Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
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 | 68 | 70 | theorem weightedDegree_apply (w : σ → M) (f : σ →₀ ℕ):
weightedDegree w f = Finsupp.sum f (fun i c => c • w i) := by |
rfl
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.Separation
import... | Mathlib/Topology/UniformSpace/Compact.lean | 69 | 75 | theorem unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ]
{u u' : UniformSpace γ} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :
u = u' := by |
refine UniformSpace.ext ?_
have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h']
|
/-
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.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
#align_import order.iterate from "leanprover-community/mathlib"@"2258b40dacd2942571... | Mathlib/Order/Iterate.lean | 68 | 71 | theorem seq_lt_seq_of_lt_of_le (hf : Monotone f) (n : ℕ) (h₀ : x 0 < y 0)
(hx : ∀ k < n, x (k + 1) < f (x k)) (hy : ∀ k < n, f (y k) ≤ y (k + 1)) : x n < y n := by |
cases n
exacts [h₀, hf.seq_pos_lt_seq_of_lt_of_le (Nat.zero_lt_succ _) h₀.le hx hy]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"... | Mathlib/RingTheory/Polynomial/Tower.lean | 37 | 38 | theorem aeval_map_algebraMap (x : B) (p : R[X]) : aeval x (map (algebraMap R A) p) = aeval x p := by |
rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.char_p.quotient from... | Mathlib/Algebra/CharP/Quotient.lean | 47 | 51 | theorem quotient_iff {R : Type*} [CommRing R] (n : ℕ) [CharP R n] (I : Ideal R) :
CharP (R ⧸ I) n ↔ ∀ x : ℕ, ↑x ∈ I → (x : R) = 0 := by |
refine ⟨fun _ x hx => ?_, CharP.quotient' n I⟩
rw [CharP.cast_eq_zero_iff R n, ← CharP.cast_eq_zero_iff (R ⧸ I) n _]
exact (Submodule.Quotient.mk_eq_zero I).mpr hx
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathl... | Mathlib/RingTheory/MatrixAlgebra.lean | 124 | 130 | theorem left_inv (M : A ⊗[R] Matrix n n R) : invFun R A n (toFunAlgHom R A n M) = M := by |
induction M using TensorProduct.induction_on with
| zero => simp
| tmul a m => simp
| add x y hx hy =>
rw [map_add]
conv_rhs => rw [← hx, ← hy, ← invFun_add]
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Order.Zorn
#align_import order.extension.linear from "leanprover-community/mathlib"@"9830a300340708eaa85d477c3fb96dd25f9468a5"
/-!
# Extend a partial ord... | Mathlib/Order/Extension/Linear.lean | 26 | 73 | theorem extend_partialOrder {α : Type u} (r : α → α → Prop) [IsPartialOrder α r] :
∃ s : α → α → Prop, IsLinearOrder α s ∧ r ≤ s := by |
let S := { s | IsPartialOrder α s }
have hS : ∀ c, c ⊆ S → IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub := by
rintro c hc₁ hc₂ s hs
haveI := (hc₁ hs).1
refine ⟨sSup c, ?_, fun z hz => le_sSup hz⟩
refine
{ refl := ?_
trans := ?_
antisymm := ?_ } <;>
simp_r... |
/-
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.Limits.Shapes.Products
import Mathlib.CategoryTheory.Functor.EpiMono
#align_import category_theory.adjunction.evaluation from "leanprover-commu... | Mathlib/CategoryTheory/Adjunction/Evaluation.lean | 81 | 86 | theorem NatTrans.mono_iff_mono_app {F G : C ⥤ D} (η : F ⟶ G) : Mono η ↔ ∀ c, Mono (η.app c) := by |
constructor
· intro h c
exact (inferInstance : Mono (((evaluation _ _).obj c).map η))
· intro _
apply NatTrans.mono_of_mono_app
|
/-
Copyright (c) 2020 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ruben Van de Velde
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import anal... | Mathlib/Analysis/NormedSpace/Extend.lean | 93 | 99 | theorem norm_extendTo𝕜'_apply_sq (fr : F →ₗ[ℝ] ℝ) (x : F) :
‖(fr.extendTo𝕜' x : 𝕜)‖ ^ 2 = fr (conj (fr.extendTo𝕜' x : 𝕜) • x) :=
calc
‖(fr.extendTo𝕜' x : 𝕜)‖ ^ 2 = re (conj (fr.extendTo𝕜' x) * fr.extendTo𝕜' x : 𝕜) := by |
rw [RCLike.conj_mul, ← ofReal_pow, ofReal_re]
_ = fr (conj (fr.extendTo𝕜' x : 𝕜) • x) := by
rw [← smul_eq_mul, ← map_smul, extendTo𝕜'_apply_re]
|
/-
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 | 52 | 55 | theorem rankD_set {arr : Array UFNode} {x v i} :
rankD (arr.set x v) i = if x.1 = i then v.rank else rankD arr i := by |
rw [rankD]; simp [Array.get_eq_getElem, rankD]
split <;> [split <;> simp [Array.get_set, *]; split <;> [(subst i; cases ‹¬_› x.2); rfl]]
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68a... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 90 | 94 | theorem compress_of_disjoint_of_le' (hva : Disjoint v a) (hua : u ≤ a) :
compress u v ((a ⊔ v) \ u) = a := by |
rw [compress_of_disjoint_of_le disjoint_sdiff_self_right
(le_sdiff.2 ⟨(le_sup_right : v ≤ a ⊔ v), hva.mono_right hua⟩),
sdiff_sup_cancel (le_sup_of_le_left hua), hva.symm.sup_sdiff_cancel_right]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
... | Mathlib/RingTheory/SimpleModule.lean | 125 | 127 | theorem toSpanSingleton_surjective {m : M} (hm : m ≠ 0) :
Function.Surjective (toSpanSingleton R M m) := by |
rw [← range_eq_top, ← span_singleton_eq_range, span_singleton_eq_top R hm]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_th... | Mathlib/NumberTheory/SumTwoSquares.lean | 86 | 94 | theorem ZMod.isSquare_neg_one_mul {m n : ℕ} (hc : m.Coprime n) (hm : IsSquare (-1 : ZMod m))
(hn : IsSquare (-1 : ZMod n)) : IsSquare (-1 : ZMod (m * n)) := by |
have : IsSquare (-1 : ZMod m × ZMod n) := by
rw [show (-1 : ZMod m × ZMod n) = ((-1 : ZMod m), (-1 : ZMod n)) from rfl]
obtain ⟨x, hx⟩ := hm
obtain ⟨y, hy⟩ := hn
rw [hx, hy]
exact ⟨(x, y), rfl⟩
simpa only [RingEquiv.map_neg_one] using this.map (ZMod.chineseRemainder hc).symm
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 93 | 95 | theorem mul_num (q₁ q₂ : ℚ) :
(q₁ * q₂).num = q₁.num * q₂.num / Nat.gcd (q₁.num * q₂.num).natAbs (q₁.den * q₂.den) := by |
rw [mul_def, normalize_eq]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 24 | 30 | theorem num_dvd (a) {b : ℤ} (b0 : b ≠ 0) : (a /. b).num ∣ a := by |
cases' e : a /. b with n d h c
rw [Rat.mk'_eq_divInt, divInt_eq_iff b0 (mod_cast h)] at e
refine Int.natAbs_dvd.1 <| Int.dvd_natAbs.1 <| Int.natCast_dvd_natCast.2 <|
c.dvd_of_dvd_mul_right ?_
have := congr_arg Int.natAbs e
simp only [Int.natAbs_mul, Int.natAbs_ofNat] at this; simp [this]
|
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e962... | Mathlib/Order/Height.lean | 77 | 77 | theorem singleton_mem_subchain_iff : [a] ∈ s.subchain ↔ a ∈ s := by | simp [cons_mem_subchain_iff]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8e... | Mathlib/Algebra/Polynomial/Expand.lean | 95 | 97 | theorem derivative_expand (f : R[X]) : Polynomial.derivative (expand R p f) =
expand R p (Polynomial.derivative f) * (p * (X ^ (p - 1) : R[X])) := by |
rw [coe_expand, derivative_eval₂_C, derivative_pow, C_eq_natCast, derivative_X, mul_one]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.LocalExtr.Basic
#align_import analysis.calcu... | Mathlib/Analysis/Calculus/Darboux.lean | 28 | 60 | theorem exists_hasDerivWithinAt_eq_of_gt_of_lt (hab : a ≤ b)
(hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a < m)
(hmb : m < f' b) : m ∈ f' '' Ioo a b := by |
rcases hab.eq_or_lt with (rfl | hab')
· exact (lt_asymm hma hmb).elim
set g : ℝ → ℝ := fun x => f x - m * x
have hg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x := by
intro x hx
simpa using (hf x hx).sub ((hasDerivWithinAt_id x _).const_mul m)
obtain ⟨c, cmem, hc⟩ : ∃ c ∈ Icc a b, IsMin... |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLat... | Mathlib/Algebra/Tropical/BigOperators.lean | 78 | 82 | theorem List.trop_minimum [LinearOrder R] (l : List R) :
trop l.minimum = List.sum (l.map (trop ∘ WithTop.some)) := by |
induction' l with hd tl IH
· simp
· simp [List.minimum_cons, ← IH]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 101 | 104 | theorem self (hp : 1 < p) : padicValNat p p = 1 := by |
have neq_one : ¬p = 1 ↔ True := iff_of_true hp.ne' trivial
have eq_zero_false : p = 0 ↔ False := iff_false_intro (zero_lt_one.trans hp).ne'
simp [padicValNat, neq_one, eq_zero_false]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-co... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 98 | 102 | theorem strongEpi_comp [StrongEpi f] [StrongEpi g] : StrongEpi (f ≫ g) :=
{ epi := epi_comp _ _
llp := by |
intros
infer_instance }
|
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Scott Morrison
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Logic.Lemmas
#align_import combinatorics.quiver.path from "leanprover-community/mathlib"@"18a5306c0... | Mathlib/Combinatorics/Quiver/Path.lean | 81 | 84 | theorem eq_of_length_zero (p : Path a b) (hzero : p.length = 0) : a = b := by |
cases p
· rfl
· cases Nat.succ_ne_zero _ hzero
|
/-
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.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Ring.Aut
import Mathlib.Algebra.Ring.CompTypeclasses
import ... | Mathlib/Algebra/Star/Basic.lean | 153 | 153 | theorem star_mul_star (x y : R) : star (x * star y) = y * star x := by | rw [star_mul, star_star]
|
/-
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 | 110 | 113 | theorem comp_differentiableOn_iff {f : G → E} {s : Set G} :
DifferentiableOn 𝕜 (iso ∘ f) s ↔ DifferentiableOn 𝕜 f s := by |
rw [DifferentiableOn, DifferentiableOn]
simp only [iso.comp_differentiableWithinAt_iff]
|
/-
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.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-communit... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 62 | 64 | theorem charmatrix_apply_ne (h : i ≠ j) : charmatrix M i j = -C (M i j) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, diagonal_apply_ne _ h,
map_apply, sub_eq_neg_self]
|
/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathl... | Mathlib/Combinatorics/SetFamily/LYM.lean | 92 | 110 | theorem card_div_choose_le_card_shadow_div_choose (hr : r ≠ 0)
(h𝒜 : (𝒜 : Set (Finset α)).Sized r) : (𝒜.card : 𝕜) / (Fintype.card α).choose r
≤ (∂ 𝒜).card / (Fintype.card α).choose (r - 1) := by |
obtain hr' | hr' := lt_or_le (Fintype.card α) r
· rw [choose_eq_zero_of_lt hr', cast_zero, div_zero]
exact div_nonneg (cast_nonneg _) (cast_nonneg _)
replace h𝒜 := card_mul_le_card_shadow_mul h𝒜
rw [div_le_div_iff] <;> norm_cast
· cases' r with r
· exact (hr rfl).elim
rw [tsub_add_eq_add_tsub h... |
/-
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.EffectiveEpi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Tactic.ApplyFun
/-!
# Effective epimorphic ... | Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean | 61 | 93 | theorem effectiveEpiFamilyStructOfEffectiveEpiDesc_aux {B : C} {α : Type*} {X : α → C}
{π : (a : α) → X a ⟶ B} [HasCoproduct X]
[∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)]
[∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)]
[∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.des... |
apply_fun ((Sigma.desc fun a ↦ pullback.fst (f := g₁) (g := (Sigma.ι X a))) ≫ ·) using
(fun a b ↦ (cancel_epi _).mp)
ext a
simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app]
rw [← Category.assoc, pullback.condition]
simp only [Category.assoc, colimit.ι_desc, Cofan.mk_pt, ... |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Dual
/-!
# Perfect pairings of modules
A perfect pairing of two (left) modules may be defined either as:
1. A bilinear map `M × N → R` such ... | Mathlib/LinearAlgebra/PerfectPairing.lean | 85 | 89 | theorem apply_apply_toDualRight_symm (x : M) (f : Dual R M) :
(p x) (p.toDualRight.symm f) = f x := by |
have h := LinearEquiv.apply_symm_apply p.toDualRight f
rw [toDualRight_apply] at h
exact congrFun (congrArg DFunLike.coe h) x
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.Fin... | Mathlib/Analysis/Analytic/Polynomial.lean | 47 | 52 | theorem AnalyticAt.aeval_mvPolynomial (hf : ∀ i, AnalyticAt 𝕜 (f · i) z) (p : MvPolynomial σ A) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by |
apply p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_ -- `refine` doesn't work
· simp_rw [aeval_C]; apply analyticAt_const
· simp_rw [map_add]; exact hp.add hq
· simp_rw [map_mul, aeval_X]; exact hp.mul (hf i)
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 77 | 77 | theorem isBounded_bot : IsBounded r ⊥ ↔ Nonempty α := by | simp [IsBounded, exists_true_iff_nonempty]
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 162 | 162 | theorem nonneg_iff {a : SignType} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by | cases a <;> decide
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Isomorphisms.lean | 67 | 70 | theorem comap_leq_ker_subToSupQuotient (p p' : Submodule R M) :
comap (Submodule.subtype p) (p ⊓ p') ≤ ker (subToSupQuotient p p') := by |
rw [LinearMap.ker_comp, Submodule.inclusion, comap_codRestrict, ker_mkQ, map_comap_subtype]
exact comap_mono (inf_le_inf_right _ le_sup_left)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theory.group_action.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-... | Mathlib/GroupTheory/GroupAction/Support.lean | 58 | 63 | theorem Supports.smul (g : H) (h : Supports G s b) : Supports G (g • s) (g • b) := by |
rintro g' hg'
rw [smul_comm, h]
rintro a ha
have := Set.forall_mem_image.1 hg' ha
rwa [smul_comm, smul_left_cancel_iff] at this
|
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 41 | 43 | theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by |
rw [derivativeFun, coeff_mk]
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 86 | 88 | theorem cons_one {α : Fin (n + 2) → Type*} (x : α 0) (p : ∀ i : Fin n.succ, α i.succ) :
cons x p 1 = p 0 := by |
rw [← cons_succ x p]; 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 Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 116 | 120 | theorem tendsto_toNNReal {a : ℝ≥0∞} (ha : a ≠ ∞) :
Tendsto ENNReal.toNNReal (𝓝 a) (𝓝 a.toNNReal) := by |
lift a to ℝ≥0 using ha
rw [nhds_coe, tendsto_map'_iff]
exact tendsto_id
|
/-
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 | 140 | 140 | theorem zero'_eval : zero'.eval = fun v => pure (0 :: v) := by | simp [eval]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-com... | Mathlib/Tactic/NormNum/Prime.lean | 67 | 82 | theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
(np : minFac n ≠ k) : MinFacHelper n k' := by |
rw [← e]
refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
· rw [add_mod, mod_self, add_zero, mod_mod]
exact h.2.1
rcases h.2.2.eq_or_lt with rfl|h2
· exact (np rfl).elim
rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
· refine ((h.1.trans_le h.2.2).ne ?_).elim
have h3 : 2 ∣ minFac n := by
rw [Nat.dvd_... |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 57 | 60 | theorem Hom.cast_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
HEq (e.cast hu hv) e := by |
subst_vars
rfl
|
/-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_t... | Mathlib/GroupTheory/DoubleCoset.lean | 44 | 45 | theorem mem_doset {s t : Set α} {a b : α} : b ∈ doset a s t ↔ ∃ x ∈ s, ∃ y ∈ t, b = x * a * y := by |
simp only [doset_eq_image2, Set.mem_image2, eq_comm]
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Ring.Commute
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f... | Mathlib/Data/Nat/Cast/Commute.lean | 24 | 27 | theorem cast_commute (n : ℕ) (x : α) : Commute (n : α) x := by |
induction n with
| zero => rw [Nat.cast_zero]; exact Commute.zero_left x
| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprov... | Mathlib/ModelTheory/Ultraproducts.lean | 83 | 91 | theorem term_realize_cast {β : Type*} (x : β → ∀ a, M a) (t : L.Term β) :
(t.realize fun i => (x i : (u : Filter α).Product M)) =
(fun a => t.realize fun i => x i a : (u : Filter α).Product M) := by |
convert @Term.realize_quotient_mk' L _ ((u : Filter α).productSetoid M)
(Ultraproduct.setoidPrestructure M u) _ t x using 2
ext a
induction t with
| var => rfl
| func _ _ t_ih => simp only [Term.realize, t_ih]; rfl
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology... | Mathlib/Topology/TietzeExtension.lean | 96 | 100 | theorem ContinuousMap.exists_forall_mem_restrict_eq {Y : Type v} [TopologicalSpace Y] (f : C(s, Y))
{t : Set Y} (hf : ∀ x, f x ∈ t) [ht : TietzeExtension.{u, v} t] :
∃ (g : C(X, Y)), (∀ x, g x ∈ t) ∧ g.restrict s = f := by |
obtain ⟨g, hg⟩ := mk _ (map_continuous f |>.codRestrict hf) |>.exists_restrict_eq hs
exact ⟨comp ⟨Subtype.val, by continuity⟩ g, by simp, by ext x; congrm(($(hg) x : Y))⟩
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6e... | Mathlib/Data/Rat/Cast/Lemmas.lean | 44 | 51 | theorem cast_nnratCast {K} [DivisionRing K] (q : ℚ≥0) :
((q : ℚ) : K) = (q : K) := by |
rw [Rat.cast_def, NNRat.cast_def, NNRat.cast_def]
have hn := @num_div_eq_of_coprime q.num q.den ?hdp q.coprime_num_den
on_goal 1 => have hd := @den_div_eq_of_coprime q.num q.den ?hdp q.coprime_num_den
case hdp => simpa only [Nat.cast_pos] using q.den_pos
simp only [Int.cast_natCast, Nat.cast_inj] at hn hd
... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 129 | 132 | theorem map_traverse' (g : α → G β) (h : β → γ) :
traverse (map h ∘ g) = (map (map h) ∘ traverse g : t α → G (t γ)) := by |
ext
rw [comp_apply, map_traverse]
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-... | Mathlib/RingTheory/Ideal/IsPrincipal.lean | 67 | 72 | theorem associatesEquivIsPrincipal_mul (x y : Associates R) :
(associatesEquivIsPrincipal R (x * y) : Ideal R) =
(associatesEquivIsPrincipal R x) * (associatesEquivIsPrincipal R y) := by |
rw [← Associates.quot_out x, ← Associates.quot_out y]
simp_rw [Associates.mk_mul_mk, ← Associates.quotient_mk_eq_mk, associatesEquivIsPrincipal_apply,
span_singleton_mul_span_singleton]
|
/-
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 | 111 | 112 | theorem dist_smul₀ (s : α) (x y : β) : dist (s • x) (s • y) = ‖s‖ * dist x y := by |
simp_rw [dist_eq_norm, (norm_smul s (x - y)).symm, smul_sub]
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 116 | 117 | theorem logb_rpow_eq_mul_logb_of_pos (hx : 0 < x) : logb b (x ^ y) = y * logb b x := by |
rw [logb, log_rpow hx, logb, mul_div_assoc]
|
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 71 | 73 | theorem smul_inv_mem_fixedBy_iff_mem_fixedBy {a : α} {g : G} :
g⁻¹ • a ∈ fixedBy α g ↔ a ∈ fixedBy α g := by |
rw [← fixedBy_inv, smul_mem_fixedBy_iff_mem_fixedBy, fixedBy_inv]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_im... | Mathlib/Data/Finset/NatAntidiagonal.lean | 58 | 58 | theorem card_antidiagonal (n : ℕ) : (antidiagonal n).card = n + 1 := by | simp [antidiagonal]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c73... | Mathlib/Topology/Maps.lean | 112 | 115 | theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
MapClusterPt (f x) l f ↔ ClusterPt x l := by |
delta MapClusterPt ClusterPt
rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff]
|
/-
Copyright (c) 2021 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.C... | Mathlib/Algebra/Category/ModuleCat/Subobject.lean | 89 | 96 | theorem toKernelSubobject_arrow {M N : ModuleCat R} {f : M ⟶ N} (x : LinearMap.ker f) :
(kernelSubobject f).arrow (toKernelSubobject x) = x.1 := by |
-- Porting note: The whole proof was just `simp [toKernelSubobject]`.
suffices ((arrow ((kernelSubobject f))) ∘ (kernelSubobjectIso f ≪≫ kernelIsoKer f).inv) x = x by
convert this
rw [Iso.trans_inv, ← coe_comp, Category.assoc]
simp only [Category.assoc, kernelSubobject_arrow', kernelIsoKer_inv_kernel_ι]
... |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Topology.Lo... | Mathlib/Geometry/Manifold/Complex.lean | 53 | 74 | theorem Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax {f : M → F} {c : M}
(hd : ∀ᶠ z in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z) (hc : IsLocalMax (norm ∘ f) c) :
∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ := by |
set e := extChartAt I c
have hI : range I = univ := ModelWithCorners.Boundaryless.range_eq_univ
have H₁ : 𝓝[range I] (e c) = 𝓝 (e c) := by rw [hI, nhdsWithin_univ]
have H₂ : map e.symm (𝓝 (e c)) = 𝓝 c := by
rw [← map_extChartAt_symm_nhdsWithin_range I c, H₁]
rw [← H₂, eventually_map]
replace hd : ∀... |
/-
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 | 112 | 114 | theorem intervalIntegrable_iff_integrableOn_Ioo_of_le [NoAtoms μ] (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ioo]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
/-!
# Action of `Mᵈᵐᵃ` on `Lᵖ` spaces
In this file we define ac... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 99 | 100 | theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g := by |
simp only [dist, ← smul_Lp_sub, norm_smul_Lp]
|
/-
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.HomologicalComplex
#align_import algebra.homology.single from "leanprover-community/mathlib"@"324a7502510e835cdbd3de1519b6c66b51fb246... | Mathlib/Algebra/Homology/Single.lean | 91 | 96 | theorem single_map_f_self (j : ι) {A B : V} (f : A ⟶ B) :
((single V c j).map f).f j = (singleObjXSelf c j A).hom ≫
f ≫ (singleObjXSelf c j B).inv := by |
dsimp [single]
rw [dif_pos rfl]
rfl
|
/-
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.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Anal... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 68 | 73 | theorem hasStrictDerivAt_cos (x : ℂ) : HasStrictDerivAt cos (-sin x) x := by |
simp only [sin, div_eq_mul_inv, neg_mul_eq_neg_mul]
convert (((hasStrictDerivAt_id x).mul_const I).cexp.add
((hasStrictDerivAt_id x).neg.mul_const I).cexp).mul_const (2 : ℂ)⁻¹ using 1
simp only [Function.comp, id]
ring
|
/-
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.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align... | Mathlib/Computability/Ackermann.lean | 82 | 85 | theorem ack_one (n : ℕ) : ack 1 n = n + 2 := by |
induction' n with n IH
· rfl
· simp [IH]
|
/-
Copyright (c) 2022 Justin Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justin Thomas
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import linear_algebra.ann... | Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean | 90 | 92 | theorem annIdealGenerator_eq_zero_iff {a : A} : annIdealGenerator 𝕜 a = 0 ↔ annIdeal 𝕜 a = ⊥ := by |
simp only [annIdealGenerator, mul_eq_zero, IsPrincipal.eq_bot_iff_generator_eq_zero,
Polynomial.C_eq_zero, inv_eq_zero, Polynomial.leadingCoeff_eq_zero, or_self_iff]
|
/-
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 | 82 | 83 | theorem toFinsupp_of (x : X) : toFinsupp (of x) = Finsupp.single x 1 := by |
simp only [toFinsupp, lift.of]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 76 | 79 | theorem card_roots' (p : R[X]) : Multiset.card p.roots ≤ natDegree p := by |
by_cases hp0 : p = 0
· simp [hp0]
exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0))
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.denoms_clearable from "leanprover-community/mathlib"@"... | Mathlib/Algebra/Polynomial/DenomsClearable.lean | 47 | 54 | theorem denomsClearable_C_mul_X_pow {N : ℕ} (a : R) (bu : bi * i b = 1) {n : ℕ} (r : R)
(nN : n ≤ N) : DenomsClearable a b N (C r * X ^ n) i := by |
refine ⟨r * a ^ n * b ^ (N - n), bi, bu, ?_⟩
rw [C_mul_X_pow_eq_monomial, map_monomial, ← C_mul_X_pow_eq_monomial, eval_mul, eval_pow, eval_C]
rw [RingHom.map_mul, RingHom.map_mul, RingHom.map_pow, RingHom.map_pow, eval_X, mul_comm]
rw [← tsub_add_cancel_of_le nN]
conv_lhs => rw [← mul_one (i a), ← bu]
sim... |
/-
Copyright (c) 2021 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Yaël Dillies
-/
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
/-!
... | Mathlib/Computability/EpsilonNFA.lean | 110 | 112 | theorem evalFrom_append_singleton (S : Set σ) (x : List α) (a : α) :
M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a := by |
rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.LinearAlgebra.Tenso... | Mathlib/Algebra/Algebra/Bilinear.lean | 134 | 136 | theorem commute_mulLeft_right (a b : A) : Commute (mulLeft R a) (mulRight R b) := by |
ext c
exact (mul_assoc a c b).symm
|
/-
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.Inverse
import Mathlib.Analysis.SpecialFunctions... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 82 | 90 | theorem differentiableWithinAt_arcsin_Ici {x : ℝ} :
DifferentiableWithinAt ℝ arcsin (Ici x) x ↔ x ≠ -1 := by |
refine ⟨?_, fun h => (hasDerivWithinAt_arcsin_Ici h).differentiableWithinAt⟩
rintro h rfl
have : sin ∘ arcsin =ᶠ[𝓝[≥] (-1 : ℝ)] id := by
filter_upwards [Icc_mem_nhdsWithin_Ici ⟨le_rfl, neg_lt_self (zero_lt_one' ℝ)⟩] with x using
sin_arcsin'
have := h.hasDerivWithinAt.sin.congr_of_eventuallyEq this.s... |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 27 | 32 | theorem Ici_add_bij : BijOn (· + d) (Ici a) (Ici (a + d)) := by |
refine
⟨fun x h => add_le_add_right (mem_Ici.mp h) _, (add_left_injective d).injOn, fun _ h => ?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ici.mp h)
rw [mem_Ici, add_right_comm, add_le_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm]⟩
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d449... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 68 | 71 | theorem Equiv.pointReflection_midpoint_left (x y : P) :
(Equiv.pointReflection (midpoint R x y)) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.LinearAlgebra.Dimension.Constructions
#align_import algebra.linear_recurrence from "leanprover-community/ma... | Mathlib/Algebra/LinearRecurrence.lean | 85 | 88 | theorem is_sol_mkSol (init : Fin E.order → α) : E.IsSolution (E.mkSol init) := by |
intro n
rw [mkSol]
simp
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Shapes.Types
#alig... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 112 | 118 | theorem isGluing_iff_pairwise {sf s} : IsGluing F U sf s ↔
∀ i, (F.mapCone (Pairwise.cocone U).op).π.app i s = objPairwiseOfFamily sf i := by |
refine ⟨fun h ↦ ?_, fun h i ↦ h (op <| Pairwise.single i)⟩
rintro (i|⟨i,j⟩)
· exact h i
· rw [← (F.mapCone (Pairwise.cocone U).op).w (op <| Pairwise.Hom.left i j)]
exact congr_arg _ (h i)
|
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 63 | 66 | theorem card_div_mul_le_card_mul_mul_card_mul (A B C : Finset α) :
(A / C).card * B.card ≤ (A * B).card * (B * C).card := by |
rw [← div_inv_eq_mul, ← card_inv B, ← card_inv (B * C), mul_inv, ← div_eq_mul_inv]
exact card_div_mul_le_card_div_mul_card_div _ _ _
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Floris van Doorn
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEqu... | Mathlib/Analysis/Convex/AmpleSet.lean | 94 | 96 | theorem preimage {s : Set F} (h : AmpleSet s) (L : E ≃ᵃL[ℝ] F) : AmpleSet (L ⁻¹' s) := by |
rw [← L.image_symm_eq_preimage]
exact h.image L.symm
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex J. Best
-/
import Mathlib.MeasureTheory.Group.Arithmetic
#align_import measure_theory.group.pointwise from "leanprover-community/mathlib"@"66f7114a1d5cba4... | Mathlib/MeasureTheory/Group/Pointwise.lean | 24 | 28 | theorem MeasurableSet.const_smul {G α : Type*} [Group G] [MulAction G α] [MeasurableSpace G]
[MeasurableSpace α] [MeasurableSMul G α] {s : Set α} (hs : MeasurableSet s) (a : G) :
MeasurableSet (a • s) := by |
rw [← preimage_smul_inv]
exact measurable_const_smul _ hs
|
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Batteries.Data.List.Lemmas
/-!
# Basic properties of `List.eraseIdx`
`List.eraseIdx l k` erases `k`-th element of `l : List α`.
If `k ≥ length l`, then it ... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 76 | 80 | theorem mem_eraseIdx_iff_get? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l.get? i = x := by |
simp only [mem_eraseIdx_iff_get, Fin.exists_iff, exists_and_left, get_eq_iff, exists_prop]
refine exists_congr fun i => and_congr_right' <| and_iff_right_of_imp fun h => ?_
obtain ⟨h, -⟩ := get?_eq_some.1 h
exact h
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_impo... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 76 | 85 | theorem exists_pos (h : LiouvilleWith p x) :
∃ (C : ℝ) (_h₀ : 0 < C),
∃ᶠ n : ℕ in atTop, 1 ≤ n ∧ ∃ m : ℤ, x ≠ m / n ∧ |x - m / n| < C / n ^ p := by |
rcases h with ⟨C, hC⟩
refine ⟨max C 1, zero_lt_one.trans_le <| le_max_right _ _, ?_⟩
refine ((eventually_ge_atTop 1).and_frequently hC).mono ?_
rintro n ⟨hle, m, hne, hlt⟩
refine ⟨hle, m, hne, hlt.trans_le ?_⟩
gcongr
apply le_max_left
|
/-
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 | 120 | 120 | theorem factorization_one : factorization 1 = 0 := by | ext; simp [factorization]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 125 | 126 | theorem measure_union_null_iff : μ (s ∪ t) = 0 ↔ μ s = 0 ∧ μ t = 0 := by |
simp [union_eq_iUnion, and_comm]
|
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Scott Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanpr... | Mathlib/Data/Matrix/Basis.lean | 139 | 140 | theorem apply_of_row_ne {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :
stdBasisMatrix i j a i' j' = 0 := by | simp [hi]
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
/-!
# Covolume of ℤ-lattices
Let `E` be a finite dimensional real vector space with an inner product.
Let `L` be a `ℤ`-l... | Mathlib/Algebra/Module/Zlattice/Covolume.lean | 67 | 76 | theorem covolume_eq_det_mul_measure {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ L)
(b₀ : Basis ι ℝ E) :
covolume L μ = |b₀.det ((↑) ∘ b)| * (μ (Zspan.fundamentalDomain b₀)).toReal := by |
rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain b μ),
Zspan.measure_fundamentalDomain _ _ b₀, measure_congr
(Zspan.fundamentalDomain_ae_parallelepiped b₀ μ), ENNReal.toReal_mul, ENNReal.toReal_ofReal
(by positivity)]
congr
ext
exact b.ofZlatticeBasis_apply ℝ L _
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 120 | 121 | theorem smul_of [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R) :
a • of f x = of f (a • x) := by | rw [of, RingHom.comp_apply, RingHom.comp_apply, smul_mk, smul_C]
|
/-
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 | 107 | 111 | theorem Heap.noSibling_tail? {s : Heap α} : s.tail? le = some s' →
s'.NoSibling := by |
simp only [Heap.tail?]; intro eq
match eq₂ : s.deleteMin le, eq with
| some (a, tl), rfl => exact noSibling_deleteMin eq₂
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e... | Mathlib/Data/Set/Opposite.lean | 92 | 96 | theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by |
ext
constructor
· apply op_injective
· apply unop_injective
|
/-
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.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 70 | 70 | theorem map_comp (g : β → γ) : (p.map f).map g = p.map (g ∘ f) := by | simp [map, Function.comp]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.ZMod.Basic
#align_import data.zmod.parity from "leanprover-community/mathlib"@"048240e809f04e2bde02482ab44bc23074... | Mathlib/Data/ZMod/Parity.lean | 28 | 29 | theorem eq_one_iff_odd {n : ℕ} : (n : ZMod 2) = 1 ↔ Odd n := by |
rw [← @Nat.cast_one (ZMod 2), ZMod.eq_iff_modEq_nat, Nat.odd_iff, Nat.ModEq]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
/-!
# Derivatives of aff... | Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 32 | 34 | theorem hasStrictDerivAt : HasStrictDerivAt f (f.linear 1) x := by |
rw [f.decomp]
exact f.linear.hasStrictDerivAt.add_const (f 0)
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 53 | 53 | theorem logb_one : logb b 1 = 0 := by | simp [logb]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411... | Mathlib/ModelTheory/ElementaryMaps.lean | 103 | 104 | theorem map_sentence (f : M ↪ₑ[L] N) (φ : L.Sentence) : M ⊨ φ ↔ N ⊨ φ := by |
rw [Sentence.Realize, Sentence.Realize, ← f.map_formula, Unique.eq_default (f ∘ default)]
|
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice
#align_import order.irreducible from "leanprover-community/mathlib"@"bf2428c9486c407ca38b5b3fb10b87dad0bc99fa"
/-!
# Irreducible and ... | Mathlib/Order/Irreducible.lean | 107 | 107 | theorem SupPrime.ne_bot (ha : SupPrime a) : a ≠ ⊥ := by | rintro rfl; exact not_supPrime_bot ha
|
/-
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.Data.Nat.ModEq
import Mathlib.Tactic.Abel
import Mathlib.Tactic.GCongr.Core
#align_import data.int.modeq from "leanprover-community/mathlib"@"47a1a73351de... | Mathlib/Data/Int/ModEq.lean | 98 | 100 | theorem modEq_iff_add_fac {a b n : ℤ} : a ≡ b [ZMOD n] ↔ ∃ t, b = a + n * t := by |
rw [modEq_iff_dvd]
exact exists_congr fun t => sub_eq_iff_eq_add'
|
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 111 | 112 | theorem finSuccEquiv'_at (i : Fin (n + 1)) : (finSuccEquiv' i) i = none := by |
simp [finSuccEquiv']
|
/-
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.Data.Finset.Option
#align_import data.fintype.option from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11... | Mathlib/Data/Fintype/Option.lean | 114 | 119 | theorem Finite.induction_empty_option {P : Type u → Prop} (of_equiv : ∀ {α β}, α ≃ β → P α → P β)
(h_empty : P PEmpty) (h_option : ∀ {α} [Fintype α], P α → P (Option α)) (α : Type u)
[Finite α] : P α := by |
cases nonempty_fintype α
refine Fintype.induction_empty_option ?_ ?_ ?_ α
exacts [fun α β _ => of_equiv, h_empty, @h_option]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "lea... | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 37 | 38 | theorem arg_expMapCircle {x : ℝ} (h₁ : -π < x) (h₂ : x ≤ π) : arg (expMapCircle x) = x := by |
rw [expMapCircle_apply, exp_mul_I, arg_cos_add_sin_mul_I ⟨h₁, h₂⟩]
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversa... | Mathlib/Control/Traversable/Instances.lean | 47 | 51 | theorem Option.naturality {α β} (f : α → F β) (x : Option α) :
η (Option.traverse f x) = Option.traverse (@η _ ∘ f) x := by |
-- Porting note: added `ApplicativeTransformation` theorems
cases' x with x <;> simp! [*, functor_norm, ApplicativeTransformation.preserves_map,
ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstance... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 122 | 125 | theorem subsingleton_iff_eq_bot : Subsingleton p ↔ p = ⊥ := by |
rw [subsingleton_iff, Submodule.eq_bot_iff]
refine ⟨fun h x hx ↦ by simpa using h ⟨x, hx⟩ ⟨0, p.zero_mem⟩,
fun h ⟨x, hx⟩ ⟨y, hy⟩ ↦ by simp [h x hx, h y hy]⟩
|
/-
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
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 92 | 100 | theorem mapAccumr_mapAccumr₂ (f₁ : γ → σ₁ → σ₁ × ζ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr f₁ (mapAccumr₂ f₂ xs ys s₂).snd s₁)
= let m := mapAccumr₂ (fun x y s =>
let r₂ := f₂ x y s.snd
let r₁ := f₁ r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂)
(m.fst.fst,... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation
/-!
# Separated maps and locally injective maps out of a topological space.
This module introduces a... | Mathlib/Topology/SeparatedMap.lean | 79 | 87 | theorem isSeparatedMap_iff_isClosed_diagonal {f : X → Y} :
IsSeparatedMap f ↔ IsClosed f.pullbackDiagonal := by |
simp_rw [isSeparatedMap_iff_nhds, ← isOpen_compl_iff, isOpen_iff_mem_nhds,
Subtype.forall, Prod.forall, nhds_induced, nhds_prod_eq]
refine forall₄_congr fun x₁ x₂ _ _ ↦ ⟨fun h ↦ ?_, fun ⟨t, ht, t_sub⟩ ↦ ?_⟩
· simp_rw [← Filter.disjoint_iff, ← compl_diagonal_mem_prod] at h
exact ⟨_, h, subset_rfl⟩
· obt... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 113 | 115 | theorem finrank_eq (f : M ≃ₗ[R] M₂) : finrank R M = finrank R M₂ := by |
unfold finrank
rw [← Cardinal.toNat_lift, f.lift_rank_eq, Cardinal.toNat_lift]
|
/-
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.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "l... | Mathlib/Algebra/Group/UniqueProds.lean | 71 | 75 | theorem of_card_le_one (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card ≤ 1) (hB1 : B.card ≤ 1) :
∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b := by |
rw [Finset.card_le_one_iff] at hA1 hB1
obtain ⟨a, ha⟩ := hA; obtain ⟨b, hb⟩ := hB
exact ⟨a, ha, b, hb, fun _ _ ha' hb' _ ↦ ⟨hA1 ha' ha, hB1 hb' hb⟩⟩
|
/-
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
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 108 | 117 | theorem mapAccumr₂_mapAccumr₂_left_left (f₁ : γ → α → σ₁ → σ₁ × φ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr₂ f₂ xs ys s₂).snd xs s₁)
= let m := mapAccumr₂ (fun x y (s₁, s₂) =>
let r₂ := f₂ x y s₂
let r₁ := f₁ r₂.snd x s₁
((r₁.fst, r₂.fst), r₁.snd)
... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
/-
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.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 105 | 107 | theorem withDensity_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by |
simpa only [add_comm] using withDensity_add_left hg f
|
/-
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.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 77 | 122 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
(hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν := by |
by_cases hE : CompleteSpace E; swap; · simp [integral, hE, stronglyMeasurable_const]
borelize E
haveI : SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
hf.separableSpace_range_union_singleton
let s : ℕ → SimpleFunc (α × β) E :=
SimpleFunc.approxOn _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by sim... |
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