Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 389 | 391 | theorem mul_le_iff_le_inv {a b r : ℝ≥0∞} (hr₀ : r ≠ 0) (hr₁ : r ≠ ∞) : r * a ≤ b ↔ a ≤ r⁻¹ * b := by |
rw [← @ENNReal.mul_le_mul_left _ a _ hr₀ hr₁, ← mul_assoc, ENNReal.mul_inv_cancel hr₀ hr₁,
one_mul]
|
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 651 | 657 | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by |
by_cases h:j = i
· subst j
simp
· simp [h]
|
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 149 | 170 | theorem exists_homeomorph_image_eq {s t : Set E}
(hsc : Convex ℝ s) (hsne : (interior s).Nonempty) (hsb : IsVonNBounded ℝ s)
(hst : Convex ℝ t) (htne : (interior t).Nonempty) (htb : IsVonNBounded ℝ t) :
∃ e : E ≃ₜ E, e '' interior s = interior t ∧ e '' closure s = closure t ∧
e '' frontier s = frontie... |
rsuffices ⟨e, h₁, h₂⟩ : ∃ e : E ≃ₜ E, e '' interior s = interior t ∧ e '' closure s = closure t
· refine ⟨e, h₁, h₂, ?_⟩
simp_rw [← closure_diff_interior, image_diff e.injective, h₁, h₂]
rcases hsne with ⟨x, hx⟩
rcases htne with ⟨y, hy⟩
set h : E ≃ₜ E := by
apply gaugeRescaleHomeomorph (-x +ᵥ s) (-y ... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatorics.hales_jewett from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open scoped Classical
universe u v
namespace ... | Mathlib/Combinatorics/HalesJewett.lean | 179 | 180 | theorem apply_of_ne_none {α ι} (l : Line α ι) (x : α) (i : ι) (h : l.idxFun i ≠ none) :
some (l x i) = l.idxFun i := by | rw [l.apply, Option.getD_of_ne_none h]
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 818 | 822 | theorem insertNth_apply_below {i j : Fin (n + 1)} (h : j < i) (x : α i)
(p : ∀ k, α (i.succAbove k)) :
i.insertNth x p j = @Eq.recOn _ _ (fun x _ ↦ α x) _
(succAbove_castPred_of_lt _ _ h) (p <| j.castPred _) := by |
rw [insertNth, succAboveCases, dif_neg h.ne, dif_pos h]
|
import Mathlib.Data.Finset.Grade
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finset.interval from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
variable {α β : Type*}
namespace Finset
section Decidable
variable [DecidableEq α] (s t : Finset α)
instance instLocally... | Mathlib/Data/Finset/Interval.lean | 110 | 111 | theorem card_Ico_finset (h : s ⊆ t) : (Ico s t).card = 2 ^ (t.card - s.card) - 1 := by |
rw [card_Ico_eq_card_Icc_sub_one, card_Icc_finset h]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 114 | 117 | theorem fib_lt_fib_succ {n : ℕ} (hn : 2 ≤ n) : fib n < fib (n + 1) := by |
rcases exists_add_of_le hn with ⟨n, rfl⟩
rw [← tsub_pos_iff_lt, add_comm 2, add_right_comm, fib_add_two, add_tsub_cancel_right, fib_pos]
exact succ_pos n
|
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Bases
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.Copy
#align_import topology.sets.opens from "leanprover-community/mathlib"@"dc6c365e751e34d100e8... | Mathlib/Topology/Sets/Opens.lean | 232 | 233 | theorem coe_iSup {ι} (s : ι → Opens α) : ((⨆ i, s i : Opens α) : Set α) = ⋃ i, s i := by |
simp [iSup]
|
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : ℕ+) : ℕ+ :=
⟨Nat.gcd (n : ℕ) (m : ℕ), Nat.gcd_pos_of_pos_left (m : ℕ) n.pos⟩
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 217 | 219 | theorem gcd_eq_right_iff_dvd {m n : ℕ+} : m ∣ n ↔ n.gcd m = m := by |
rw [gcd_comm]
apply gcd_eq_left_iff_dvd
|
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 | 70 | 72 | theorem eraseLead_add_C_mul_X_pow (f : R[X]) :
f.eraseLead + C f.leadingCoeff * X ^ f.natDegree = f := by |
rw [C_mul_X_pow_eq_monomial, eraseLead_add_monomial_natDegree_leadingCoeff]
|
import Mathlib.CategoryTheory.Adjunction.Opposites
import Mathlib.CategoryTheory.Comma.Presheaf
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.CategoryTheory.Limits.Over... | Mathlib/CategoryTheory/Limits/Presheaf.lean | 121 | 126 | theorem restrictYonedaHomEquiv_natural (P : Cᵒᵖ ⥤ Type u₁) (E₁ E₂ : ℰ) (g : E₁ ⟶ E₂) {c : Cocone _}
(t : IsColimit c) (k : c.pt ⟶ E₁) :
restrictYonedaHomEquiv A P E₂ t (k ≫ g) =
restrictYonedaHomEquiv A P E₁ t k ≫ (restrictedYoneda A).map g := by |
ext x X
apply (assoc _ _ _).symm
|
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 257 | 261 | theorem pi_apply_single [Fintype ι] [DecidableEq ι]
(Q : ∀ i, QuadraticForm R (Mᵢ i)) (i : ι) (m : Mᵢ i) :
pi Q (Pi.single i m) = Q i m := by |
rw [pi_apply, Fintype.sum_eq_single i fun j hj => ?_, Pi.single_eq_same]
rw [Pi.single_eq_of_ne hj, map_zero]
|
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 139 | 144 | theorem disjoint_splitLower_splitUpper (I : Box ι) (i : ι) (x : ℝ) :
Disjoint (I.splitLower i x) (I.splitUpper i x) := by |
rw [← disjoint_withBotCoe, coe_splitLower, coe_splitUpper]
refine (Disjoint.inf_left' _ ?_).inf_right' _
rw [Set.disjoint_left]
exact fun y (hle : y i ≤ x) hlt => not_lt_of_le hle hlt
|
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 | 129 | 132 | theorem col_mulVec [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) :
Matrix.col (M *ᵥ v) = M * Matrix.col v := by |
ext
rfl
|
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
variable {ι B F X : Type*} [TopologicalSpace X]
open TopologicalSpace Filter Set Bundle Topology
... | Mathlib/Topology/FiberBundle/Basic.lean | 528 | 540 | theorem localTrivAsPartialEquiv_trans (i j : ι) :
(Z.localTrivAsPartialEquiv i).symm.trans (Z.localTrivAsPartialEquiv j) ≈
(Z.trivChange i j).toPartialEquiv := by |
constructor
· ext x
simp only [mem_localTrivAsPartialEquiv_target, mfld_simps]
rfl
· rintro ⟨x, v⟩ hx
simp only [trivChange, localTrivAsPartialEquiv, PartialEquiv.symm, true_and_iff,
Prod.mk.inj_iff, prod_mk_mem_set_prod_eq, PartialEquiv.trans_source, mem_inter_iff,
and_true_iff, mem_prei... |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 129 | 133 | theorem equivariantProjection_condition (v : V) : (π.equivariantProjection G) (i v) = v := by |
rw [equivariantProjection_apply]
simp only [conjugate_i π i h]
rw [Finset.sum_const, Finset.card_univ, nsmul_eq_smul_cast k, smul_smul,
Invertible.invOf_mul_self, one_smul]
|
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 287 | 293 | theorem Derivation.liftKaehlerDifferential_comp (D : Derivation R S M) :
D.liftKaehlerDifferential.compDer (KaehlerDifferential.D R S) = D := by |
ext a
dsimp [KaehlerDifferential.D_apply]
refine (D.liftKaehlerDifferential_apply _).trans ?_
rw [Subtype.coe_mk, map_sub, Derivation.tensorProductTo_tmul, Derivation.tensorProductTo_tmul,
one_smul, D.map_one_eq_zero, smul_zero, sub_zero]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 553 | 558 | theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β}
(hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y) :
ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y) := by |
unfold ContinuousWithinAt at *
rw [nhdsWithin_prod_eq, Prod.map, nhds_prod_eq]
exact hf.prod_map hg
|
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 586 | 590 | theorem has_diff {s₁ s₂ : Set α} (h₁ : d.Has s₁) (h₂ : d.Has s₂) (h : s₂ ⊆ s₁) :
d.Has (s₁ \ s₂) := by |
apply d.has_compl_iff.1
simp only [diff_eq, compl_inter, compl_compl]
exact d.has_union (d.has_compl h₁) h₂ (disjoint_compl_left.mono_right h)
|
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 440 | 442 | theorem Prime.prod_divisors {α : Type*} [CommMonoid α] {p : ℕ} {f : ℕ → α} (h : p.Prime) :
∏ x ∈ p.divisors, f x = f p * f 1 := by |
rw [← cons_self_properDivisors h.ne_zero, prod_cons, h.prod_properDivisors]
|
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
| Mathlib/GroupTheory/ClassEquation.lean | 31 | 35 | theorem sum_conjClasses_card_eq_card [Fintype <| ConjClasses G] [Fintype G]
[∀ x : ConjClasses G, Fintype x.carrier] :
∑ x : ConjClasses G, x.carrier.toFinset.card = Fintype.card G := by |
suffices (Σ x : ConjClasses G, x.carrier) ≃ G by simpa using (Fintype.card_congr this)
simpa [carrier_eq_preimage_mk] using Equiv.sigmaFiberEquiv ConjClasses.mk
|
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 801 | 806 | theorem split_add_lt {o e n a m} [NF o] (h : split o = (oadd e n a, m)) :
repr a + m < ω ^ repr e := by |
cases' nf_repr_split h with h₁ h₂
cases' h₁.of_dvd_omega (split_dvd h) with e0 d
apply principal_add_omega_opow _ h₁.snd'.repr_lt (lt_of_lt_of_le (nat_lt_omega _) _)
simpa using opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0)
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 160 | 161 | theorem mul_add_one [LeftDistribClass α] (a b : α) : a * (b + 1) = a * b + a := by |
rw [mul_add, mul_one]
|
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 422 | 423 | theorem pure_prod {a : α} {f : Filter β} : pure a ×ˢ f = map (Prod.mk a) f := by |
rw [prod_eq, map_pure, pure_seq_eq_map]
|
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import ring_theory.witt_vector.structure_polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open MvPolynomial Set
open Finset (range)
o... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 164 | 175 | theorem wittStructureRat_rec_aux (Φ : MvPolynomial idx ℚ) (n : ℕ) :
wittStructureRat p Φ n * C ((p : ℚ) ^ n) =
bind₁ (fun b => rename (fun i => (b, i)) (W_ ℚ n)) Φ -
∑ i ∈ range n, C ((p : ℚ) ^ i) * wittStructureRat p Φ i ^ p ^ (n - i) := by |
have := xInTermsOfW_aux p ℚ n
replace := congr_arg (bind₁ fun k : ℕ => bind₁ (fun i => rename (Prod.mk i) (W_ ℚ k)) Φ) this
rw [AlgHom.map_mul, bind₁_C_right] at this
rw [wittStructureRat, this]; clear this
conv_lhs => simp only [AlgHom.map_sub, bind₁_X_right]
rw [sub_right_inj]
simp only [AlgHom.map_sum... |
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import ring_theory.witt_vector.structure_polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open MvPolynomial Set
open Finset (range)
o... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 229 | 267 | theorem C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum (Φ : MvPolynomial idx ℤ) (n : ℕ)
(IH :
∀ m : ℕ,
m < n →
map (Int.castRingHom ℚ) (wittStructureInt p Φ m) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) m) :
(C ((p ^ n :) : ℤ) : MvPolynomial (idx × ℕ) ℤ) ∣
bind₁ ... |
cases' n with n
· simp only [isUnit_one, Int.ofNat_zero, Int.ofNat_succ, zero_add, pow_zero, C_1, IsUnit.dvd,
Nat.cast_one, Nat.zero_eq]
-- prepare a useful equation for rewriting
have key := bind₁_rename_expand_wittPolynomial Φ n IH
apply_fun map (Int.castRingHom (ZMod (p ^ (n + 1)))) at key
conv_lh... |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 235 | 238 | theorem comap_bracket_eq (hf₁ : f.ker = ⊥) (hf₂ : N₂ ≤ f.range) :
comap f ⁅I, N₂⁆ = ⁅I, comap f N₂⁆ := by |
conv_lhs => rw [← map_comap_eq N₂ f hf₂]
rw [← map_bracket_eq, comap_map_eq _ f hf₁]
|
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analysis.locally_convex.polar from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
variable {𝕜 E F : Type*}
open Topology
namespace Li... | Mathlib/Analysis/LocallyConvex/Polar.lean | 73 | 75 | theorem polar_eq_iInter {s : Set E} : B.polar s = ⋂ x ∈ s, { y : F | ‖B x y‖ ≤ 1 } := by |
ext
simp only [polar_mem_iff, Set.mem_iInter, Set.mem_setOf_eq]
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 155 | 157 | theorem trace_eq_contract_of_basis' [Fintype ι] [DecidableEq ι] (b : Basis ι R M) :
LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquivOfBasis b).symm.toLinearMap := by |
simp [LinearEquiv.eq_comp_toLinearMap_symm, trace_eq_contract_of_basis b]
|
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : ∀ {b₁ b₂ : Buckets α β}, b₁.1.data = b₂.1.data → b₁ = b₂
| ⟨⟨_⟩, _⟩, ⟨⟨_⟩, _⟩, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 29 | 33 | theorem update_update (self : Buckets α β) (i d d' h h') :
(self.update i d h).update i d' h' = self.update i d' h := by |
simp only [update, Array.uset, Array.data_length]
congr 1
rw [Array.set_set]
|
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 177 | 181 | theorem constant_on {f : α → E} {s : Set α} (hf : (f '' s).Subsingleton) :
eVariationOn f s = 0 := by |
rw [eq_zero_iff]
rintro x xs y ys
rw [hf ⟨x, xs, rfl⟩ ⟨y, ys, rfl⟩, edist_self]
|
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 | 92 | 99 | theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z := by |
by_contra! H
rw [lt_min_iff] at H
refine apply_firstDiff_ne h ?_
calc
x (firstDiff x z) = y (firstDiff x z) := apply_eq_of_lt_firstDiff H.1
_ = z (firstDiff x z) := apply_eq_of_lt_firstDiff H.2
|
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
noncomputable section
open Polynomial
universe u v w
variable {R : Type*} {F... | Mathlib/Algebra/Polynomial/Splits.lean | 146 | 151 | theorem splits_prod {ι : Type u} {s : ι → K[X]} {t : Finset ι} :
(∀ j ∈ t, (s j).Splits i) → (∏ x ∈ t, s x).Splits i := by |
classical
refine Finset.induction_on t (fun _ => splits_one i) fun a t hat ih ht => ?_
rw [Finset.forall_mem_insert] at ht; rw [Finset.prod_insert hat]
exact splits_mul i ht.1 (ih ht.2)
|
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.clifford_algebra.grading from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
namespace CliffordAlgebra
variable {R M : Type*} [Co... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 162 | 201 | theorem evenOdd_induction (n : ZMod 2) {motive : ∀ x, x ∈ evenOdd Q n → Prop}
(range_ι_pow : ∀ (v) (h : v ∈ LinearMap.range (ι Q) ^ n.val),
motive v (Submodule.mem_iSup_of_mem ⟨n.val, n.natCast_zmod_val⟩ h))
(add : ∀ x y hx hy, motive x hx → motive y hy → motive (x + y) (Submodule.add_mem _ hx hy))
... |
apply Submodule.iSup_induction' (C := motive) _ (range_ι_pow 0 (Submodule.zero_mem _)) add
refine Subtype.rec ?_
simp_rw [ZMod.natCast_eq_iff, add_comm n.val]
rintro n' ⟨k, rfl⟩ xv
simp_rw [pow_add, pow_mul]
intro hxv
induction hxv using Submodule.mul_induction_on' with
| mem_mul_mem a ha b hb =>
i... |
import Mathlib.Algebra.Module.PID
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.finite_abelian from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915347dafd749ad6"
open scoped DirectSum
private def directSumNeZeroMulHom {ι : Type} [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
(⨁ i : {i ... | Mathlib/GroupTheory/FiniteAbelian.lean | 131 | 143 | theorem equiv_directSum_zmod_of_finite [Finite G] :
∃ (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
Nonempty <| G ≃+ ⨁ i : ι, ZMod (p i ^ e i) := by |
cases nonempty_fintype G
obtain ⟨n, ι, fι, p, hp, e, ⟨f⟩⟩ := equiv_free_prod_directSum_zmod G
cases' n with n
· have : Unique (Fin Nat.zero →₀ ℤ) :=
{ uniq := by simp only [Nat.zero_eq, eq_iff_true_of_subsingleton]; trivial }
exact ⟨ι, fι, p, hp, e, ⟨f.trans AddEquiv.uniqueProd⟩⟩
· haveI := @Fintyp... |
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type*}
namespace Set
section LT
varia... | Mathlib/Order/Height.lean | 142 | 144 | theorem chainHeight_eq_zero_iff : s.chainHeight = 0 ↔ s = ∅ := by |
rw [← not_iff_not, ← Ne, ← ENat.one_le_iff_ne_zero, one_le_chainHeight_iff,
nonempty_iff_ne_empty]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 194 | 195 | theorem two_nsmul_eq_iff {ψ θ : Angle} : (2 : ℕ) • ψ = (2 : ℕ) • θ ↔ ψ = θ ∨ ψ = θ + ↑π := by |
simp_rw [← natCast_zsmul, Nat.cast_ofNat, two_zsmul_eq_iff]
|
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 129 | 132 | theorem ker_toSpanSingleton_isMaximal {m : M} (hm : m ≠ 0) :
Ideal.IsMaximal (ker (toSpanSingleton R M m)) := by |
rw [Ideal.isMaximal_def, ← isSimpleModule_iff_isCoatom]
exact congr (quotKerEquivOfSurjective _ <| toSpanSingleton_surjective R hm)
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 389 | 401 | theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by |
rcases le_or_lt 0 (re z) with hre | hre
· simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
simp only [hre.not_le, false_or_iff]
rcases le_or_lt 0 (im z) with him | him
· simp only [him]
rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him]
exact (Real.neg_pi_div_two_l... |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 315 | 317 | theorem log_sqrt {x : ℝ} (hx : 0 ≤ x) : log (√x) = log x / 2 := by |
rw [eq_div_iff, mul_comm, ← Nat.cast_two, ← log_pow, sq_sqrt hx]
exact two_ne_zero
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 806 | 811 | theorem set_lintegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞) :
∫⁻ x in { x | f x = r }, f x ∂μ = r * μ { x | f x = r } := by |
have : ∀ᵐ x ∂μ, x ∈ { x | f x = r } → f x = r := ae_of_all μ fun _ hx => hx
rw [set_lintegral_congr_fun _ this]
· rw [lintegral_const, Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
· exact hf (measurableSet_singleton r)
|
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 424 | 425 | theorem inner_rotation_pi_div_two_left (x : V) : ⟪o.rotation (π / 2 : ℝ) x, x⟫ = 0 := by |
rw [rotation_pi_div_two, inner_rightAngleRotation_self]
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 839 | 845 | theorem _root_.AffineSubspace.SOppSide.oangle_sign_eq_neg {s : AffineSubspace ℝ P} {p₁ p₂ p₃ p₄ : P}
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃p₄ : s.SOppSide p₃ p₄) :
(∡ p₁ p₄ p₂).sign = -(∡ p₁ p₃ p₂).sign := by |
have hp₁p₃ : p₁ ≠ p₃ := by rintro rfl; exact hp₃p₄.left_not_mem hp₁
rw [← (hp₃p₄.symm.trans (sOppSide_pointReflection hp₁ hp₃p₄.left_not_mem)).oangle_sign_eq hp₁ hp₂,
← oangle_rotate_sign p₁, ← oangle_rotate_sign p₁, oangle_swap₁₃_sign,
(sbtw_pointReflection_of_ne ℝ hp₁p₃).symm.oangle_sign_eq _]
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 306 | 312 | theorem self_act_eq_iff_eq {x y : R} : x ◃ x = y ◃ y ↔ x = y := by |
constructor; swap
· rintro rfl; rfl
intro h
trans (x ◃ x) ◃⁻¹ x ◃ x
· rw [← left_cancel (x ◃ x), right_inv, self_act_act_eq]
· rw [h, ← left_cancel (y ◃ y), right_inv, self_act_act_eq]
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.measure.content from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
universe u v w
noncomputable section
open Set Topologic... | Mathlib/MeasureTheory/Measure/Content.lean | 341 | 345 | theorem outerMeasure_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
(h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
(hK : μ K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty) : 0 < μ.outerMeasure U := by |
convert μ.innerContent_pos_of_is_mul_left_invariant h3 K hK ⟨U, h1U⟩ h2U
exact μ.outerMeasure_opens ⟨U, h1U⟩
|
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
noncomputable section
section OrderedCommSemiring
variable (R : Type*) [StrictOrderedCommSemiring R]
variable (M : Typ... | Mathlib/LinearAlgebra/Orientation.lean | 100 | 101 | theorem Orientation.reindex_refl : (Orientation.reindex R M <| Equiv.refl ι) = Equiv.refl _ := by |
rw [Orientation.reindex, AlternatingMap.domDomCongrₗ_refl, Module.Ray.map_refl]
|
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 114 | 119 | theorem det_rotation (θ : Real.Angle) : LinearMap.det (o.rotation θ).toLinearMap = 1 := by |
haveI : Nontrivial V :=
FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
rw [o.rotation_eq_matrix_toLin θ hx]
simpa [sq] using θ.cos_sq_add_sin_sq
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 212 | 218 | theorem eval₂_list_prod_noncomm (ps : List R[X])
(hf : ∀ p ∈ ps, ∀ (k), Commute (f <| coeff p k) x) :
eval₂ f x ps.prod = (ps.map (Polynomial.eval₂ f x)).prod := by |
induction' ps using List.reverseRecOn with ps p ihp
· simp
· simp only [List.forall_mem_append, List.forall_mem_singleton] at hf
simp [eval₂_mul_noncomm _ _ hf.2, ihp hf.1]
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 588 | 593 | theorem uniformEquicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'}
{S : Set β} : UniformEquicontinuousOn (uα := ⨅ k, u k) F S ↔
∀ k, UniformEquicontinuousOn (uα := u k) F S := by |
simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uα := _)]
unfold UniformContinuousOn
rw [UniformFun.iInf_eq, iInf_uniformity, tendsto_iInf]
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 79 | 82 | theorem incMatrix_apply_mul_incMatrix_apply : G.incMatrix R a e * G.incMatrix R b e =
(G.incidenceSet a ∩ G.incidenceSet b).indicator 1 e := by |
classical simp only [incMatrix, Set.indicator_apply, ite_zero_mul_ite_zero, Pi.one_apply, mul_one,
Set.mem_inter_iff]
|
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 336 | 343 | theorem widePushout_exists_rep' {B : C} {α : Type _} [Nonempty α] {X : α → C}
(f : ∀ j : α, B ⟶ X j) [HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]
(x : ↑(widePushout B X f)) : ∃ (i : α) (y : X i), ι f i y = x := by |
rcases Concrete.widePushout_exists_rep f x with (⟨y, rfl⟩ | ⟨i, y, rfl⟩)
· inhabit α
use default, f _ y
simp only [← arrow_ι _ default, comp_apply]
· use i, y
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 156 | 173 | theorem mem_sigmaLift (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (a : Sigma α) (b : Sigma β)
(x : Sigma γ) :
x ∈ sigmaLift f a b ↔ ∃ (ha : a.1 = x.1) (hb : b.1 = x.1), x.2 ∈ f (ha ▸ a.2) (hb ▸ b.2) := by |
obtain ⟨⟨i, a⟩, j, b⟩ := a, b
obtain rfl | h := Decidable.eq_or_ne i j
· constructor
· simp_rw [sigmaLift]
simp only [dite_eq_ite, ite_true, mem_map, Embedding.sigmaMk_apply, forall_exists_index,
and_imp]
rintro x hx rfl
exact ⟨rfl, rfl, hx⟩
· rintro ⟨⟨⟩, ⟨⟩, hx⟩
rw [sigma... |
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 80 | 83 | theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.prod ≤ (l.map natDegree).sum := by |
induction' l with hd tl IH
· simp
· simpa using natDegree_mul_le.trans (add_le_add_left IH _)
|
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 359 | 359 | theorem mul_one (a : R) : a * (nat_lit 1).rawCast = a := by | simp [Nat.rawCast]
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 288 | 292 | theorem pentagon_hom_inv_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv =
(α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i := by |
rw [← cancel_epi (α_ f g (h ≫ i)).inv, ← cancel_mono ((α_ f g h).inv ▷ i)]
simp
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {α : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 264 | 265 | theorem eventually_filterAt_measurableSet (x : α) : ∀ᶠ a in v.filterAt x, MeasurableSet a := by |
filter_upwards [v.eventually_filterAt_mem_setsAt x] with _ ha using v.measurableSet _ _ ha
|
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 114 | 119 | theorem δ_comp_δ' {n} {i : Fin (n + 2)} {j : Fin (n + 3)} (H : Fin.castSucc i < j) :
X.δ j ≫ X.δ i =
X.δ (Fin.castSucc i) ≫
X.δ (j.pred fun (hj : j = 0) => by simp [hj, Fin.not_lt_zero] at H) := by |
dsimp [δ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ' H]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 129 | 130 | theorem right_vsub_midpoint (p₁ p₂ : P) : p₂ -ᵥ midpoint R p₁ p₂ = (⅟ 2 : R) • (p₂ -ᵥ p₁) := by |
rw [midpoint_comm, left_vsub_midpoint]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 121 | 123 | theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : Set α) := by |
simp_rw [mem_nhdsWithin_iff_eventually, eventuallyEq_set, mem_inter_iff, iff_self_and]
|
import Mathlib.Data.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.basic from "leanprover-community/mathlib"@"f0b3759a8ef0bd8239ecdaa5e1089add5feebe1a"
-- Porting note: Lean 3 uses `Set` for `ZFSet`.
set_option linter... | Mathlib/SetTheory/ZFC/Basic.lean | 370 | 370 | theorem not_nonempty_empty : ¬PSet.Nonempty ∅ := by | simp [PSet.Nonempty]
|
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 37 | 42 | theorem continuantsAux_stable_of_terminated (n_lt_m : n < m) (terminated_at_n : g.TerminatedAt n) :
g.continuantsAux m = g.continuantsAux (n + 1) := by |
refine Nat.le_induction rfl (fun k hnk hk => ?_) _ n_lt_m
rcases Nat.exists_eq_add_of_lt hnk with ⟨k, rfl⟩
refine (continuantsAux_stable_step_of_terminated ?_).trans hk
exact terminated_stable (Nat.le_add_right _ _) terminated_at_n
|
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 | 871 | 875 | theorem image_affine_Icc' {a : α} (h : 0 < a) (b c d : α) :
(a * · + b) '' Icc c d = Icc (a * c + b) (a * d + b) := by |
suffices (· + b) '' ((a * ·) '' Icc c d) = Icc (a * c + b) (a * d + b) by
rwa [Set.image_image] at this
rw [image_mul_left_Icc' h, image_add_const_Icc]
|
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
set_option autoImplicit true
namespace Mathlib
open Lean hiding Rat mkRat
open Meta
namespace Meta.NormNum
open Qq
theorem natPow_zero : Nat.pow a (nat_lit 0) = nat_lit 1 := rfl
theorem natPow_one : Nat.pow a (nat_lit 1) = a := Nat.pow_one _... | Mathlib/Tactic/NormNum/Pow.lean | 216 | 220 | theorem isRat_zpow_pos {α : Type*} [DivisionRing α] {a : α} {b : ℤ} {nb : ℕ}
{num : ℤ} {den : ℕ}
(pb : IsNat b nb) (pe' : IsRat (a^nb) num den) :
IsRat (a^b) num den := by |
rwa [pb.out, zpow_natCast]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 126 | 130 | theorem infEdist_le_infEdist_add_edist : infEdist x s ≤ infEdist y s + edist x y :=
calc
⨅ z ∈ s, edist x z ≤ ⨅ z ∈ s, edist y z + edist x y :=
iInf₂_mono fun z _ => (edist_triangle _ _ _).trans_eq (add_comm _ _)
_ = (⨅ z ∈ s, edist y z) + edist x y := by | simp only [ENNReal.iInf_add]
|
import Mathlib.Data.Matroid.Dual
open Set
namespace Matroid
variable {α : Type*} {M : Matroid α} {R I J X Y : Set α}
section restrict
@[simps] def restrictIndepMatroid (M : Matroid α) (R : Set α) : IndepMatroid α where
E := R
Indep I := M.Indep I ∧ I ⊆ R
indep_empty := ⟨M.empty_indep, empty_subset _⟩
i... | Mathlib/Data/Matroid/Restrict.lean | 198 | 203 | theorem restrict_eq_restrict_iff (M M' : Matroid α) (X : Set α) :
M ↾ X = M' ↾ X ↔ ∀ I, I ⊆ X → (M.Indep I ↔ M'.Indep I) := by |
refine ⟨fun h I hIX ↦ ?_, fun h ↦ eq_of_indep_iff_indep_forall rfl fun I (hI : I ⊆ X) ↦ ?_⟩
· rw [← and_iff_left (a := (M.Indep I)) hIX, ← and_iff_left (a := (M'.Indep I)) hIX,
← restrict_indep_iff, h, restrict_indep_iff]
rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI]... |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 181 | 185 | theorem mem_L1_inner (f g : α →₂[μ] E) :
AEEqFun.mk (fun x => ⟪f x, g x⟫)
((Lp.aestronglyMeasurable f).inner (Lp.aestronglyMeasurable g)) ∈
Lp 𝕜 1 μ := by |
simp_rw [mem_Lp_iff_snorm_lt_top, snorm_aeeqFun]; exact snorm_inner_lt_top f g
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 349 | 352 | theorem _root_.Real.toNNReal_rpow_of_nonneg {x y : ℝ} (hx : 0 ≤ x) :
Real.toNNReal (x ^ y) = Real.toNNReal x ^ y := by |
nth_rw 1 [← Real.coe_toNNReal x hx]
rw [← NNReal.coe_rpow, Real.toNNReal_coe]
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 27 | 33 | theorem isPrimePow_of_minFac_pow_factorization_eq {n : ℕ}
(h : n.minFac ^ n.factorization n.minFac = n) (hn : n ≠ 1) : IsPrimePow n := by |
rcases eq_or_ne n 0 with (rfl | hn')
· simp_all
refine ⟨_, _, (Nat.minFac_prime hn).prime, ?_, h⟩
simp [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.support_factorization, hn',
Nat.minFac_prime hn, Nat.minFac_dvd]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 819 | 820 | theorem some_append_some [Append α] (a b : α) : some a ++ some b = some (a ++ b) := by |
simp [append_def]
|
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 150 | 153 | theorem quadratic_reciprocity_one_mod_four (hp : p % 4 = 1) (hq : q ≠ 2) :
legendreSym q p = legendreSym p q := by |
rw [quadratic_reciprocity' (Prime.mod_two_eq_one_iff_ne_two.mp (odd_of_mod_four_eq_one hp)) hq,
pow_mul, neg_one_pow_div_two_of_one_mod_four hp, one_pow, one_mul]
|
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 134 | 164 | theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
(H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p := by |
have := H.1.1
let f' := (Ideal.Quotient.mk I).comp f
have e : RingHom.ker f' = I.comap f := by
ext1
exact Submodule.Quotient.mk_eq_zero _
have : RingHom.ker (Ideal.Quotient.mk <| RingHom.ker f') ≤ p := by
rw [Ideal.mk_ker, e]
exact H.1.2
suffices _ by
have ⟨p', hp₁, hp₂⟩ := Ideal.exists_c... |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 233 | 236 | theorem withDensity_eq_zero {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (h : μ.withDensity f = 0) :
f =ᵐ[μ] 0 := by |
rw [← lintegral_eq_zero_iff' hf, ← set_lintegral_univ, ← withDensity_apply _ MeasurableSet.univ,
h, Measure.coe_zero, Pi.zero_apply]
|
import Mathlib.Topology.Connected.Basic
open Set Function
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section TotallyDisconnected
def IsTotallyDisconnected (s : Set α) : Prop :=
∀ t, t ⊆ s → IsPreconnected t → t.Subsingleton
#align is_t... | Mathlib/Topology/Connected/TotallyDisconnected.lean | 93 | 104 | theorem isTotallyDisconnected_of_isClopen_set {X : Type*} [TopologicalSpace X]
(hX : Pairwise fun x y => ∃ (U : Set X), IsClopen U ∧ x ∈ U ∧ y ∉ U) :
IsTotallyDisconnected (Set.univ : Set X) := by |
rintro S - hS
unfold Set.Subsingleton
by_contra! h_contra
rcases h_contra with ⟨x, hx, y, hy, hxy⟩
obtain ⟨U, hU, hxU, hyU⟩ := hX hxy
specialize
hS U Uᶜ hU.2 hU.compl.2 (fun a _ => em (a ∈ U)) ⟨x, hx, hxU⟩ ⟨y, hy, hyU⟩
rw [inter_compl_self, Set.inter_empty] at hS
exact Set.not_nonempty_empty hS
|
import Mathlib.Data.List.Nodup
#align_import data.list.dedup from "leanprover-community/mathlib"@"d9e96a3e3e0894e93e10aff5244f4c96655bac1c"
universe u
namespace List
variable {α : Type u} [DecidableEq α]
@[simp]
theorem dedup_nil : dedup [] = ([] : List α) :=
rfl
#align list.dedup_nil List.dedup_nil
theorem... | Mathlib/Data/List/Dedup.lean | 126 | 132 | theorem dedup_append (l₁ l₂ : List α) : dedup (l₁ ++ l₂) = l₁ ∪ dedup l₂ := by |
induction' l₁ with a l₁ IH; · rfl
simp only [cons_union] at *
rw [← IH, cons_append]
by_cases h : a ∈ dedup (l₁ ++ l₂)
· rw [dedup_cons_of_mem' h, insert_of_mem h]
· rw [dedup_cons_of_not_mem' h, insert_of_not_mem h]
|
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Order.Monoid.Defs
#align_import algebra.order.monoid.canonical.defs from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
universe u
variable {α : Type u}
class ExistsMulOf... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 242 | 243 | theorem one_lt_mul_iff : 1 < a * b ↔ 1 < a ∨ 1 < b := by |
simp only [one_lt_iff_ne_one, Ne, mul_eq_one_iff, not_and_or]
|
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Topology.Algebra.Module.Simple
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.RingTheory.Ideal.LocalRing
#align_import topology.algebra.module.finite_dimension from "leanprove... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 245 | 260 | theorem LinearMap.continuous_of_finiteDimensional [T2Space E] [FiniteDimensional 𝕜 E]
(f : E →ₗ[𝕜] F') : Continuous f := by |
-- for the proof, go to a model vector space `b → 𝕜` thanks to `continuous_equivFun_basis`, and
-- argue that all linear maps there are continuous.
let b := Basis.ofVectorSpace 𝕜 E
have A : Continuous b.equivFun := continuous_equivFun_basis_aux b
have B : Continuous (f.comp (b.equivFun.symm : (Basis.ofVect... |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 214 | 221 | theorem _root_.IntermediateField.AdjoinSimple.norm_gen_eq_one {x : L} (hx : ¬IsIntegral K x) :
norm K (AdjoinSimple.gen K x) = 1 := by |
rw [norm_eq_one_of_not_exists_basis]
contrapose! hx
obtain ⟨s, ⟨b⟩⟩ := hx
refine .of_mem_of_fg K⟮x⟯.toSubalgebra ?_ x ?_
· exact (Submodule.fg_iff_finiteDimensional _).mpr (of_fintype_basis b)
· exact IntermediateField.subset_adjoin K _ (Set.mem_singleton x)
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 438 | 440 | theorem monic_mul_leadingCoeff_inv {p : K[X]} (h : p ≠ 0) : Monic (p * C (leadingCoeff p)⁻¹) := by |
rw [Monic, leadingCoeff_mul, leadingCoeff_C,
mul_inv_cancel (show leadingCoeff p ≠ 0 from mt leadingCoeff_eq_zero.1 h)]
|
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 671 | 679 | theorem sizeOf_dedupKeys [DecidableEq α] [SizeOf (Sigma β)]
(xs : List (Sigma β)) : SizeOf.sizeOf (dedupKeys xs) ≤ SizeOf.sizeOf xs := by |
simp only [SizeOf.sizeOf, _sizeOf_1]
induction' xs with x xs
· simp [dedupKeys]
· simp only [dedupKeys_cons, kinsert_def, Nat.add_le_add_iff_left, Sigma.eta]
trans
· apply sizeOf_kerase
· assumption
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 294 | 297 | theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) :
(n : ℕ∞) ≤ trailingDegree (C a * X ^ n) := by |
rw [C_mul_X_pow_eq_monomial]
exact le_trailingDegree_monomial
|
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed.mul_action from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
variable {α β : Type*}
section SeminormedAddGroup
variable [SeminormedAddGroup α] [SeminormedAddGroup β] ... | Mathlib/Analysis/Normed/MulAction.lean | 29 | 30 | theorem norm_smul_le (r : α) (x : β) : ‖r • x‖ ≤ ‖r‖ * ‖x‖ := by |
simpa [smul_zero] using dist_smul_pair r 0 x
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 61 | 64 | theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by |
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
|
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.Data.List.Chain
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theor... | Mathlib/GroupTheory/CoprodI.lean | 393 | 403 | theorem mem_rcons_iff {i j : ι} (p : Pair M i) (m : M j) :
⟨_, m⟩ ∈ (rcons p).toList ↔ ⟨_, m⟩ ∈ p.tail.toList ∨
m ≠ 1 ∧ (∃ h : i = j, m = h ▸ p.head) := by |
simp only [rcons, cons, ne_eq]
by_cases hij : i = j
· subst i
by_cases hm : m = p.head
· subst m
split_ifs <;> simp_all
· split_ifs <;> simp_all
· split_ifs <;> simp_all [Ne.symm hij]
|
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 98 | 100 | theorem polar_add (f g : M → R) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by |
simp only [polar, Pi.add_apply]
abel
|
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 60 | 61 | theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑(toNat c) = (0 : Cardinal) := by |
rw [toNat_apply_of_aleph0_le h, Nat.cast_zero]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
#align_import linear_algebra.vandermonde from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Vandermonde.lean | 59 | 64 | theorem vandermonde_succ {n : ℕ} (v : Fin n.succ → R) :
vandermonde v =
Fin.cons (fun (j : Fin n.succ) => v 0 ^ (j : ℕ)) fun i =>
Fin.cons 1 fun j => v i.succ * vandermonde (Fin.tail v) i j := by |
conv_lhs => rw [← Fin.cons_self_tail v, vandermonde_cons]
rfl
|
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
variable {R : Type*} {R₂ : Type*} {R₃ : Type*}
variable {K : Type*} {K₂ : Type*}
variable {M : Type*} {M₂ : Type*} {M₃ : Type*}
variable {V : ... | Mathlib/Algebra/Module/Submodule/Range.lean | 179 | 180 | theorem _root_.Submodule.map_comap_eq_self [RingHomSurjective τ₁₂] {f : F} {q : Submodule R₂ M₂}
(h : q ≤ range f) : map f (comap f q) = q := by | rwa [Submodule.map_comap_eq, inf_eq_right]
|
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 159 | 162 | theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
IsCoprime (r ^ 2 + s ^ 2) (r * s) := by |
apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
rw [add_comm]; apply Int.coprime_of_sq_sum h
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 64 | 66 | theorem ascPochhammer_succ_left (n : ℕ) :
ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1) := by |
rw [ascPochhammer]
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 369 | 374 | theorem _root_.MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α}
(hA : MeasurableSet A) (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε := by |
rcases A.exists_isOpen_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩
use U, hAU, hUo, hU.trans_le le_top
exact measure_diff_lt_of_lt_add hA hAU hA' hU
|
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
set_option autoImplicit true
open Set
namespace OrderIso
variable [Preorder α] [Preorder β] (f : α ≃o β)
theorem upperBounds_image {... | Mathlib/Order/Bounds/OrderIso.lean | 41 | 42 | theorem isLUB_image' {s : Set α} {x : α} : IsLUB (f '' s) (f x) ↔ IsLUB s x := by |
rw [isLUB_image, f.symm_apply_apply]
|
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 68 | 92 | theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
(∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v := by |
have hiw : ⟪i w, w⟫ = 1 / 2 := by
rw [inner_smul_left, inner_self_eq_norm_sq_to_K, RCLike.ofReal_real_eq_id, id,
RCLike.conj_to_real, ← div_div, div_mul_cancel₀]
rwa [Ne, sq_eq_zero_iff, norm_eq_zero]
have :
(fun v : V => 𝐞 (-⟪v, w⟫) • f (v + i w)) =
fun v : V => (fun x : V => -(𝐞 (-⟪x, w... |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 161 | 165 | theorem differentiable_tsum' (hu : Summable u) (hg : ∀ n y, HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, ‖g' n y‖ ≤ u n) : Differentiable 𝕜 fun z => ∑' n, g n z := by |
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg
refine differentiable_tsum hu hg ?_
simpa? says simpa only [ContinuousLinearMap.norm_smulRight_apply, norm_one, one_mul]
|
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 352 | 358 | theorem rightDistributor_ext₂_left {J : Type} [Fintype J]
{f : J → C} {X Y Z : C} {g h : ((⨁ f) ⊗ X) ⊗ Y ⟶ Z}
(w : ∀ j, ((biproduct.ι f j ▷ X) ▷ Y) ≫ g = ((biproduct.ι f j ▷ X) ▷ Y) ≫ h) :
g = h := by |
apply (cancel_epi (α_ _ _ _).inv).mp
ext
simp [w]
|
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 36 | 42 | theorem P_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((P (q + 1)).f n : X _[n] ⟶ _) = (P q).f n := by |
rcases n with (_|n)
· simp only [Nat.zero_eq, P_f_0_eq]
· simp only [P_succ, add_right_eq_self, comp_add, HomologicalComplex.comp_f,
HomologicalComplex.add_f_apply, comp_id]
exact (HigherFacesVanish.of_P q n).comp_Hσ_eq_zero (Nat.succ_le_iff.mp hqn)
|
import Mathlib.Order.Lattice
#align_import order.min_max from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v
variable {α : Type u} {β : Type v}
attribute [simp] max_eq_left max_eq_right min_eq_left min_eq_right
section
variable [LinearOrder α] [LinearOrder β] {f : α → β... | Mathlib/Order/MinMax.lean | 196 | 197 | theorem min_lt_min_right_iff : min a b < min a c ↔ b < c ∧ b < a := by |
simp_rw [min_comm a, min_lt_min_left_iff]
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 585 | 587 | theorem oangle_midpoint_left (p₁ p₂ p₃ : P) : ∡ (midpoint ℝ p₁ p₂) p₂ p₃ = ∡ p₁ p₂ p₃ := by |
by_cases h : p₁ = p₂; · simp [h]
exact (sbtw_midpoint_of_ne ℝ h).symm.oangle_eq_left
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 352 | 354 | theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : Angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π := by |
induction ψ using Real.Angle.induction_on
exact sin_eq_real_sin_iff_eq_or_add_eq_pi
|
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 131 | 131 | theorem lie_bot : ⁅I, (⊥ : LieSubmodule R L M)⁆ = ⊥ := by | rw [eq_bot_iff]; apply lie_le_right
|
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 132 | 132 | theorem boundary_idem (a : α) : ∂ ∂ a = ∂ a := by | rw [boundary, hnot_boundary, inf_top_eq]
|
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