx hy).2 hxy
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# The rank of a linear map
## Main Definit... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 58 | 60 | theorem lift_rank_comp_le_right (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') :
Cardinal.lift.{v'} (rank (f.comp g)) ≤ Cardinal.lift.{v''} (rank g) := by |
rw [rank, rank, LinearMap.range_comp]; exact lift_rank_map_le _ _
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 415 | 440 | theorem isSheaf_of_sep (P : Cᵒᵖ ⥤ D)
(hsep :
∀ (X : C) (S : J.Cover X) (x y : P.obj (op X)),
(∀ I : S.Arrow, P.map I.f.op x = P.map I.f.op y) → x = y) :
Presheaf.IsSheaf J (J.plusObj P) := by |
rw [Presheaf.isSheaf_iff_multiequalizer]
intro X S
apply @isIso_of_reflects_iso _ _ _ _ _ _ _ (forget D) ?_
rw [isIso_iff_bijective]
constructor
· intro x y h
apply sep P S _ _
intro I
apply_fun Meq.equiv _ _ at h
apply_fun fun e => e I at h
convert h <;> erw [Meq.equiv_apply, ← comp_ap... |
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 112 | 112 | theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by | simp [laverage]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 432 | 434 | theorem finsum_weightedHomogeneousComponent :
(finsum fun m => weightedHomogeneousComponent w m φ) = φ := by |
rw [sum_weightedHomogeneousComponent]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 889 | 891 | theorem boundary_zero : (c.boundary ⟨0, c.card_boundaries_pos⟩ : Fin (n + 1)) = 0 := by |
rw [boundary, Finset.orderEmbOfFin_zero rfl c.card_boundaries_pos]
exact le_antisymm (Finset.min'_le _ _ c.zero_mem) (Fin.zero_le _)
|
/-
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 | 1,799 | 1,801 | theorem mem_closure_range_iff {e : β → α} {a : α} :
a ∈ closure (range e) ↔ ∀ ε > 0, ∃ k : β, dist a (e k) < ε := by |
simp only [mem_closure_iff, exists_range_iff]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 133 | 134 | theorem arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) := by |
rw [← arcsin_projIcc, projIcc_of_le_left _ hx, Subtype.coe_mk, arcsin_neg_one]
|
/-
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.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 114 | 115 | theorem sinh_log {x : ℝ} (hx : 0 < x) : sinh (log x) = (x - x⁻¹) / 2 := by |
rw [sinh_eq, exp_neg, exp_log hx]
|
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
/-... | Mathlib/Data/Real/Pi/Wallis.lean | 62 | 75 | theorem W_eq_factorial_ratio (n : ℕ) :
W n = 2 ^ (4 * n) * n ! ^ 4 / ((2 * n)! ^ 2 * (2 * n + 1)) := by |
induction' n with n IH
· simp only [W, prod_range_zero, Nat.factorial_zero, mul_zero, pow_zero,
algebraMap.coe_one, one_pow, mul_one, algebraMap.coe_zero, zero_add, div_self, Ne,
one_ne_zero, not_false_iff]
norm_num
· unfold W at IH ⊢
rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.d... |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Ta... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 562 | 564 | theorem map_id : map (fun i ↦ (LinearMap.id : s i →ₗ[R] s i)) = .id := by |
ext
simp only [LinearMap.compMultilinearMap_apply, map_tprod, LinearMap.id_coe, id_eq]
|
/-
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.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 444 | 444 | theorem sdiff_le_iff' : a \ b ≤ c ↔ a ≤ c ⊔ b := by | rw [sdiff_le_iff, sup_comm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 548 | 553 | theorem toNNReal_sInf (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toNNReal = sInf (ENNReal.toNNReal '' s) := by |
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
-- Porting note: `← sInf_image'` had to be replaced by `← image_eq_range` as the lemmas are used
-- in a different order.
simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Reid Barton
-/
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types f... | Mathlib/CategoryTheory/Limits/Types.lean | 585 | 589 | theorem colimit_sound' {j j' : J} {x : F.obj j} {x' : F.obj j'} {j'' : J}
(f : j ⟶ j'') (f' : j' ⟶ j'') (w : F.map f x = F.map f' x') :
colimit.ι F j x = colimit.ι F j' x' := by |
rw [← colimit.w _ f, ← colimit.w _ f']
rw [types_comp_apply, types_comp_apply, w]
|
/-
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
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 75 | 79 | theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
rw [closure_Ioo hab]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the language o... | Mathlib/Tactic/Abel.lean | 132 | 134 | theorem const_add_termg {α} [AddCommGroup α] (k n x a a') (h : k + a = a') :
k + @termg α _ n x a = termg n x a' := by |
simp [h.symm, termg, add_comm, add_assoc]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "lean... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 286 | 290 | theorem restrict_toMeasure_support [MeasurableSingletonClass α] (p : PMF α) :
Measure.restrict (toMeasure p) (support p) = toMeasure p := by |
ext s hs
apply (MeasureTheory.Measure.restrict_apply hs).trans
apply toMeasure_apply_inter_support p s hs p.support_countable.measurableSet
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 451 | 452 | theorem dualBasis_repr (l : Dual R M) (i : ι) : b.dualBasis.repr l i = l (b i) := by |
rw [← total_dualBasis b, Basis.total_repr b.dualBasis l]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 483 | 493 | theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by |
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add]
· rw [(ext rfl hi : x = x.re)]
rcases lt_trichotomy x.re 0 with (hr | hr | hr)
· rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ←
Real.Angle.coe_... |
/-
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.Topology.LocalAtTarget
import Mathlib.AlgebraicGeometry.Morphisms.Basic
#align_import algebraic_geometry.morphisms.open_immersion from "leanprover-community... | Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean | 34 | 38 | theorem isOpenImmersion_iff_stalk {f : X ⟶ Y} : IsOpenImmersion f ↔
OpenEmbedding f.1.base ∧ ∀ x, IsIso (PresheafedSpace.stalkMap f.1 x) := by |
constructor
· intro h; exact ⟨h.1, inferInstance⟩
· rintro ⟨h₁, h₂⟩; exact IsOpenImmersion.of_stalk_iso f h₁
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 192 | 194 | theorem Fintype.not_linearIndependent_iff [Fintype ι] :
¬LinearIndependent R v ↔ ∃ g : ι → R, ∑ i, g i • v i = 0 ∧ ∃ i, g i ≠ 0 := by |
simpa using not_iff_not.2 Fintype.linearIndependent_iff
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | Mathlib/RingTheory/Kaehler.lean | 242 | 261 | theorem KaehlerDifferential.span_range_derivation :
Submodule.span S (Set.range <| KaehlerDifferential.D R S) = ⊤ := by |
rw [_root_.eq_top_iff]
rintro x -
obtain ⟨⟨x, hx⟩, rfl⟩ := Ideal.toCotangent_surjective _ x
have : x ∈ (KaehlerDifferential.ideal R S).restrictScalars S := hx
rw [← KaehlerDifferential.submodule_span_range_eq_ideal] at this
suffices ∃ hx, (KaehlerDifferential.ideal R S).toCotangent ⟨x, hx⟩ ∈
Submodul... |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 359 | 361 | theorem ContinuousLinearEquiv.comp_contDiff_iff (e : F ≃L[𝕜] G) :
ContDiff 𝕜 n (e ∘ f) ↔ ContDiff 𝕜 n f := by |
simp only [← contDiffOn_univ, e.comp_contDiffOn_iff]
|
/-
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
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 1,168 | 1,173 | theorem coequalizerComparison_map_desc [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)]
{Z : C} {h : Y ⟶ Z} (w : f ≫ h = g ≫ h) :
coequalizerComparison f g G ≫ G.map (coequalizer.desc h w) =
coequalizer.desc (G.map h) (by simp only [← G.map_comp, w]) := by |
apply coequalizer.hom_ext
simp [← G.map_comp]
|
/-
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.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925... | Mathlib/SetTheory/Cardinal/Continuum.lean | 90 | 90 | theorem mk_set_nat : #(Set ℕ) = 𝔠 := by | simp
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 415 | 418 | theorem map_coe (γ : Path x y) {f : X → Y} (h : Continuous f) :
(γ.map h : I → Y) = f ∘ γ := by |
ext t
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder
#align_import order.disjoint from "leanprover-community/mathlib"@"22c4d2ff43714b6ff724b2745ccfdc0f236a4a76"
/-!
# Dis... | Mathlib/Order/Disjoint.lean | 648 | 648 | theorem eq_top_of_isCompl_bot (h : IsCompl x ⊥) : x = ⊤ := by | rw [← sup_bot_eq x, h.sup_eq_top]
|
/-
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 | 122 | 127 | theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 134 | 137 | theorem toGraph_compl_eq [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) :
h.compl.toGraph = h.toGraphᶜ := by |
ext v w
cases' h.zero_or_one v w with h h <;> by_cases hvw : v = w <;> simp [Matrix.compl, h, hvw]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 458 | 469 | theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K)
(h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) :
f = (basisModTorsion K).repr (Additive.ofMul ↑x) := by |
suffices Additive.ofMul ↑x = ∑ i, (f i) • (basisModTorsion K i) by
rw [← (basisModTorsion K).repr_sum_self f, ← this]
calc
Additive.ofMul ↑x
_ = ∑ i, (f i) • Additive.ofMul ↑(fundSystem K i) := by
rw [h, QuotientGroup.mk_mul, (QuotientGroup.eq_one_iff _).mpr hζ, one_mul,
QuotientG... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 221 | 234 | theorem comp_right_hasFDerivWithinAt_iff {f : F → G} {s : Set F} {x : E} {f' : F →L[𝕜] G} :
HasFDerivWithinAt (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) (iso ⁻¹' s) x ↔
HasFDerivWithinAt f f' s (iso x) := by |
refine ⟨fun H => ?_, fun H => H.comp x iso.hasFDerivWithinAt (mapsTo_preimage _ s)⟩
rw [← iso.symm_apply_apply x] at H
have A : f = (f ∘ iso) ∘ iso.symm := by
rw [Function.comp.assoc, iso.self_comp_symm]
rfl
have B : f' = (f'.comp (iso : E →L[𝕜] F)).comp (iso.symm : F →L[𝕜] E) := by
rw [Continuou... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_seri... | Mathlib/RingTheory/PowerSeries/Order.lean | 339 | 341 | theorem order_X_pow (n : ℕ) : order ((X : R⟦X⟧) ^ n) = n := by |
rw [X_pow_eq, order_monomial_of_ne_zero]
exact one_ne_zero
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 608 | 610 | theorem preimage_ball (h : α ≃ᵢ β) (x : β) (r : ℝ) :
h ⁻¹' Metric.ball x r = Metric.ball (h.symm x) r := by |
rw [← h.isometry.preimage_ball (h.symm x) r, h.apply_symm_apply]
|
/-
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, Mario Carneiro
-/
/-!
# Definitions and properties of `coprime`
-/
namespace Nat
/-!
### `coprime`
See also `nat.coprime_of_dvd` and `nat.coprime_o... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 65 | 75 | theorem exists_coprime (m n : Nat) :
∃ m' n', Coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n := by |
cases eq_zero_or_pos (gcd m n) with
| inl h0 =>
rw [gcd_eq_zero_iff] at h0
refine ⟨1, 1, gcd_one_left 1, ?_⟩
simp [h0]
| inr hpos =>
exact ⟨_, _, coprime_div_gcd_div_gcd hpos,
(Nat.div_mul_cancel (gcd_dvd_left m n)).symm,
(Nat.div_mul_cancel (gcd_dvd_right m n)).symm⟩
|
/-
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.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 304 | 312 | theorem isBigO_rpow_top_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : b < a)
(hf : f =O[atTop] (· ^ (-a))) :
(fun t : ℝ => log t • f t) =O[atTop] (· ^ (-b)) := by |
refine
((isLittleO_log_rpow_atTop (sub_pos.mpr hab)).isBigO.smul hf).congr'
(eventually_of_forall fun t => by rfl)
((eventually_gt_atTop 0).mp (eventually_of_forall fun t ht => ?_))
simp only
rw [smul_eq_mul, ← rpow_add ht, ← sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel_left]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 566 | 567 | theorem single_blocksFun {n : ℕ} (h : 0 < n) (i : Fin (single n h).length) :
(single n h).blocksFun i = n := by | simp [blocksFun, single, blocks, i.2]
|
/-
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.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 632 | 633 | theorem Perm.subtypeCongr.left_apply {a : ε} (h : p a) : ep.subtypeCongr en a = ep ⟨a, h⟩ := by |
simp [Perm.subtypeCongr.apply, h]
|
/-
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, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 437 | 441 | theorem FiniteDimensional.finrank_eq_zero_iff_isTorsion {R} [CommRing R] [StrongRankCondition R]
[IsDomain R] [Module R M] [Module.Finite R M] :
finrank R M = 0 ↔ Module.IsTorsion R M := by |
rw [← rank_eq_zero_iff_isTorsion (R := R), ← finrank_eq_rank]
norm_cast
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 274 | 279 | theorem IndepFun.integrable_exp_mul_add {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ)
(h_int_X : Integrable (fun ω => exp (t * X ω)) μ)
(h_int_Y : Integrable (fun ω => exp (t * Y ω)) μ) :
Integrable (fun ω => exp (t * (X + Y) ω)) μ := by |
simp_rw [Pi.add_apply, mul_add, exp_add]
exact (h_indep.exp_mul t t).integrable_mul h_int_X h_int_Y
|
/-
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 | 2,706 | 2,710 | theorem map_le_map_iff {f : α → β} (hf : Function.Injective f) {s t : Multiset α} :
s.map f ≤ t.map f ↔ s ≤ t := by |
classical
refine ⟨fun h => le_iff_count.mpr fun a => ?_, map_le_map⟩
simpa [count_map_eq_count' f _ hf] using le_iff_count.mp h (f a)
|
/-
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 | 709 | 710 | theorem Ici_eq_cons_Ioi (a : α) : Ici a = (Ioi a).cons a not_mem_Ioi_self := by |
classical rw [cons_eq_insert, Ioi_insert]
|
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanpro... | Mathlib/Algebra/Polynomial/Splits.lean | 481 | 491 | theorem prod_roots_eq_coeff_zero_of_monic_of_split {P : K[X]} (hmo : P.Monic)
(hP : P.Splits (RingHom.id K)) : coeff P 0 = (-1) ^ P.natDegree * P.roots.prod := by |
nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]
rw [coeff_zero_eq_eval_zero, eval_multiset_prod, Multiset.map_map]
simp_rw [Function.comp_apply, eval_sub, eval_X, zero_sub, eval_C]
conv_lhs =>
congr
congr
ext
rw [neg_eq_neg_one_mul]
simp only [splits_iff_card_roots.1 hP, neg_mul, one_mu... |
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