Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
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import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 120 | 124 | theorem monomial_mem_lifts {s : S} (n : ℕ) (h : s ∈ Set.range f) : monomial n s ∈ lifts f := by |
obtain ⟨r, rfl⟩ := Set.mem_range.1 h
use monomial n r
simp only [coe_mapRingHom, Set.mem_univ, map_monomial, Subsemiring.coe_top, eq_self_iff_true,
and_self_iff]
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 162 | 164 | theorem cons_dotProduct (x : α) (v : Fin n → α) (w : Fin n.succ → α) :
dotProduct (vecCons x v) w = x * vecHead w + dotProduct v (vecTail w) := by |
simp [dotProduct, Fin.sum_univ_succ, vecHead, vecTail]
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 113 | 115 | theorem QuotientAddGroup.norm_eq_infDist {S : AddSubgroup M} (x : M ⧸ S) :
‖x‖ = infDist 0 { m : M | (m : M ⧸ S) = x } := by |
simp only [AddSubgroup.quotient_norm_eq, infDist_eq_iInf, sInf_image', dist_zero_left]
|
import Mathlib.Algebra.Polynomial.DenomsClearable
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Data.Real.Irrational
import Mathlib.Topology.Algebra.Polynomial
#align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"04e80bb7e851095... | Mathlib/NumberTheory/Liouville/Basic.lean | 95 | 120 | theorem exists_one_le_pow_mul_dist {Z N R : Type*} [PseudoMetricSpace R] {d : N → ℝ}
{j : Z → N → R} {f : R → R} {α : R} {ε M : ℝ}
-- denominators are positive
(d0 : ∀ a : N, 1 ≤ d a)
(e0 : 0 < ε)
-- function is Lipschitz at α
(B : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M)... |
-- A useful inequality to keep at hand
have me0 : 0 < max (1 / ε) M := lt_max_iff.mpr (Or.inl (one_div_pos.mpr e0))
-- The maximum between `1 / ε` and `M` works
refine ⟨max (1 / ε) M, me0, fun z a => ?_⟩
-- First, let's deal with the easy case in which we are far away from `α`
by_cases dm1 : 1 ≤ dist α (j ... |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 81 | 83 | theorem flip' (n ν : ℕ) (h : ν ≤ n) :
bernsteinPolynomial R n ν = (bernsteinPolynomial R n (n - ν)).comp (1 - X) := by |
simp [← flip _ _ _ h, Polynomial.comp_assoc]
|
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import linear_algebra.matrix.charpoly.eigs from "leanprover-community/mathlib"@"48dc6abe71248bd6f4bffc9703dc87bdd4e37d0b"
variable {n : Type*} [Fintype n] [DecidableEq n]
variable {R : Type*} [Field R]
variable {A : Matrix... | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 60 | 64 | theorem det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
A.det = (Matrix.charpoly A).roots.prod := by |
rw [det_eq_sign_charpoly_coeff, ← charpoly_natDegree_eq_dim A,
Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split A.charpoly_monic hAps, ← mul_assoc,
← pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul]
|
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 169 | 177 | theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i ∈ s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i ∈ s, w i * z i = x :=
calc
∑ i ∈ s, w i * z i = ∑ i ∈ s, w i * x := by |
refine sum_congr rfl fun i hi => ?_
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 320 | 321 | theorem primitiveRoots_zero : primitiveRoots 0 R = ∅ := by |
rw [primitiveRoots, nthRoots_zero, Multiset.toFinset_zero, Finset.filter_empty]
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Sigma
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Vector
import Mathlib.Algebra.BigOperators.Option
#align_import data.fintype.big_operators from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde5525935... | Mathlib/Data/Fintype/BigOperators.lean | 83 | 86 | theorem prod_eq_mul {f : α → M} (a b : α) (h₁ : a ≠ b) (h₂ : ∀ x, x ≠ a ∧ x ≠ b → f x = 1) :
∏ x, f x = f a * f b := by |
apply Finset.prod_eq_mul a b h₁ fun x _ hx => h₂ x hx <;>
exact fun hc => (hc (Finset.mem_univ _)).elim
|
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
#align_import order.iterate from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
open Function
open Function (Commute)
namespace Monotone
variable {α : Type*} [Preorder α] {f : α → α} {x y : ℕ → α}
theorem se... | Mathlib/Order/Iterate.lean | 51 | 60 | theorem seq_pos_lt_seq_of_lt_of_le (hf : Monotone f) {n : ℕ} (hn : 0 < 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 |
induction' n with n ihn
· exact hn.false.elim
suffices x n ≤ y n from (hx n n.lt_succ_self).trans_le ((hf this).trans <| hy n n.lt_succ_self)
cases n with
| zero => exact h₀
| succ n =>
refine (ihn n.zero_lt_succ (fun k hk => hx _ ?_) fun k hk => hy _ ?_).le <;>
exact hk.trans n.succ.lt_succ_self
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 115 | 117 | theorem mem_const_vadd_affineSegment {x y z : P} (v : V) :
v +ᵥ z ∈ affineSegment R (v +ᵥ x) (v +ᵥ y) ↔ z ∈ affineSegment R x y := by |
rw [← affineSegment_const_vadd_image, (AddAction.injective v).mem_set_image]
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 47 | 53 | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a := by |
classical
by_cases ha : a = 0
· rw [ha, monomial_zero_right, mirror_zero]
· rw [mirror, reverse, natDegree_monomial n a, if_neg ha, natTrailingDegree_monomial ha, ←
C_mul_X_pow_eq_monomial, reflect_C_mul_X_pow, revAt_le (le_refl n), tsub_self, pow_zero,
mul_one]
|
import Mathlib.CategoryTheory.Comma.Basic
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Small.Set
#align_import category_theory.structured_arrow from "leanprover-community/mathlib"@"8a318021995877a44630c898d... | Mathlib/CategoryTheory/Comma/StructuredArrow.lean | 90 | 91 | theorem w {A B : StructuredArrow S T} (f : A ⟶ B) : A.hom ≫ T.map f.right = B.hom := by |
have := f.w; aesop_cat
|
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 93 | 101 | theorem not_integrableOn_Ioi_rpow (s : ℝ) : ¬ IntegrableOn (fun x ↦ x ^ s) (Ioi (0 : ℝ)) := by |
intro h
rcases le_or_lt s (-1) with hs|hs
· have : IntegrableOn (fun x ↦ x ^ s) (Ioo (0 : ℝ) 1) := h.mono Ioo_subset_Ioi_self le_rfl
rw [integrableOn_Ioo_rpow_iff zero_lt_one] at this
exact hs.not_lt this
· have : IntegrableOn (fun x ↦ x ^ s) (Ioi (1 : ℝ)) := h.mono (Ioi_subset_Ioi zero_le_one) le_rfl
... |
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite Topolog... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 126 | 130 | theorem quasiSeparated_eq_affineProperty_diagonal :
@QuasiSeparated = targetAffineLocally QuasiCompact.affineProperty.diagonal := by |
rw [quasiSeparated_eq_diagonal_is_quasiCompact, quasiCompact_eq_affineProperty]
exact
diagonal_targetAffineLocally_eq_targetAffineLocally _ QuasiCompact.affineProperty_isLocal
|
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 46 | 48 | theorem continuum_le_lift {c : Cardinal.{u}} : 𝔠 ≤ lift.{v} c ↔ 𝔠 ≤ c := by |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_le]
|
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
open MeasureTheory ProbabilityTheory
open scoped MeasureTheory ENNReal NNReal
namesp... | Mathlib/Probability/Kernel/WithDensity.lean | 116 | 122 | theorem integral_withDensity {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
(hg : Measurable (Function.uncurry g)) :
∫ b, f b ∂withDensity κ (fun a b => g a b) a = ∫ b, g a b • f b ∂κ a := by |
rw [kernel.withDensity_apply, integral_withDensity_eq_integral_smul]
· exact Measurable.of_uncurry_left hg
· exact measurable_coe_nnreal_ennreal.comp hg
|
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 28 | 33 | theorem Filter.IsBoundedUnder.isLittleO_sub_self_inv {𝕜 E : Type*} [NormedField 𝕜] [Norm E] {a : 𝕜}
{f : 𝕜 → E} (h : IsBoundedUnder (· ≤ ·) (𝓝[≠] a) (norm ∘ f)) :
f =o[𝓝[≠] a] fun x => (x - a)⁻¹ := by |
refine (h.isBigO_const (one_ne_zero' ℝ)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr ?_)
simp only [(· ∘ ·), norm_inv]
exact (tendsto_norm_sub_self_punctured_nhds a).inv_tendsto_zero
|
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section UnusedInput
variable {xs : Vector α n} {ys : Vector β n}
@[simp]
theorem mapAccumr₂_unused_input_left [Inhabited α] (f : α → β → σ → σ × γ)
(h : ∀ a b s, f default b s =... | Mathlib/Data/Vector/MapLemmas.lean | 354 | 359 | theorem mapAccumr₂_unused_input_right [Inhabited β] (f : α → β → σ → σ × γ)
(h : ∀ a b s, f a default s = f a b s) :
mapAccumr₂ f xs ys s = mapAccumr (fun a s => f a default s) xs s := by |
induction xs, ys using Vector.revInductionOn₂ generalizing s with
| nil => rfl
| snoc xs ys x y ih => simp [h x y s, ih]
|
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 204 | 209 | theorem pnatCast_one [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] : ((1 : ℕ+) : Rˣ) = 1 := by |
apply Units.ext
rw [Units.val_one]
change ((PNat.isUnit_natCast (R := R) 1).unit : R) = 1
rw [IsUnit.unit_spec (PNat.isUnit_natCast 1)]
rw [PNat.one_coe, Nat.cast_one]
|
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 60 | 63 | theorem Squarefree.ne_zero [MonoidWithZero R] [Nontrivial R] {m : R} (hm : Squarefree (m : R)) :
m ≠ 0 := by |
rintro rfl
exact not_squarefree_zero hm
|
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 31 | 34 | theorem preimage_image_sigmaMk_of_ne (h : i ≠ j) (s : Set (α j)) :
Sigma.mk i ⁻¹' (Sigma.mk j '' s) = ∅ := by |
ext x
simp [h.symm]
|
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 121 | 126 | theorem isPrime_of_isPrime_prod_top' {I : Ideal S} (h : (Ideal.prod (⊤ : Ideal R) I).IsPrime) :
I.IsPrime := by |
apply isPrime_of_isPrime_prod_top (S := R)
rw [← map_prodComm_prod]
-- Note: couldn't synthesize the right instances without the `R` and `S` hints
exact map_isPrime_of_equiv (RingEquiv.prodComm (R := R) (S := S))
|
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Function Set Metric TopologicalSpace intervalIntegral MeasureTheory
open MeasureTh... | Mathlib/Analysis/ODE/PicardLindelof.lean | 146 | 147 | theorem proj_of_mem {t : ℝ} (ht : t ∈ Icc v.tMin v.tMax) : ↑(v.proj t) = t := by |
simp only [proj, projIcc_of_mem v.tMin_le_tMax ht]
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.comp from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncomputable sect... | Mathlib/Analysis/Calculus/FDeriv/Comp.lean | 53 | 59 | theorem HasFDerivAtFilter.comp {g : F → G} {g' : F →L[𝕜] G} {L' : Filter F}
(hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
HasFDerivAtFilter (g ∘ f) (g'.comp f') x L := by |
let eq₁ := (g'.isBigO_comp _ _).trans_isLittleO hf.isLittleO
let eq₂ := (hg.isLittleO.comp_tendsto hL).trans_isBigO hf.isBigO_sub
refine .of_isLittleO <| eq₂.triangle <| eq₁.congr_left fun x' => ?_
simp
|
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 105 | 124 | theorem exists_finset_card_le_mul [FiniteIndex H] {S : Finset G} (hS : closure (S : Set G) = ⊤) :
∃ T : Finset H, T.card ≤ H.index * S.card ∧ closure (T : Set H) = ⊤ := by |
letI := H.fintypeQuotientOfFiniteIndex
haveI : DecidableEq G := Classical.decEq G
obtain ⟨R₀, hR, hR1⟩ := H.exists_right_transversal 1
haveI : Fintype R₀ := Fintype.ofEquiv _ (toEquiv hR)
let R : Finset G := Set.toFinset R₀
replace hR : (R : Set G) ∈ rightTransversals (H : Set G) := by rwa [Set.coe_toFinse... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 58 | 69 | theorem add_modByMonic (p₁ p₂ : R[X]) : (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q := by |
by_cases hq : q.Monic
· cases' subsingleton_or_nontrivial R with hR hR
· simp only [eq_iff_true_of_subsingleton]
· exact
(div_modByMonic_unique (p₁ /ₘ q + p₂ /ₘ q) _ hq
⟨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div _ hq, ← add_assoc,
add_comm (q * _... |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 340 | 344 | theorem hasFDerivWithinAt_euclidean :
HasFDerivWithinAt f f' t y ↔
∀ i, HasFDerivWithinAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
rfl
|
import Mathlib.Geometry.Manifold.MFDeriv.Atlas
noncomputable section
open scoped Manifold
open Set
section UniqueMDiff
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type*}
[Topolog... | Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean | 98 | 107 | theorem UniqueMDiffOn.uniqueDiffOn_inter_preimage (hs : UniqueMDiffOn I s) (x : M) (y : M')
{f : M → M'} (hf : ContinuousOn f s) :
UniqueDiffOn 𝕜
((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) :=
haveI : UniqueMDiffOn I (s ∩ f ⁻¹' (extChartAt I' y).source) :=... |
intro z hz
apply (hs z hz.1).inter'
apply (hf z hz.1).preimage_mem_nhdsWithin
exact (isOpen_extChartAt_source I' y).mem_nhds hz.2
this.uniqueDiffOn_target_inter _
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v₁ v₂ u₁ u₂
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 158 | 168 | theorem pullbackCompatible_iff (x : FamilyOfElements P R) [R.hasPullbacks] :
x.Compatible ↔ x.PullbackCompatible := by |
constructor
· intro t Y₁ Y₂ f₁ f₂ hf₁ hf₂
apply t
haveI := hasPullbacks.has_pullbacks hf₁ hf₂
apply pullback.condition
· intro t Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ comm
haveI := hasPullbacks.has_pullbacks hf₁ hf₂
rw [← pullback.lift_fst _ _ comm, op_comp, FunctorToTypes.map_comp_apply, t hf₁ hf₂,
... |
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0"
universe u v
variable {α : Type*} (M : α → Type*) (u : Ultrafilter α)
open FirstOr... | Mathlib/ModelTheory/Ultraproducts.lean | 77 | 80 | theorem funMap_cast {n : ℕ} (f : L.Functions n) (x : Fin n → ∀ a, M a) :
(funMap f fun i => (x i : (u : Filter α).Product M)) =
(fun a => funMap f fun i => x i a : (u : Filter α).Product M) := by |
apply funMap_quotient_mk'
|
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 70 | 78 | theorem Sphere.secondInter_eq_self_iff {s : Sphere P} {p : P} {v : V} :
s.secondInter p v = p ↔ ⟪v, p -ᵥ s.center⟫ = 0 := by |
refine ⟨fun hp => ?_, fun hp => ?_⟩
· by_cases hv : v = 0
· simp [hv]
rwa [Sphere.secondInter, eq_comm, eq_vadd_iff_vsub_eq, vsub_self, eq_comm, smul_eq_zero,
or_iff_left hv, div_eq_zero_iff, inner_self_eq_zero, or_iff_left hv, mul_eq_zero,
or_iff_right (by norm_num : (-2 : ℝ) ≠ 0)] at hp
· r... |
import Mathlib.Init.Align
import Mathlib.Topology.PartialHomeomorph
#align_import geometry.manifold.charted_space from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open TopologicalSpace Topology
universe u
variable {H : Type u} {H' : Type*} {M : Type*} {M' : Ty... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 691 | 695 | theorem ChartedSpace.secondCountable_of_sigma_compact [SecondCountableTopology H]
[SigmaCompactSpace M] : SecondCountableTopology M := by |
obtain ⟨s, hsc, hsU⟩ : ∃ s, Set.Countable s ∧ ⋃ (x) (_ : x ∈ s), (chartAt H x).source = univ :=
countable_cover_nhds_of_sigma_compact fun x : M ↦ chart_source_mem_nhds H x
exact ChartedSpace.secondCountable_of_countable_cover H hsU hsc
|
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 54 | 56 | theorem charmatrix_apply_natDegree_le (i j : n) :
(charmatrix M i j).natDegree ≤ ite (i = j) 1 0 := by |
split_ifs with h <;> simp [h, natDegree_X_le]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 118 | 120 | theorem leadingCoeff_opRingEquiv (p : R[X]ᵐᵒᵖ) :
(opRingEquiv R p).leadingCoeff = op (unop p).leadingCoeff := by |
rw [leadingCoeff, coeff_opRingEquiv, natDegree_opRingEquiv, leadingCoeff]
|
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 70 | 74 | theorem char_tensor' (V W : FdRep k G) :
character (Action.FunctorCategoryEquivalence.inverse.obj
(Action.FunctorCategoryEquivalence.functor.obj V ⊗
Action.FunctorCategoryEquivalence.functor.obj W)) = V.character * W.character := by |
simp [← char_tensor]
|
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.List.AList
#align_import data.finsupp.alist from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
namespace AList
variable {α M : Type*} [Zero M]
open List
noncomputable def lookupFinsupp (l : AList fun _x : α => M) : α →₀ M where
... | Mathlib/Data/Finsupp/AList.lean | 116 | 120 | theorem singleton_lookupFinsupp (a : α) (m : M) :
(singleton a m).lookupFinsupp = Finsupp.single a m := by |
classical
-- porting note (#10745): was `simp [← AList.insert_empty]` but timeout issues
simp only [← AList.insert_empty, insert_lookupFinsupp, empty_lookupFinsupp, Finsupp.zero_update]
|
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import number_theory.frobenius_number from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open Nat
def FrobeniusNumber (n : ℕ) (s : Set ℕ) : Pro... | Mathlib/NumberTheory/FrobeniusNumber.lean | 55 | 82 | theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
FrobeniusNumber (m * n - m - n) {m, n} := by |
simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
have hmn : m + n ≤ m * n := add_le_mul hm hn
constructor
· push_neg
intro a b h
apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
simp only [Nat.sub_sub, smul_eq_mul] at h
zify [hmn] at h ⊢
rw [← sub_eq_zero] at h ⊢... |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
| Mathlib/Data/Vector/MapLemmas.lean | 60 | 68 | theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂ xs s₂).snd ys s₁)
= let m := (mapAccumr₂ (fun x y s =>
let r₂ := f₂ x s.snd
let r₁ := f₁ r₂.snd y s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂))
(m.f... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 153 | 159 | theorem drop_take_succ_join_eq_get' (L : List (List α)) (i : Fin L.length) :
(L.join.take (Nat.sum ((L.map length).take (i + 1)))).drop (Nat.sum ((L.map length).take i)) =
get L i := by |
have : (L.map length).take i = ((L.take (i + 1)).map length).take i := by
simp [map_take, take_take, Nat.min_eq_left]
simp only [this, length_map, take_sum_join', drop_sum_join', drop_take_succ_eq_cons_get,
join, append_nil]
|
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Basic
#align_import init.control.lawful from "leanprover-community/lean"@"9af482290ef68e8aaa5ead01aa7b09b7be7019fd"
set_option autoImplicit true
universe u v
#align is_lawful_functor LawfulFunctor
#align is_lawful_functor.map_const_eq LawfulFunctor.map_const
... | Mathlib/Init/Control/Lawful.lean | 213 | 219 | theorem run_map (f : α → β) [LawfulMonad m] : (f <$> x).run = Option.map f <$> x.run := by |
rw [← bind_pure_comp _ x.run]
change x.run >>= (fun
| some a => OptionT.run (pure (f a))
| none => pure none) = _
apply bind_congr
intro a; cases a <;> simp [Option.map, Option.bind]
|
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.Order.Copy
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.sites.grothendieck fr... | Mathlib/CategoryTheory/Sites/Grothendieck.lean | 158 | 162 | theorem intersection_covering (rj : R ∈ J X) (sj : S ∈ J X) : R ⊓ S ∈ J X := by |
apply J.transitive rj _ fun Y f Hf => _
intros Y f hf
rw [Sieve.pullback_inter, R.pullback_eq_top_of_mem hf]
simp [sj]
|
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Function PNat
namespace PNat
variable (a b : ℕ+)
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ+ := Subtype.... | Mathlib/Data/PNat/Interval.lean | 118 | 119 | theorem card_fintype_Ioc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [← card_Ioc, Fintype.card_ofFinset]
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 71 | 77 | theorem mul_left_strictMono (h0 : a ≠ 0) (hinf : a ≠ ∞) : StrictMono (a * ·) := by |
lift a to ℝ≥0 using hinf
rw [coe_ne_zero] at h0
intro x y h
contrapose! h
simpa only [← mul_assoc, ← coe_mul, inv_mul_cancel h0, coe_one, one_mul]
using mul_le_mul_left' h (↑a⁻¹)
|
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 110 | 117 | theorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b := by |
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun a => id
|
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
open Set Submodule
namespace Basis
section Semiring
variable {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ}
def flag (b : Basis (Fin n) R M) (k : Fin (n + 1)) : Submodule R M :=
.span R <... | Mathlib/LinearAlgebra/Basis/Flag.lean | 35 | 36 | theorem flag_last (b : Basis (Fin n) R M) : b.flag (.last n) = ⊤ := by |
simp [flag, Fin.castSucc_lt_last]
|
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 48 | 49 | theorem destutter'_cons_pos (h : R b a) : (a :: l).destutter' R b = b :: l.destutter' R a := by |
rw [destutter', if_pos h]
|
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 164 | 169 | theorem coe_vanishingIdeal (t : Set (PrimeSpectrum R)) :
(vanishingIdeal t : Set R) = { f : R | ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, Submodule.mem_iInf]
apply forall_congr'; intro x
rw [Submodule.mem_iInf]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 181 | 184 | theorem hasStrictDerivAt_arsinh (x : ℝ) : HasStrictDerivAt arsinh (√(1 + x ^ 2))⁻¹ x := by |
convert sinhHomeomorph.toPartialHomeomorph.hasStrictDerivAt_symm (mem_univ x) (cosh_pos _).ne'
(hasStrictDerivAt_sinh _) using 2
exact (cosh_arsinh _).symm
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 240 | 248 | theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a b ≤ slope f c b := by |
rw [← lineMap_apply_one_sub, ← lineMap_apply_one_sub _ _ r]
revert h; generalize 1 - r = r'; clear! r; intro h
simp_rw [lineMap_apply, slope, vsub_eq_sub, vadd_eq_add, smul_eq_mul]
rw [sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_mul_neg, neg_sub,
le_inv_smul_iff_of_pos h, smul_smul, mul_inv_can... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 93 | 94 | theorem mem_rootsOfUnity' (k : ℕ+) (ζ : Mˣ) : ζ ∈ rootsOfUnity k M ↔ (ζ : M) ^ (k : ℕ) = 1 := by |
rw [mem_rootsOfUnity]; norm_cast
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 100 | 108 | theorem nat_coe_dvd_iff : (n : Cardinal) ∣ m ↔ n ∣ m := by |
refine ⟨?_, fun ⟨h, ht⟩ => ⟨h, mod_cast ht⟩⟩
rintro ⟨k, hk⟩
have : ↑m < ℵ₀ := nat_lt_aleph0 m
rw [hk, mul_lt_aleph0_iff] at this
rcases this with (h | h | ⟨-, hk'⟩)
iterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk; simp [hk]
lift k to ℕ using hk'
exact ⟨k, mod_cast hk⟩
|
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 104 | 108 | theorem map_permutationsAux2 (t : α) (ts : List α) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts [] ys id).2.map f = (permutationsAux2 t ts [] ys f).2 := by |
rw [map_permutationsAux2' id, map_id, map_id]
· rfl
simp
|
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 211 | 212 | theorem pred_eval (v) : pred.eval v = pure [v.headI.pred] := by |
simp [pred]; cases v.headI <;> simp
|
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 88 | 90 | theorem smeval_X :
(X : R[X]).smeval x = x ^ 1 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_X_index, one_smul]
|
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
namespace CategoryTheory
namespace Bicategory
open Category
open scoped Bicategory
open Mathlib.Tactic.BicategoryCoherence (bicategoricalComp bicategoricalIsoComp)
universe w v u
variable {B : Type u} [Bicategory... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 79 | 91 | theorem rightZigzag_idempotent_of_left_triangle
(η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) (h : leftZigzag η ε = (λ_ _).hom ≫ (ρ_ _).inv) :
rightZigzag η ε ⊗≫ rightZigzag η ε = rightZigzag η ε := by |
dsimp only [rightZigzag]
calc
_ = g ◁ η ⊗≫ ((ε ▷ g ▷ 𝟙 a) ≫ (𝟙 b ≫ g) ◁ η) ⊗≫ ε ▷ g := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫ g ◁ (η ▷ 𝟙 a ≫ (f ≫ g) ◁ η) ⊗≫ (ε ▷ (g ≫ f) ≫ 𝟙 b ◁ ε) ▷ g ⊗≫ 𝟙 _ := by
rw [← whisker_exchange]; simp [bicategoricalComp]; coherence
_ = g ◁ η ⊗≫ g ... |
import Mathlib.Data.Set.Lattice
#align_import data.semiquot from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
-- Porting note: removed universe parameter
structure Semiquot (α : Type*) where mk' ::
s : Set α
val : Trunc s
#align semiquot Semiquot
namespace Semiquot
vari... | Mathlib/Data/Semiquot.lean | 47 | 50 | theorem ext_s {q₁ q₂ : Semiquot α} : q₁ = q₂ ↔ q₁.s = q₂.s := by |
refine ⟨congr_arg _, fun h => ?_⟩
cases' q₁ with _ v₁; cases' q₂ with _ v₂; congr
exact Subsingleton.helim (congrArg Trunc (congrArg Set.Elem h)) v₁ v₂
|
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 79 | 80 | theorem length_ndinsert_of_not_mem {a : α} {s : Multiset α} (h : a ∉ s) :
card (ndinsert a s) = card s + 1 := by | simp [h]
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 119 | 121 | theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by |
obtain rfl | rfl := subset_singleton_iff_eq.1 h
exacts [gauge_empty, gauge_zero']
|
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 109 | 114 | theorem div_neg_eq_neg_div (a b : K) : b / -a = -(b / a) :=
calc
b / -a = b * (1 / -a) := by | rw [← inv_eq_one_div, division_def]
_ = b * -(1 / a) := by rw [one_div_neg_eq_neg_one_div]
_ = -(b * (1 / a)) := by rw [neg_mul_eq_mul_neg]
_ = -(b / a) := by rw [mul_one_div]
|
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 110 | 116 | theorem lmarginal_update_of_mem {i : δ} (hi : i ∈ s)
(f : (∀ i, π i) → ℝ≥0∞) (x : ∀ i, π i) (y : π i) :
(∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∂μ) x := by |
apply lmarginal_congr
intro j hj
have : j ≠ i := by rintro rfl; exact hj hi
apply update_noteq this
|
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 92 | 93 | theorem one_lt_rank_of_one_lt_finrank (h : 1 < finrank R M) : 1 < Module.rank R M := by |
simpa using lt_rank_of_lt_finrank h
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 109 | 118 | theorem weightedVSubOfPoint_eq_of_sum_eq_zero (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 0)
(b₁ b₂ : P) : s.weightedVSubOfPoint p b₁ w = s.weightedVSubOfPoint p b₂ w := by |
apply eq_of_sub_eq_zero
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, ← sum_sub_distrib]
conv_lhs =>
congr
· skip
· ext
rw [← smul_sub, vsub_sub_vsub_cancel_left]
rw [← sum_smul, h, zero_smul]
|
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.PowerSeries.Order
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 54 | 81 | theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
coeff R n (inv.aux a φ) =
if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux]
simp only [Finsupp.single_eq_zero]
split_ifs; · rfl
congr 1
symm
apply Finset.sum_nbij' (fun (a, b) ↦ (single () a, single () b))
fun (f, g) ↦ (f (), g ())
· aesop
· aesop
· ... |
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
namespace Sigma
variable {ι : Type*} {E : ι → Type... | Mathlib/Topology/MetricSpace/Gluing.lean | 342 | 343 | theorem dist_same (i : ι) (x y : E i) : dist (Sigma.mk i x) ⟨i, y⟩ = dist x y := by |
simp [Dist.dist, Sigma.dist]
|
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 111 | 111 | theorem birthday_star : birthday star = 1 := by | rw [birthday_def]; simp
|
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 153 | 153 | theorem not_mem_top {n m : ℕ} : m ∉ Ico n m := by | simp
|
import Mathlib.Algebra.Group.Defs
#align_import group_theory.eckmann_hilton from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
universe u
namespace EckmannHilton
variable {X : Type u}
local notation a " <" m:51 "> " b => m a b
structure IsUnital (m : X → X → X) (e : X) extends Std... | Mathlib/GroupTheory/EckmannHilton.lean | 56 | 57 | theorem one : e₁ = e₂ := by |
simpa only [h₁.left_id, h₁.right_id, h₂.left_id, h₂.right_id] using distrib e₂ e₁ e₁ e₂
|
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
section
open scoped Classical
theorem Algebra.fg_trans' {R S... | Mathlib/RingTheory/Adjoin/Tower.lean | 92 | 135 | theorem exists_subalgebra_of_fg (hAC : (⊤ : Subalgebra A C).FG) (hBC : (⊤ : Submodule B C).FG) :
∃ B₀ : Subalgebra A B, B₀.FG ∧ (⊤ : Submodule B₀ C).FG := by |
cases' hAC with x hx
cases' hBC with y hy
have := hy
simp_rw [eq_top_iff', mem_span_finset] at this
choose f hf using this
let s : Finset B := Finset.image₂ f (x ∪ y * y) y
have hxy :
∀ xi ∈ x, xi ∈ span (Algebra.adjoin A (↑s : Set B)) (↑(insert 1 y : Finset C) : Set C) :=
fun xi hxi =>
hf xi... |
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.LocallyFinite
import Mathlib.Topology.ProperMap
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.compact_convergence from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
univer... | Mathlib/Topology/UniformSpace/CompactConvergence.lean | 103 | 144 | theorem tendsto_iff_forall_compact_tendstoUniformlyOn
{ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f} :
Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K := by |
rw [tendsto_nhds_compactOpen]
constructor
· -- Let us prove that convergence in the compact-open topology
-- implies uniform convergence on compacts.
-- Consider a compact set `K`
intro h K hK
-- Since `K` is compact, it suffices to prove locally uniform convergence
rw [← tendstoLocallyUnifor... |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 160 | 165 | theorem weightedVSubOfPoint_indicator_subset (w : ι → k) (p : ι → P) (b : P) {s₁ s₂ : Finset ι}
(h : s₁ ⊆ s₂) :
s₁.weightedVSubOfPoint p b w = s₂.weightedVSubOfPoint p b (Set.indicator (↑s₁) w) := by |
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]
exact Eq.symm <|
sum_indicator_subset_of_eq_zero w (fun i wi => wi • (p i -ᵥ b : V)) h fun i => zero_smul k _
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 88 | 89 | theorem taylor_coeff_zero : (taylor r f).coeff 0 = f.eval r := by |
rw [taylor_coeff, hasseDeriv_zero, LinearMap.id_apply]
|
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 100 | 102 | theorem transpose_row (v : m → α) : (Matrix.row v)ᵀ = Matrix.col v := by |
ext
rfl
|
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 103 | 108 | theorem foldr_loop (f : Fin (n+1) → α → α) (x) (h : m+1 ≤ n+1) :
foldr.loop (n+1) f ⟨m+1, h⟩ x =
f 0 (foldr.loop n (fun i => f i.succ) ⟨m, Nat.le_of_succ_le_succ h⟩ x) := by |
induction m generalizing x with
| zero => simp [foldr_loop_zero, foldr_loop_succ]
| succ m ih => rw [foldr_loop_succ, ih, foldr_loop_succ, Fin.succ]
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 131 | 131 | theorem self_mem_cylinder (x : ∀ n, E n) (n : ℕ) : x ∈ cylinder x n := by | simp
|
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 107 | 109 | theorem dart_edge_eq_iff : ∀ d₁ d₂ : G.Dart, d₁.edge = d₂.edge ↔ d₁ = d₂ ∨ d₁ = d₂.symm := by |
rintro ⟨p, hp⟩ ⟨q, hq⟩
simp
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 208 | 208 | theorem getVert_zero {u v} (w : G.Walk u v) : w.getVert 0 = u := by | cases w <;> rfl
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 134 | 139 | theorem succ_mul_binomial [DecidableEq α] (h : a ≠ b) :
(f a + f b).succ * multinomial {a, b} f =
(f a).succ * multinomial {a, b} (Function.update f a (f a).succ) := by |
rw [binomial_eq_choose h, binomial_eq_choose h, mul_comm (f a).succ, Function.update_same,
Function.update_noteq (ne_comm.mp h)]
rw [succ_mul_choose_eq (f a + f b) (f a), succ_add (f a) (f b)]
|
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 127 | 129 | theorem nat_mul_inj {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) : n = 0 ∨ a = b := by |
rw [← sub_eq_zero, ← mul_sub, mul_eq_zero, sub_eq_zero] at h
exact mod_cast h
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 162 | 162 | theorem one_re : re (1 : K) = 1 := by | rw [← ofReal_one, ofReal_re]
|
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 560 | 565 | theorem frobenius_nnnorm_def (A : Matrix m n α) :
‖A‖₊ = (∑ i, ∑ j, ‖A i j‖₊ ^ (2 : ℝ)) ^ (1 / 2 : ℝ) := by |
-- Porting note: added, along with `WithLp.equiv_symm_pi_apply` below
change ‖(WithLp.equiv 2 _).symm fun i => (WithLp.equiv 2 _).symm fun j => A i j‖₊ = _
simp_rw [PiLp.nnnorm_eq_of_L2, NNReal.sq_sqrt, NNReal.sqrt_eq_rpow, NNReal.rpow_two,
WithLp.equiv_symm_pi_apply]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 120 | 129 | theorem sublistsAux_eq_array_foldl :
sublistsAux = fun (a : α) (r : List (List α)) =>
(r.toArray.foldl (init := #[])
fun r l => (r.push l).push (a :: l)).toList := by |
funext a r
simp only [sublistsAux, Array.foldl_eq_foldl_data, Array.mkEmpty]
have := foldl_hom Array.toList (fun r l => (r.push l).push (a :: l))
(fun (r : List (List α)) l => r ++ [l, a :: l]) r #[]
(by simp)
simpa using this
|
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.Lattice
import Mathlib.Order.CompleteLattice
#align_import category_theory.sites.sieves from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef... | Mathlib/CategoryTheory/Sites/Sieves.lean | 141 | 148 | theorem ofArrows_pUnit : (ofArrows _ fun _ : PUnit => f) = singleton f := by |
funext Y
ext g
constructor
· rintro ⟨_⟩
apply singleton.mk
· rintro ⟨_⟩
exact ofArrows.mk PUnit.unit
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 57 | 57 | theorem dichotomy (b : Bool) : b = false ∨ b = true := by | cases b <;> simp
|
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 123 | 126 | theorem Dense.inter_of_Gδ {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by |
rw [inter_eq_iInter]
apply dense_iInter_of_Gδ <;> simp [Bool.forall_bool, *]
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 152 | 153 | theorem preimage_const_add_Ico : (fun x => a + x) ⁻¹' Ico b c = Ico (b - a) (c - a) := by |
simp [← Ici_inter_Iio]
|
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 92 | 97 | theorem content_C {r : R} : (C r).content = normalize r := by |
rw [content]
by_cases h0 : r = 0
· simp [h0]
have h : (C r).support = {0} := support_monomial _ h0
simp [h]
|
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
variable {α β : Type*}
open Function
namespace Option
def toFinset (o : Option α) : Finset α :=
o.elim ∅ singleton
#align option.to_finset Option.toFinset
@[simp]
... | Mathlib/Data/Finset/Option.lean | 55 | 55 | theorem card_toFinset (o : Option α) : o.toFinset.card = o.elim 0 1 := by | cases o <;> rfl
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 81 | 86 | theorem wittPolynomial_eq_sum_C_mul_X_pow (n : ℕ) :
wittPolynomial p R n = ∑ i ∈ range (n + 1), C ((p : R) ^ i) * X i ^ p ^ (n - i) := by |
apply sum_congr rfl
rintro i -
rw [monomial_eq, Finsupp.prod_single_index]
rw [pow_zero]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : ℕ → ℕ
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 113 | 118 | theorem factorial_eq_one : n ! = 1 ↔ n ≤ 1 := by |
constructor
· intro h
rw [← not_lt, ← one_lt_factorial, h]
apply lt_irrefl
· rintro (_|_|_) <;> rfl
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 130 | 134 | theorem taylor_injective {R} [CommRing R] (r : R) : Function.Injective (taylor r) := by |
intro f g h
apply_fun taylor (-r) at h
simpa only [taylor_apply, comp_assoc, add_comp, X_comp, C_comp, C_neg, neg_add_cancel_right,
comp_X] using h
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Index
import Mathlib.Order.Interval.Finset.Nat
import Mat... | Mathlib/GroupTheory/OrderOfElement.lean | 126 | 129 | theorem IsOfFinOrder.apply {η : Type*} {Gs : η → Type*} [∀ i, Monoid (Gs i)] {x : ∀ i, Gs i}
(h : IsOfFinOrder x) : ∀ i, IsOfFinOrder (x i) := by |
obtain ⟨n, npos, hn⟩ := h.exists_pow_eq_one
exact fun _ => isOfFinOrder_iff_pow_eq_one.mpr ⟨n, npos, (congr_fun hn.symm _).symm⟩
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 89 | 92 | theorem eraseLead_ne_zero (f0 : 2 ≤ f.support.card) : eraseLead f ≠ 0 := by |
rw [Ne, ← card_support_eq_zero, eraseLead_support]
exact
(zero_lt_one.trans_le <| (tsub_le_tsub_right f0 1).trans Finset.pred_card_le_card_erase).ne.symm
|
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 188 | 190 | theorem refl_symm {a : X} : (Path.refl a).symm = Path.refl a := by |
ext
rfl
|
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 78 | 80 | theorem eigenvectorUnitary_mulVec (j : n) :
eigenvectorUnitary hA *ᵥ Pi.single j 1 = ⇑(hA.eigenvectorBasis j) := by |
simp only [mulVec_single, eigenvectorUnitary_apply, mul_one]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathlib"@"148aefbd371a25f1cff33c85f20c661ce3155def"
noncomputable section
open Set Filter
open scoped Topology
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
structure D... | Mathlib/Topology/DenseEmbedding.lean | 65 | 72 | theorem closure_image_mem_nhds {s : Set α} {a : α} (di : DenseInducing i) (hs : s ∈ 𝓝 a) :
closure (i '' s) ∈ 𝓝 (i a) := by |
rw [di.nhds_eq_comap a, ((nhds_basis_opens _).comap _).mem_iff] at hs
rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩
refine mem_of_superset (hUo.mem_nhds haU) ?_
calc
U ⊆ closure (i '' (i ⁻¹' U)) := di.dense.subset_closure_image_preimage_of_isOpen hUo
_ ⊆ closure (i '' s) := closure_mono (image_subs... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 99 | 103 | theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) :
trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) := by |
rcases s.eq_empty_or_nonempty with (rfl | h)
· simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, trop_top]
rw [← inf'_eq_csInf_image _ h, inf'_eq_inf, s.trop_inf]
|
import Mathlib.Order.Chain
#align_import order.zorn from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
open scoped Classical
open Set
variable {α β : Type*} {r : α → α → Prop} {c : Set α}
local infixl:50 " ≺ " => r
theorem exists_maximal_of_chains_bounded (h : ∀ c, IsChain r c → ∃... | Mathlib/Order/Zorn.lean | 144 | 149 | theorem zorn_nonempty_Ici₀ (a : α)
(ih : ∀ c ⊆ Ici a, IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub, ∀ z ∈ c, z ≤ ub)
(x : α) (hax : a ≤ x) : ∃ m, x ≤ m ∧ ∀ z, m ≤ z → z ≤ m := by |
let ⟨m, _, hxm, hm⟩ := zorn_nonempty_preorder₀ (Ici a) (fun c hca hc y hy ↦ ?_) x hax
· exact ⟨m, hxm, fun z hmz => hm _ (hax.trans <| hxm.trans hmz) hmz⟩
· have ⟨ub, hub⟩ := ih c hca hc y hy; exact ⟨ub, (hca hy).trans (hub y hy), hub⟩
|
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 120 | 123 | theorem formPerm_apply_mem_eq_next (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x = next l x hx := by |
obtain ⟨k, rfl⟩ := get_of_mem hx
rw [next_get _ hl, formPerm_apply_get _ hl]
|
import Mathlib.Data.DFinsupp.WellFounded
import Mathlib.Data.Finsupp.Lex
#align_import data.finsupp.well_founded from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
variable {α N : Type*}
namespace Finsupp
variable [Zero N] {r : α → α → Prop} {s : N → N → Prop} (hbot : ∀ ⦃n⦄, ¬s n 0)
... | Mathlib/Data/Finsupp/WellFounded.lean | 37 | 42 | theorem Lex.acc (x : α →₀ N) (h : ∀ a ∈ x.support, Acc (rᶜ ⊓ (· ≠ ·)) a) :
Acc (Finsupp.Lex r s) x := by |
rw [lex_eq_invImage_dfinsupp_lex]
classical
refine InvImage.accessible toDFinsupp (DFinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ ?_)
simpa only [toDFinsupp_support] using h
|
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