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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 358 | 359 | theorem testAgainstNN_zero (μ : FiniteMeasure Ω) : μ.testAgainstNN 0 = 0 := by |
simpa only [zero_mul] using μ.testAgainstNN_const 0
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 785 | 787 | theorem coe_bot : ((⊥ : NonUnitalStarSubalgebra R A) : Set A) = {0} := by |
simp only [Set.ext_iff, NonUnitalStarAlgebra.mem_bot, SetLike.mem_coe, Set.mem_singleton_iff,
iff_self_iff, forall_const]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 146 | 147 | theorem map₂_snoc : map₂ f (xs.snoc x) (ys.snoc y) = (map₂ f xs ys).snoc (f x y) := by |
induction xs, ys using Vector.inductionOn₂ <;> simp_all
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 862 | 864 | theorem contained_proj (o : Ordinal) : contained (π C (ord I · < o)) o := by |
intro x ⟨_, _, h⟩ j hj
aesop (add simp Proj)
|
/-
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 | 749 | 751 | theorem continuous_iSup_dom {t₁ : ι → TopologicalSpace α} {t₂ : TopologicalSpace β} :
Continuous[iSup t₁, t₂] f ↔ ∀ i, Continuous[t₁ i, t₂] f := by |
simp only [continuous_iff_le_induced, iSup_le_iff]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 969 | 978 | theorem cons_diff (a : α) (l₁ l₂ : List α) :
(a :: l₁).diff l₂ = if a ∈ l₂ then l₁.diff (l₂.erase a) else a :: l₁.diff l₂ := by |
induction l₂ generalizing l₁ with
| nil => rfl
| cons b l₂ ih =>
by_cases h : a = b
next => simp [*]
next =>
have := Ne.symm h
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 | 1,719 | 1,721 | theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by |
rw [← nhds_comap_dist a, tendsto_comap_iff, Function.comp_def]
|
/-
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 | 140 | 162 | theorem applyComposition_update (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n)
(j : Fin n) (v : Fin n → E) (z : E) :
p.applyComposition c (Function.update v j z) =
Function.update (p.applyComposition c v) (c.index j)
(p (c.blocksFun (c.index j))
(Function.update (v ∘ c.... |
ext k
by_cases h : k = c.index j
· rw [h]
let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
simp only [Function.update_same]
change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
let j' := c.invEmbedding j
suffices B : Function.update v j z ∘ r = Function.u... |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.PreservesLocallyBijec... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 67 | 82 | theorem coverPreserving : CoverPreserving J (e.locallyCoverDense J).inducedTopology e.functor where
cover_preserve {U S} h := by |
change _ ∈ J.sieves (e.inverse.obj (e.functor.obj U))
convert J.pullback_stable (e.unitInv.app U) h
ext Z f
rw [← Sieve.functorPushforward_comp]
simp only [Sieve.functorPushforward_apply, Presieve.functorPushforward, exists_and_left, id_obj,
comp_obj, Sieve.pullback_apply]
constructor
... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,299 | 1,299 | theorem sin_sub_pi_div_two (x : ℂ) : sin (x - π / 2) = -cos x := by | simp [sub_eq_add_neg, sin_add]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 247 | 248 | theorem relindex_top_right : H.relindex ⊤ = H.index := by |
rw [← relindex_mul_index (show H ≤ ⊤ from le_top), index_top, mul_one]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.Numb... | Mathlib/NumberTheory/PellMatiyasevic.lean | 944 | 1,013 | theorem eq_pow_of_pell {m n k} :
n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
2 * a * n = t + (n * n + 1) ∧ m < t ∧
n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by |
constructor
· rintro rfl
refine k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, ?_⟩
refine n.eq_zero_or_pos.imp (fun n0 : n = 0 ↦ n0.symm ▸ ⟨rfl, zero_pow hk.ne'⟩)
fun hn ↦ ⟨hn, ?_⟩
set w := max n k
have nw : n ≤ w := le_max_left _ _
have kw : k ≤ w := le_max_... |
/-
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 | 677 | 681 | theorem mem_prodMap {f : α →. γ} {g : β →. δ} {x : α × β} {y : γ × δ} :
y ∈ f.prodMap g x ↔ y.1 ∈ f x.1 ∧ y.2 ∈ g x.2 := by |
trans ∃ hp hq, (f x.1).get hp = y.1 ∧ (g x.2).get hq = y.2
· simp only [prodMap, Part.mem_mk_iff, And.exists, Prod.ext_iff]
· simp only [exists_and_left, exists_and_right, (· ∈ ·), Part.Mem]
|
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | Mathlib/LinearAlgebra/Matrix/Block.lean | 63 | 69 | theorem blockTriangular_reindex_iff {b : n → α} {e : m ≃ n} :
(reindex e e M).BlockTriangular b ↔ M.BlockTriangular (b ∘ e) := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· convert h.submatrix
simp only [reindex_apply, submatrix_submatrix, submatrix_id_id, Equiv.symm_comp_self]
· convert h.submatrix
simp only [comp.assoc b e e.symm, Equiv.self_comp_symm, comp_id]
|
/-
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 | 95 | 96 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by | cases a <;> rfl
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,183 | 1,183 | theorem trNat_zero : trNat 0 = [] := by | rw [trNat, Nat.cast_zero]; rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 560 | 572 | theorem support_gen_of_gen {S : Set (MonoidAlgebra R M)} (hS : Algebra.adjoin R S = ⊤) :
Algebra.adjoin R (⋃ f ∈ S, of R M '' (f.support : Set M)) = ⊤ := by |
refine le_antisymm le_top ?_
rw [← hS, adjoin_le_iff]
intro f hf
-- Porting note: ⋃ notation did not work here. Was
-- ⋃ (g : MonoidAlgebra R M) (H : g ∈ S), (of R M '' g.support)
have hincl : (of R M '' f.support) ⊆
Set.iUnion fun (g : MonoidAlgebra R M)
=> Set.iUnion fun (_ : g ∈ S) => (of ... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 218 | 219 | theorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by | simp only [← mul_apply, f.mul_inv, coe_one, id]
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#alig... | Mathlib/LinearAlgebra/Dimension/Free.lean | 111 | 118 | theorem nonempty_linearEquiv_of_lift_rank_eq
(cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :
Nonempty (M ≃ₗ[R] M') := by |
obtain ⟨⟨α, B⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
obtain ⟨⟨β, B'⟩⟩ := Module.Free.exists_basis (R := R) (M := M')
have : Cardinal.lift.{v', v} #α = Cardinal.lift.{v, v'} #β := by
rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']
exact (Cardinal.lift_mk_eq.{v, v', 0}.1 this).map (B.equiv B')
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.quotient_pi from "leanprover-community/mathlib"@"398f60f60... | Mathlib/LinearAlgebra/QuotientPi.lean | 42 | 46 | theorem piQuotientLift_mk [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
(q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (x : ∀ i, Ms i) :
(piQuotientLift p q f hf fun i => Quotient.mk (x i)) = Quotient.mk (lsum _ _ R f x) := by |
rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]
simp only [coe_proj, mapQ_apply, mkQ_apply, comp_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
-/
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da... | Mathlib/Data/Nat/GCD/BigOperators.lean | 52 | 54 | theorem coprime_fintype_prod_left_iff [Fintype ι] {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i, s i) x ↔ ∀ i, Coprime (s i) x := by |
simp [coprime_prod_left_iff]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 417 | 420 | theorem image_eq_image [HasImages V] [HasEqualizers V] {i i' j : ι} (r : c.Rel i j)
(r' : c.Rel i' j) : imageSubobject (C.d i j) = imageSubobject (C.d i' j) := by |
rw [← eqToHom_comp_d C r r']
apply imageSubobject_iso_comp
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 124 | 134 | theorem ortho_smul_right {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₂} {ha : a ≠ 0} :
IsOrtho B x y ↔ IsOrtho B x (a • y) := by |
dsimp only [IsOrtho]
constructor <;> intro H
· rw [map_smulₛₗ, H, smul_zero]
· rw [map_smulₛₗ, smul_eq_zero] at H
cases' H with H H
· simp at H
exfalso
exact ha H
· exact H
|
/-
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 | 1,526 | 1,528 | theorem map_concat (f : α → β) (a : α) (l : List α) :
map f (concat l a) = concat (map f l) (f a) := by |
induction l <;> [rfl; simp only [*, concat_eq_append, cons_append, map, map_append]]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 530 | 531 | theorem pullback_comp {f : Y ⟶ X} {g : Z ⟶ Y} (S : Sieve X) :
S.pullback (g ≫ f) = (S.pullback f).pullback g := by | simp [Sieve.ext_iff]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
/-!
# Flag of submodules defined by a basis
... | Mathlib/LinearAlgebra/Basis/Flag.lean | 78 | 81 | theorem flag_le_ker_coord (b : Basis (Fin n) R M) {k : Fin (n + 1)} {l : Fin n}
(h : k ≤ l.castSucc) : b.flag k ≤ LinearMap.ker (b.coord l) := by |
nontriviality R
exact b.flag_le_ker_coord_iff.2 h
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 557 | 564 | theorem norm_apply_le_norm (hp : p ≠ 0) (f : lp E p) (i : α) : ‖f i‖ ≤ ‖f‖ := by |
rcases eq_or_ne p ∞ with (rfl | hp')
· haveI : Nonempty α := ⟨i⟩
exact (isLUB_norm f).1 ⟨i, rfl⟩
have hp'' : 0 < p.toReal := ENNReal.toReal_pos hp hp'
have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i => Real.rpow_nonneg (norm_nonneg _) _
rw [← Real.rpow_le_rpow_iff (norm_nonneg _) (norm_nonneg' _) hp'']
conve... |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 295 | 304 | theorem WithSeminorms.hasBasis_zero_ball (hp : WithSeminorms p) :
(𝓝 (0 : E)).HasBasis
(fun sr : Finset ι × ℝ => 0 < sr.2) fun sr => (sr.1.sup p).ball 0 sr.2 := by |
refine ⟨fun V => ?_⟩
simp only [hp.hasBasis.mem_iff, SeminormFamily.basisSets_iff, Prod.exists]
constructor
· rintro ⟨-, ⟨s, r, hr, rfl⟩, hV⟩
exact ⟨s, r, hr, hV⟩
· rintro ⟨s, r, hr, hV⟩
exact ⟨_, ⟨s, r, hr, rfl⟩, hV⟩
|
/-
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 | 716 | 718 | theorem affineCombinationLineMapWeights_apply_right [DecidableEq ι] {i j : ι} (h : i ≠ j) (c : k) :
affineCombinationLineMapWeights i j c j = c := by |
simp [affineCombinationLineMapWeights, h.symm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 606 | 607 | theorem mem_toFinset {s : Set α} [Fintype s] {a : α} : a ∈ s.toFinset ↔ a ∈ s := by |
simp [toFinset]
|
/-
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 | 132 | 134 | theorem card_support_eraseLead' {c : ℕ} (fc : f.support.card = c + 1) :
f.eraseLead.support.card = c := by |
rw [card_support_eraseLead, fc, add_tsub_cancel_right]
|
/-
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.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 800 | 804 | theorem ofFractionRing_mk' (x : K[X]) (y : K[X]⁰) :
-- Porting note: I gave explicitly the argument `(FractionRing K[X])`
ofFractionRing (IsLocalization.mk' (FractionRing K[X]) x y) =
IsLocalization.mk' (RatFunc K) x y := by |
rw [IsFractionRing.mk'_eq_div, IsFractionRing.mk'_eq_div, ← mk_eq_div', ← mk_eq_div]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 206 | 212 | theorem HasDerivWithinAt.mul (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x) :
HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x := by |
have := (HasFDerivWithinAt.mul' hc hd).hasDerivWithinAt
rwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul, one_smul,
add_comm] at this... |
/-
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 | 930 | 960 | theorem lift_word_prod_nontrivial_of_not_empty {i j} (w : NeWord H i j) : lift f w.prod ≠ 1 := by |
classical
cases' hcard with hcard hcard
· obtain ⟨i, h1, h2⟩ := Cardinal.three_le hcard i j
exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w h1 h2
· cases' hcard with k hcard
by_cases hh : i = k <;> by_cases hl : j = k
· subst hh
subst hl
exact lift... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 331 | 335 | theorem natDegree_monomial [DecidableEq R] (i : ℕ) (r : R) :
natDegree (monomial i r) = if r = 0 then 0 else i := by |
split_ifs with hr
· simp [hr]
· rw [← C_mul_X_pow_eq_monomial, natDegree_C_mul_X_pow i r hr]
|
/-
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.Topology.MetricSpace.Algebra
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed.mul_action from "leanprover-community/mathlib"@"bc91ed... | Mathlib/Analysis/Normed/MulAction.lean | 37 | 38 | theorem dist_smul_le (s : α) (x y : β) : dist (s • x) (s • y) ≤ ‖s‖ * dist x y := by |
simpa only [dist_eq_norm, sub_zero] using dist_smul_pair s x y
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 491 | 496 | theorem integral_sub_zpow_of_undef {n : ℤ} {c w : ℂ} {R : ℝ} (hn : n < 0)
(hw : w ∈ sphere c |R|) : (∮ z in C(c, R), (z - w) ^ n) = 0 := by |
rcases eq_or_ne R 0 with (rfl | h0)
· apply integral_radius_zero
· apply integral_undef
simpa [circleIntegrable_sub_zpow_iff, *, not_or]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... | Mathlib/Data/Nat/Fib/Basic.lean | 187 | 194 | theorem fib_two_mul_add_two (n : ℕ) :
fib (2 * n + 2) = fib (n + 1) * (2 * fib n + fib (n + 1)) := by |
rw [fib_add_two, fib_two_mul, fib_two_mul_add_one]
-- Porting note: A bunch of issues similar to [this zulip thread](https://github.com/leanprover-community/mathlib4/pull/1576) with `zify`
have : fib n ≤ 2 * fib (n + 1) :=
le_trans fib_le_fib_succ (mul_comm 2 _ ▸ Nat.le_mul_of_pos_right _ two_pos)
zify [th... |
/-
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.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 106 | 109 | theorem exists_rat_eq_nth_numerator : ∃ q : ℚ, (of v).numerators n = (q : K) := by |
rcases exists_gcf_pair_rat_eq_nth_conts v n with ⟨⟨a, _⟩, nth_cont_eq⟩
use a
simp [num_eq_conts_a, nth_cont_eq]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 47 | 50 | theorem parentD_set {arr : Array UFNode} {x v i} :
parentD (arr.set x v) i = if x.1 = i then v.parent else parentD arr i := by |
rw [parentD]; simp [Array.get_eq_getElem, parentD]
split <;> [split <;> simp [Array.get_set, *]; split <;> [(subst i; cases ‹¬_› x.2); rfl]]
|
/-
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 | 1,769 | 1,770 | theorem union_le (h₁ : s ≤ u) (h₂ : t ≤ u) : s ∪ t ≤ u := by |
rw [← eq_union_left h₂]; exact union_le_union_right h₁ t
|
/-
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.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520... | Mathlib/Data/ENNReal/Inv.lean | 634 | 646 | theorem Ioo_zero_top_eq_iUnion_Ico_zpow {y : ℝ≥0∞} (hy : 1 < y) (h'y : y ≠ ⊤) :
Ioo (0 : ℝ≥0∞) (∞ : ℝ≥0∞) = ⋃ n : ℤ, Ico (y ^ n) (y ^ (n + 1)) := by |
ext x
simp only [mem_iUnion, mem_Ioo, mem_Ico]
constructor
· rintro ⟨hx, h'x⟩
exact exists_mem_Ico_zpow hx.ne' h'x.ne hy h'y
· rintro ⟨n, hn, h'n⟩
constructor
· apply lt_of_lt_of_le _ hn
exact ENNReal.zpow_pos (zero_lt_one.trans hy).ne' h'y _
· apply lt_trans h'n _
exact ENNReal.z... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 533 | 535 | theorem finiteType_iff_group_fg {G : Type*} [AddCommGroup G] [CommRing R] [Nontrivial R] :
FiniteType R R[G] ↔ AddGroup.FG G := by |
simpa [AddGroup.fg_iff_addMonoid_fg] using finiteType_iff_fg
|
/-
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 | 1,969 | 1,970 | theorem sub_mulVec [Fintype n] (A B : Matrix m n α) (x : n → α) :
(A - B) *ᵥ x = A *ᵥ x - B *ᵥ x := by | simp [sub_eq_add_neg, add_mulVec, neg_mulVec]
|
/-
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 | 204 | 206 | theorem dotProduct_adjMatrix [NonAssocSemiring α] (v : V) (vec : V → α) :
dotProduct vec (G.adjMatrix α v) = ∑ u ∈ G.neighborFinset v, vec u := by |
simp [neighborFinset_eq_filter, dotProduct, sum_filter, Finset.sum_apply]
|
/-
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.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 701 | 702 | theorem mem_coclosed_compact' : s ∈ coclosedCompact X ↔ ∃ t, IsClosed t ∧ IsCompact t ∧ sᶜ ⊆ t := by |
simp only [hasBasis_coclosedCompact.mem_iff, compl_subset_comm, and_assoc]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 148 | 151 | theorem prod_assoc_preimage {α β γ} {s : Set α} {t : Set β} {u : Set γ} :
Equiv.prodAssoc α β γ ⁻¹' s ×ˢ t ×ˢ u = (s ×ˢ t) ×ˢ u := by |
ext
simp [and_assoc]
|
/-
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 | 2,905 | 2,917 | theorem monotone_filter_right (l : List α) ⦃p q : α → Bool⦄
(h : ∀ a, p a → q a) : l.filter p <+ l.filter q := by |
induction' l with hd tl IH
· rfl
· by_cases hp : p hd
· rw [filter_cons_of_pos _ hp, filter_cons_of_pos _ (h _ hp)]
exact IH.cons_cons hd
· rw [filter_cons_of_neg _ hp]
by_cases hq : q hd
· rw [filter_cons_of_pos _ hq]
exact sublist_cons_of_sublist hd IH
· rw [filter_cons_... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,165 | 2,166 | theorem biprod.symmetry (P Q : C) :
(biprod.braiding P Q).hom ≫ (biprod.braiding Q P).hom = 𝟙 _ := by | simp
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Math... | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | 716 | 718 | theorem pullbackIsoPullback_inv_fst_apply (x : (Types.pullbackCone f g).pt) :
(pullback.fst : _ ⟶ X) ((pullbackIsoPullback f g).inv x) = (fun p => (p.1 : X × Y).fst) x := by |
rw [elementwise_of% pullbackIsoPullback_inv_fst]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Tactic.Ring
#align_import data.fintype.perm from "leanprover-community/mathlib"... | Mathlib/Data/Fintype/Perm.lean | 47 | 74 | theorem mem_permsOfList_of_mem {l : List α} {f : Perm α} (h : ∀ x, f x ≠ x → x ∈ l) :
f ∈ permsOfList l := by |
induction l generalizing f with
| nil =>
-- Porting note: applied `not_mem_nil` because it is no longer true definitionally.
simp only [not_mem_nil] at h
exact List.mem_singleton.2 (Equiv.ext fun x => Decidable.by_contradiction <| h x)
| cons a l IH =>
by_cases hfa : f a = a
· refine mem_append_l... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 347 | 348 | theorem right_distrib (x y z : X q) : (x + y) * z = x * z + y * z := by |
ext <;> dsimp <;> ring
|
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 361 | 363 | theorem smul_eq_C_mul (f : R⟦X⟧) (a : R) : a • f = C R a * f := by |
ext
simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 448 | 449 | theorem ofFinsupp_single (n : ℕ) (r : R) : (⟨Finsupp.single n r⟩ : R[X]) = monomial n r := by |
simp [monomial]
|
/-
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.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 998 | 1,000 | theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t := by |
let ⟨x, ⟨xs, xt⟩⟩ := h
simpa using diam_union xs xt
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 536 | 539 | theorem kernel_from_eq_kernel [HasKernels V] {i j : ι} (r : c.Rel i j) :
kernelSubobject (C.dFrom i) = kernelSubobject (C.d i j) := by |
rw [C.dFrom_eq r]
apply kernelSubobject_comp_mono
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.un... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 211 | 214 | theorem angle_eq_zero_iff {x y : V} : angle x y = 0 ↔ x ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • x := by |
rw [angle, ← real_inner_div_norm_mul_norm_eq_one_iff, Real.arccos_eq_zero, LE.le.le_iff_eq,
eq_comm]
exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).2
|
/-
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.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "lea... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 247 | 250 | theorem biproduct_ι_comp_rightDistributor_inv {J : Type} [Fintype J] (f : J → C) (X : C) (j : J) :
biproduct.ι _ j ≫ (rightDistributor f X).inv = biproduct.ι _ j ▷ X := by |
simp [rightDistributor_inv, Preadditive.comp_sum, ← id_tensor_comp, biproduct.ι_π_assoc,
dite_comp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 886 | 890 | theorem infinite_of_mem_nhds {X} [TopologicalSpace X] [T1Space X] (x : X) [hx : NeBot (𝓝[≠] x)]
{s : Set X} (hs : s ∈ 𝓝 x) : Set.Infinite s := by |
refine fun hsf => hx.1 ?_
rw [← isOpen_singleton_iff_punctured_nhds]
exact isOpen_singleton_of_finite_mem_nhds x hs hsf
|
/-
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 | 188 | 191 | theorem map_map₂_antidistrib_left {g : γ → δ} {f' : β' → α → δ} {g' : β → β'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g' b) a) :
(map₂ f a b).map g = map₂ f' (b.map g') a := by |
cases a <;> cases b <;> simp [h_antidistrib]
|
/-
Copyright (c) 2021 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak, Kyle Miller, Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Math... | Mathlib/LinearAlgebra/CrossProduct.lean | 146 | 148 | theorem leibniz_cross (u v w : Fin 3 → R) : u ×₃ (v ×₃ w) = u ×₃ v ×₃ w + v ×₃ (u ×₃ w) := by |
simp_rw [cross_apply, vec3_add]
apply vec3_eq <;> norm_num <;> ring
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kyle Miller
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align... | Mathlib/Data/Set/Finite.lean | 894 | 896 | theorem finite_of_finite_preimage (h : (f ⁻¹' s).Finite) (hs : s ⊆ range f) : s.Finite := by |
rw [← image_preimage_eq_of_subset hs]
exact Finite.image f h
|
/-
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.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 81 | 82 | theorem mem_balancedCore_iff : x ∈ balancedCore 𝕜 s ↔ ∃ t, Balanced 𝕜 t ∧ t ⊆ s ∧ x ∈ t := by |
simp_rw [balancedCore, mem_sUnion, mem_setOf_eq, and_assoc]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing... | Mathlib/Topology/Sheaves/Stalks.lean | 562 | 578 | theorem app_surjective_of_stalkFunctor_map_bijective {F G : Sheaf C X} (f : F ⟶ G) (U : Opens X)
(h : ∀ x : U, Function.Bijective ((stalkFunctor C x.val).map f.1)) :
Function.Surjective (f.1.app (op U)) := by |
refine app_surjective_of_injective_of_locally_surjective f U (fun x => (h x).1) fun t x => ?_
-- Now we need to prove our initial claim: That we can find preimages of `t` locally.
-- Since `f` is surjective on stalks, we can find a preimage `s₀` of the germ of `t` at `x`
obtain ⟨s₀, hs₀⟩ := (h x).2 (G.presheaf... |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.P... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 497 | 521 | theorem exp_mem_exp [RCLike 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (a : A)
{z : 𝕜} (hz : z ∈ spectrum 𝕜 a) : exp 𝕜 z ∈ spectrum 𝕜 (exp 𝕜 a) := by |
have hexpmul : exp 𝕜 a = exp 𝕜 (a - ↑ₐ z) * ↑ₐ (exp 𝕜 z) := by
rw [algebraMap_exp_comm z, ← exp_add_of_commute (Algebra.commutes z (a - ↑ₐ z)).symm,
sub_add_cancel]
let b := ∑' n : ℕ, ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ^ n
have hb : Summable fun n : ℕ => ((n + 1).factorial⁻¹ : 𝕜) • (a - ↑ₐ z) ... |
/-
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 | 880 | 881 | theorem integral_comp_sub_right (d) : (∫ x in a..b, f (x - d)) = ∫ x in a - d..b - d, f x := by |
simpa only [sub_eq_add_neg] using integral_comp_add_right f (-d)
|
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.List.Sym
/-! # Unordered tuples of elements of a multiset
Defines `Multiset.sym` and the specialized `Multiset.sym2` for computing multisets of all
un... | Mathlib/Data/Multiset/Sym.lean | 63 | 66 | theorem sym2_mono {m m' : Multiset α} (h : m ≤ m') : m.sym2 ≤ m'.sym2 := by |
refine Quotient.inductionOn₂ m m' (fun xs ys h => ?_) h
suffices xs <+~ ys from this.sym2
simpa only [quot_mk_to_coe, coe_le, sym2_coe] using h
|
/-
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.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 248 | 252 | theorem cardinal_lt_aleph0_of_finiteDimensional [Finite K] [Module.Free K V] [Module.Finite K V] :
#V < ℵ₀ := by |
rw [← lift_lt_aleph0.{v, u}, lift_cardinal_mk_eq_lift_cardinal_mk_field_pow_lift_rank K V]
exact power_lt_aleph0 (lift_lt_aleph0.2 (lt_aleph0_of_finite K))
(lift_lt_aleph0.2 (rank_lt_aleph0 K V))
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 641 | 645 | theorem iterate_derivative_comp_one_sub_X (p : R[X]) (k : ℕ) :
derivative^[k] (p.comp (1 - X)) = (-1) ^ k * (derivative^[k] p).comp (1 - X) := by |
induction' k with k ih generalizing p
· simp
· simp [ih (derivative p), iterate_derivative_neg, derivative_comp, pow_succ]
|
/-
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 | 102 | 106 | theorem nfpFamily_le_apply [Nonempty ι] (H : ∀ i, IsNormal (f i)) {a b} :
(∃ i, nfpFamily.{u, v} f a ≤ f i b) ↔ nfpFamily.{u, v} f a ≤ b := by |
rw [← not_iff_not]
push_neg
exact apply_lt_nfpFamily_iff H
|
/-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 103 | 104 | theorem independent_iff_not_dependent : Independent f ↔ ¬Dependent f := by |
rw [dependent_iff_not_independent, Classical.not_not]
|
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Martin Dvorak
-/
import Mathlib.Algebra.Order.Kleene
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.List.Join
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.... | Mathlib/Computability/Language.lean | 274 | 276 | theorem one_add_self_mul_kstar_eq_kstar (l : Language α) : 1 + l * l∗ = l∗ := by |
simp only [kstar_eq_iSup_pow, mul_iSup, ← pow_succ', ← pow_zero l]
exact sup_iSup_nat_succ _
|
/-
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 | 937 | 940 | theorem tendstoUniformly_iff {F : ι → β → α} {f : β → α} {p : Filter ι} :
TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, dist (f x) (F n x) < ε := by |
rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff]
simp
|
/-
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.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 314 | 320 | theorem homology'.map_desc (p : α.right = β.left) {D : V} (k : (kernelSubobject g' : V) ⟶ D)
(z : imageToKernel f' g' w' ≫ k = 0) :
homology'.map w w' α β p ≫ homology'.desc f' g' w' k z =
homology'.desc f g w (kernelSubobjectMap β ≫ k)
(by simp only [imageSubobjectMap_comp_imageToKernel_assoc w w... |
ext
simp only [homology'.π_desc, homology'.π_map_assoc]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 877 | 883 | theorem isCoprime_span_singleton_iff (x y : R) :
IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by |
simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup,
mem_span_singleton]
constructor
· rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩
· rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,757 | 1,757 | theorem pow_eq_prod_const (b : β) : ∀ n, b ^ n = ∏ _k ∈ range n, b := by | simp
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import meas... | Mathlib/MeasureTheory/Function/L2Space.lean | 81 | 83 | theorem Integrable.const_inner (c : E) (hf : Integrable f μ) :
Integrable (fun x => ⟪c, f x⟫) μ := by |
rw [← memℒp_one_iff_integrable] at hf ⊢; exact hf.const_inner c
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 690 | 693 | theorem realize_graph {f : L.Functions n} {x : Fin n → M} {y : M} :
(Formula.graph f).Realize (Fin.cons y x : _ → M) ↔ funMap f x = y := by |
simp only [Formula.graph, Term.realize, realize_equal, Fin.cons_zero, Fin.cons_succ]
rw [eq_comm]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#a... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 786 | 786 | theorem isCycleOn_singleton : f.IsCycleOn {a} ↔ f a = a := by | simp [IsCycleOn, SameCycle.rfl]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 575 | 588 | theorem nmul_one (a : Ordinal) : a ⨳ 1 = a := by |
rw [nmul]
simp only [lt_one_iff_zero, forall_eq, nmul_zero, nadd_zero]
convert csInf_Ici (α := Ordinal)
ext b
-- Porting note: added this `simp` line, as the result from `convert`
-- is slightly different.
simp only [Set.mem_setOf_eq, Set.mem_Ici]
refine ⟨fun H => le_of_forall_lt fun c hc => ?_, fun ha... |
/-
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.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 160 | 161 | theorem support_zero : support (0 : FreeAbelianGroup X) = ∅ := by |
simp only [support, Finsupp.support_zero, AddMonoidHom.map_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.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 80 | 91 | theorem χ₄_eq_neg_one_pow {n : ℕ} (hn : n % 2 = 1) : χ₄ n = (-1) ^ (n / 2) := by |
rw [χ₄_nat_eq_if_mod_four]
simp only [hn, Nat.one_ne_zero, if_false]
conv_rhs => -- Porting note: was `nth_rw`
arg 2; rw [← Nat.div_add_mod n 4]
enter [1, 1, 1]; rw [(by norm_num : 4 = 2 * 2)]
rw [mul_assoc, add_comm, Nat.add_mul_div_left _ _ (by norm_num : 0 < 2), pow_add, pow_mul,
neg_one_sq, one... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,359 | 1,360 | theorem not_frequently {p : α → Prop} {f : Filter α} : (¬∃ᶠ x in f, p x) ↔ ∀ᶠ x in f, ¬p x := by |
simp only [Filter.Frequently, not_not]
|
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 659 | 662 | theorem rpow_le_rpow_left_iff (hx : 1 < x) : x ^ y ≤ x ^ z ↔ y ≤ z := by |
have x_pos : 0 < x := lt_trans zero_lt_one hx
rw [← log_le_log_iff (rpow_pos_of_pos x_pos y) (rpow_pos_of_pos x_pos z), log_rpow x_pos,
log_rpow x_pos, mul_le_mul_right (log_pos hx)]
|
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Constructions.Prod.Integral
/-!
# Integration with respect to a finite product of measures
... | Mathlib/MeasureTheory/Integral/Pi.lean | 87 | 93 | theorem integral_fintype_prod_eq_prod (ι : Type*) [Fintype ι] {E : ι → Type*}
(f : (i : ι) → E i → 𝕜) [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))] :
∫ x : (i : ι) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x := by |
let e := (equivFin ι).symm
rw [← (volume_measurePreserving_piCongrLeft _ e).integral_comp']
simp_rw [← e.prod_comp, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft_apply_apply,
MeasureTheory.integral_fin_nat_prod_eq_prod]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 349 | 350 | theorem coeff_X_add_one_pow (R : Type*) [Semiring R] (n k : ℕ) :
((X + 1) ^ n).coeff k = (n.choose k : R) := by | rw [← C_1, coeff_X_add_C_pow, one_pow, one_mul]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 869 | 873 | theorem map₂_eq_range_lift_comp_mapIncl (f : P →ₗ[R] Q →ₗ[R] M)
(p : Submodule R P) (q : Submodule R Q) :
Submodule.map₂ f p q = LinearMap.range (lift f ∘ₗ mapIncl p q) := by |
simp_rw [LinearMap.range_comp, range_mapIncl, Submodule.map_span,
Set.image_image2, Submodule.map₂_eq_span_image2, lift.tmul]
|
/-
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.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 407 | 408 | theorem circumradius_reindex {m n : ℕ} (s : Simplex ℝ P m) (e : Fin (m + 1) ≃ Fin (n + 1)) :
(s.reindex e).circumradius = s.circumradius := by | simp_rw [circumradius, circumsphere_reindex]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 1,074 | 1,094 | theorem nrRealPlaces_eq_zero_of_two_lt (hk : 2 < k) (hζ : IsPrimitiveRoot ζ k) :
NumberField.InfinitePlace.NrRealPlaces K = 0 := by |
refine (@Fintype.card_eq_zero_iff _ (_)).2 ⟨fun ⟨w, hwreal⟩ ↦ ?_⟩
rw [NumberField.InfinitePlace.isReal_iff] at hwreal
let f := w.embedding
have hζ' : IsPrimitiveRoot (f ζ) k := hζ.map_of_injective f.injective
have him : (f ζ).im = 0 := by
rw [← Complex.conj_eq_iff_im, ← NumberField.ComplexEmbedding.conju... |
/-
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.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.f... | Mathlib/Topology/FiberBundle/Trivialization.lean | 458 | 469 | theorem tendsto_nhds_iff {α : Type*} {l : Filter α} {f : α → Z} {z : Z} (hz : z ∈ e.source) :
Tendsto f l (𝓝 z) ↔
Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x ↦ (e (f x)).2) l (𝓝 (e z).2) := by |
rw [e.nhds_eq_comap_inf_principal hz, tendsto_inf, tendsto_comap_iff, Prod.tendsto_iff, coe_coe,
tendsto_principal, coe_fst _ hz]
by_cases hl : ∀ᶠ x in l, f x ∈ e.source
· simp only [hl, and_true]
refine (tendsto_congr' ?_).and Iff.rfl
exact hl.mono fun x ↦ e.coe_fst
· simp only [hl, and_false, fal... |
/-
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.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Spec... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 297 | 315 | theorem hasSum_log_sub_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
HasSum (fun k : ℕ => (2 : ℝ) * (1 / (2 * k + 1)) * x ^ (2 * k + 1))
(log (1 + x) - log (1 - x)) := by |
set term := fun n : ℕ => -1 * ((-x) ^ (n + 1) / ((n : ℝ) + 1)) + x ^ (n + 1) / (n + 1)
have h_term_eq_goal :
term ∘ (2 * ·) = fun k : ℕ => 2 * (1 / (2 * k + 1)) * x ^ (2 * k + 1) := by
ext n
dsimp only [term, (· ∘ ·)]
rw [Odd.neg_pow (⟨n, rfl⟩ : Odd (2 * n + 1)) x]
push_cast
ring_nf
rw ... |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombinat... | Mathlib/NumberTheory/FLT/Four.lean | 309 | 313 | theorem fermatLastTheoremFour : FermatLastTheoremFor 4 := by |
rw [fermatLastTheoremFor_iff_int]
intro a b c ha hb _ heq
apply @not_fermat_42 _ _ (c ^ 2) ha hb
rw [heq]; ring
|
/-
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.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_characte... | Mathlib/NumberTheory/MulChar/Basic.lean | 138 | 143 | theorem ext {χ χ' : MulChar R R'} (h : ∀ a : Rˣ, χ a = χ' a) : χ = χ' := by |
apply ext'
intro a
by_cases ha : IsUnit a
· exact h ha.unit
· rw [map_nonunit χ ha, map_nonunit χ' ha]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.Analysis.BoxIntegral.Partition.Additive
import Mathlib.Analysis.Calculus.FDeriv.Prod
#align_import analysi... | Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean | 65 | 136 | theorem norm_volume_sub_integral_face_upper_sub_lower_smul_le {f : (Fin (n + 1) → ℝ) → E}
{f' : (Fin (n + 1) → ℝ) →L[ℝ] E} (hfc : ContinuousOn f (Box.Icc I)) {x : Fin (n + 1) → ℝ}
(hxI : x ∈ (Box.Icc I)) {a : E} {ε : ℝ} (h0 : 0 < ε)
(hε : ∀ y ∈ (Box.Icc I), ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0}
... |
-- Porting note: Lean fails to find `α` in the next line
set e : ℝ → (Fin n → ℝ) → (Fin (n + 1) → ℝ) := i.insertNth (α := fun _ ↦ ℝ)
/- **Plan of the proof**. The difference of the integrals of the affine function
`fun y ↦ a + f' (y - x)` over the faces `x i = I.upper i` and `x i = I.lower i` is equal to the... |
/-
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.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 691 | 693 | theorem zero_comp (f : NormedAddGroupHom V₁ V₂) : (0 : NormedAddGroupHom V₂ V₃).comp f = 0 := by |
ext
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.Init.Align
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.Algebra.Category.ModuleCat.EpiMono
#align... | Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | 302 | 310 | theorem pseudo_injective_of_mono {P Q : C} (f : P ⟶ Q) [Mono f] : Function.Injective f := by |
intro abar abar'
refine Quotient.inductionOn₂ abar abar' fun a a' ha => ?_
apply Quotient.sound
have : (⟦(a.hom ≫ f : Over Q)⟧ : Quotient (setoid Q)) = ⟦↑(a'.hom ≫ f)⟧ := by convert ha
have ⟨R, p, q, ep, Eq, comm⟩ := Quotient.exact this
exact ⟨R, p, q, ep, Eq, (cancel_mono f).1 <| by
simp only [Categor... |
/-
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 | 1,280 | 1,281 | theorem cast_ofZNum [AddGroupWithOne α] (n : ZNum) : (ofZNum n : α) = Int.toNat n := by |
rw [← cast_to_nat, ofZNum_toNat]
|
/-
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.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368c... | Mathlib/Data/Nat/Squarefree.lean | 28 | 30 | theorem squarefree_iff_nodup_factors {n : ℕ} (h0 : n ≠ 0) : Squarefree n ↔ n.factors.Nodup := by |
rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0, Nat.factors_eq]
simp
|
/-
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 | 315 | 315 | theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 := by | rw [average, integral_zero]
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
... | Mathlib/Control/Bifunctor.lean | 92 | 93 | theorem fst_snd {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
fst f (snd f' x) = bimap f f' x := by | simp [fst, bimap_bimap]
|
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