Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [Algebra F K]
section per... | Mathlib/FieldTheory/PurelyInseparable.lean | 645 | 648 | theorem isPurelyInseparable_adjoin_iff_pow_mem (q : ℕ) [hF : ExpChar F q] {S : Set E} :
IsPurelyInseparable F (adjoin F S) ↔ ∀ x ∈ S, ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by |
simp_rw [← le_perfectClosure_iff, adjoin_le_iff, ← mem_perfectClosure_iff_pow_mem q,
Set.le_iff_subset, Set.subset_def, SetLike.mem_coe]
|
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 251 | 253 | theorem bit_coe_nat (b) (n : ℕ) : bit b n = Nat.bit b n := by |
rw [bit_val, Nat.bit_val]
cases b <;> rfl
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 49 | 49 | theorem logb_zero : logb b 0 = 0 := by | simp [logb]
|
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.NonemptyFinLinOrd
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import algebraic_topology.simplex_category from "leanprover-community/mathlib"@"e8ac6315bcfcbaf2d19a046719c3b5... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 272 | 289 | theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ Fin.castSucc j) :
δ (Fin.castSucc i) ≫ σ j.succ = σ j ≫ δ i := by |
ext k : 3
dsimp [σ, δ]
rcases le_or_lt i k with (hik | hik)
· rw [Fin.succAbove_of_le_castSucc _ _ (Fin.castSucc_le_castSucc_iff.mpr hik),
Fin.succ_predAbove_succ, Fin.succAbove_of_le_castSucc]
rcases le_or_lt k (j.castSucc) with (hjk | hjk)
· rwa [Fin.predAbove_of_le_castSucc _ _ hjk, Fin.castSucc... |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section CommGroup
variable [CommGroup α] (e : α) (x : F... | Mathlib/Combinatorics/Additive/ETransform.lean | 75 | 81 | theorem mulDysonETransform_idem :
mulDysonETransform e (mulDysonETransform e x) = mulDysonETransform e x := by |
ext : 1 <;> dsimp
· rw [smul_finset_inter, smul_inv_smul, inter_comm, union_eq_left]
exact inter_subset_union
· rw [smul_finset_union, inv_smul_smul, union_comm, inter_eq_left]
exact inter_subset_union
|
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 192 | 207 | theorem vars_sum_of_disjoint [DecidableEq σ] (h : Pairwise <| (Disjoint on fun i => (φ i).vars)) :
(∑ i ∈ t, φ i).vars = Finset.biUnion t fun i => (φ i).vars := by |
classical
induction t using Finset.induction_on with
| empty => simp
| insert has hsum =>
rw [Finset.biUnion_insert, Finset.sum_insert has, vars_add_of_disjoint, hsum]
unfold Pairwise onFun at h
rw [hsum]
simp only [Finset.disjoint_iff_ne] at h ⊢
intro v hv v2 hv2
rw [Finset.mem_biUnion... |
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#align_import measure_theory.integral.peak_function from "leanprover-community/mathlib"@"13b0d72fd8533ba459ac66e9a885e35ffabb32b2"
open Set Filter MeasureTheory MeasureTheory.Measure TopologicalSpace Metric
open scoped Topology ENNReal
open Set
variable... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 235 | 247 | theorem tendsto_integral_peak_smul_of_integrable_of_tendsto
{t : Set α} (ht : MeasurableSet t) (h'ts : t ∈ 𝓝 x₀)
(h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l uᶜ)
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
(h'iφ : ∀ᶠ i ... |
suffices Tendsto (fun i : ι ↦ ∫ x in univ, φ i x • g x ∂μ) l (𝓝 a) by simpa
exact tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto MeasurableSet.univ ht (x₀ := x₀)
(subset_univ _) (by simpa [nhdsWithin_univ]) h't (by simpa)
(by simpa [← compl_eq_univ_diff] using hlφ) hiφ
(by simpa) (by simpa) ... |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section tprod
variable [CommMonoid α] [TopologicalSpace α] {f g : β → α} {a a₁ a₂ : ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 475 | 476 | theorem Finset.tprod_subtype' (s : Finset β) (f : β → α) :
∏' x : (s : Set β), f x = ∏ x ∈ s, f x := by | simp
|
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 | 210 | 212 | theorem kahler_rotation_left' (x y : V) (θ : Real.Angle) :
o.kahler (o.rotation θ x) y = (-θ).expMapCircle * o.kahler x y := by |
simp only [Real.Angle.expMapCircle_neg, coe_inv_circle_eq_conj, kahler_rotation_left]
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_character from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace MulChar
section Group
-- The domain of our multipli... | Mathlib/NumberTheory/MulChar/Basic.lean | 383 | 386 | theorem pow_apply_coe (χ : MulChar R R') (n : ℕ) (a : Rˣ) : (χ ^ n) a = χ a ^ n := by |
induction' n with n ih
· rw [pow_zero, pow_zero, one_apply_coe]
· rw [pow_succ, pow_succ, mul_apply, ih]
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 161 | 161 | theorem refl : IntervalIntegrable f μ a a := by | constructor <;> simp
|
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import number_theory.sum_four_squares from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
open Finset Polynomial FiniteField Equiv
the... | Mathlib/NumberTheory/SumFourSquares.lean | 46 | 59 | theorem sq_add_sq_of_two_mul_sq_add_sq {m x y : ℤ} (h : 2 * m = x ^ 2 + y ^ 2) :
m = ((x - y) / 2) ^ 2 + ((x + y) / 2) ^ 2 :=
have : Even (x ^ 2 + y ^ 2) := by | simp [← h, even_mul]
have hxaddy : Even (x + y) := by simpa [sq, parity_simps]
have hxsuby : Even (x - y) := by simpa [sq, parity_simps]
mul_right_injective₀ (show (2 * 2 : ℤ) ≠ 0 by decide) <|
calc
2 * 2 * m = (x - y) ^ 2 + (x + y) ^ 2 := by rw [mul_assoc, h]; ring
_ = (2 * ((x - y) / 2)) ^ 2 + ... |
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 280 | 281 | theorem coeSubtype : (SubgroupClass.subtype H : H → G) = ((↑) : H → G) := by |
rfl
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Finsupp
#align_import algebra.big_operators.associated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β γ δ : Type*}
-- the same local notation used in `Algebra.Associated`
local infixl:50 " ~ᵤ " => ... | Mathlib/Algebra/BigOperators/Associated.lean | 124 | 130 | theorem finset_prod_mk {p : Finset β} {f : β → α} :
(∏ i ∈ p, Associates.mk (f i)) = Associates.mk (∏ i ∈ p, f i) := by |
-- Porting note: added
have : (fun i => Associates.mk (f i)) = Associates.mk ∘ f :=
funext fun x => Function.comp_apply
rw [Finset.prod_eq_multiset_prod, this, ← Multiset.map_map, prod_mk,
← Finset.prod_eq_multiset_prod]
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 630 | 631 | theorem Icc_eq_cons_Ioc (h : a ≤ b) : Icc a b = (Ioc a b).cons a left_not_mem_Ioc := by |
classical rw [cons_eq_insert, Ioc_insert_left h]
|
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_ideal from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
open SetLike Direc... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 416 | 418 | theorem toIdeal_iInf₂ {κ : Sort*} {κ' : κ → Sort*} (s : ∀ i, κ' i → HomogeneousIdeal 𝒜) :
(⨅ (i) (j), s i j).toIdeal = ⨅ (i) (j), (s i j).toIdeal := by |
simp_rw [toIdeal_iInf]
|
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.adjunction.basic from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
namespace CategoryTheory
open Category
-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ u₁ u₂ u₃
... | Mathlib/CategoryTheory/Adjunction/Basic.lean | 318 | 320 | theorem homEquiv_naturality_left_aux (f : X' ⟶ X) (g : F.obj X ⟶ Y) :
(adj.homEquiv X' (F.obj X)) (F.map f) ≫ G.map g = f ≫ (adj.homEquiv X Y) g := by |
rw [← homEquiv_naturality_right, ← Equiv.eq_symm_apply]; simp
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 693 | 693 | theorem tan_ofReal_im (x : ℝ) : (tan x).im = 0 := by | rw [← ofReal_tan_ofReal_re, ofReal_im]
|
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 220 | 236 | theorem norm_comp_le (g : W₂ →ᴬ[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ := by |
rw [norm_def, max_le_iff]
constructor
· calc
‖f.comp g 0‖ = ‖f (g 0)‖ := by simp
_ = ‖f.contLinear (g 0) + f 0‖ := by rw [f.decomp]; simp
_ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ :=
((norm_add_le _ _).trans (add_le_add_right (f.contLinear.le_opNorm _) _))
_ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ :=
... |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 195 | 196 | theorem mul_right_apply_of_ne (a b : n) (hbj : b ≠ j) (M : Matrix n n α) :
(M * stdBasisMatrix i j c) a b = 0 := by | simp [mul_apply, hbj.symm]
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 776 | 782 | theorem dirac_prod (x : α) : (dirac x).prod ν = map (Prod.mk x) ν := by |
rw [← sum_sFiniteSeq ν, prod_sum_right, map_sum measurable_prod_mk_left.aemeasurable]
congr
ext1 i
refine prod_eq fun s t hs ht => ?_
simp_rw [map_apply measurable_prod_mk_left (hs.prod ht), mk_preimage_prod_right_eq_if, measure_if,
dirac_apply' _ hs, ← indicator_mul_left _ _ fun _ => sFiniteSeq ν i t, P... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 326 | 329 | theorem nnnorm_zero : ‖(0 : Lp E p μ)‖₊ = 0 := by |
rw [nnnorm_def]
change (snorm (⇑(0 : α →ₘ[μ] E)) p μ).toNNReal = 0
simp [snorm_congr_ae AEEqFun.coeFn_zero, snorm_zero]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {α : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,295 | 1,296 | theorem toWord_injective : Function.Injective (toWord : FreeGroup α → List (α × Bool)) := by |
rintro ⟨L₁⟩ ⟨L₂⟩; exact reduce.exact
|
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Order.DenselyOrdered
open Set Filter TopologicalSpace Bornology
open scoped ENNReal NNReal Uniformity Topology
universe u v w
variable {α : Type u} {β : Typ... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,360 | 1,361 | theorem Real.dist_left_le_of_mem_uIcc {x y z : ℝ} (h : y ∈ uIcc x z) : dist x y ≤ dist x z := by |
simpa only [dist_comm x] using abs_sub_left_of_mem_uIcc h
|
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-community/mathlib"@"9822b65bfc4ac74537d77ae318d27df1df662471"
universe u v w w₁ w₂
variable {R M}
theorem Submodule.exists_lieSubmodule_coe_eq_iff (p... | Mathlib/Algebra/Lie/Submodule.lean | 632 | 636 | theorem subsingleton_iff : Subsingleton (LieSubmodule R L M) ↔ Subsingleton M :=
have h : Subsingleton (LieSubmodule R L M) ↔ Subsingleton (Submodule R M) := by |
rw [← subsingleton_iff_bot_eq_top, ← subsingleton_iff_bot_eq_top, ← coe_toSubmodule_eq_iff,
top_coeSubmodule, bot_coeSubmodule]
h.trans <| Submodule.subsingleton_iff R
|
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib.Data.Subtype
import Mathlib.Data.Sum.Basic
import Mathlib.Init.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Logic.Function.Conjugate
import Mathlib.Tactic.Lift... | Mathlib/Logic/Equiv/Basic.lean | 1,665 | 1,667 | theorem eq_or_eq_of_swap_apply_ne_self {a b x : α} (h : swap a b x ≠ x) : x = a ∨ x = b := by |
contrapose! h
exact swap_apply_of_ne_of_ne h.1 h.2
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,391 | 2,394 | theorem prod_erase_attach [DecidableEq ι] {s : Finset ι} (f : ι → α) (i : ↑s) :
∏ j ∈ s.attach.erase i, f ↑j = ∏ j ∈ s.erase ↑i, f j := by |
rw [← Function.Embedding.coe_subtype, ← prod_map]
simp [attach_map_val]
|
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal.cofinality from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputable section
open Function Cardinal Set Order
open scoped Classical
open Cardinal Ordinal
un... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 377 | 386 | theorem nfpFamily_lt_ord_lift {ι} {f : ι → Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c)
(hc' : Cardinal.lift.{v, u} #ι < cof c) (hf : ∀ (i), ∀ b < c, f i b < c) {a} (ha : a < c) :
nfpFamily.{u, v} f a < c := by |
refine sup_lt_ord_lift ((Cardinal.lift_le.2 (mk_list_le_max ι)).trans_lt ?_) fun l => ?_
· rw [lift_max]
apply max_lt _ hc'
rwa [Cardinal.lift_aleph0]
· induction' l with i l H
· exact ha
· exact hf _ _ H
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 253 | 257 | theorem toIocDiv_add_zsmul' (a b : α) (m : ℤ) :
toIocDiv hp (a + m • p) b = toIocDiv hp a b - m := by |
refine toIocDiv_eq_of_sub_zsmul_mem_Ioc _ ?_
rw [sub_smul, ← sub_add, add_right_comm]
simpa using sub_toIocDiv_zsmul_mem_Ioc hp a b
|
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 403 | 419 | theorem mfderiv_prod_eq_add {f : M × M' → M''} {p : M × M'}
(hf : MDifferentiableAt (I.prod I') I'' f p) :
mfderiv (I.prod I') I'' f p =
show E × E' →L[𝕜] E'' from
mfderiv (I.prod I') I'' (fun z : M × M' => f (z.1, p.2)) p +
mfderiv (I.prod I') I'' (fun z : M × M' => f (p.1, z.2)) p := ... |
dsimp only
erw [mfderiv_comp_of_eq hf ((mdifferentiableAt_fst I I').prod_mk (mdifferentiableAt_const _ _))
rfl,
mfderiv_comp_of_eq hf ((mdifferentiableAt_const _ _).prod_mk (mdifferentiableAt_snd I I')) rfl,
← ContinuousLinearMap.comp_add,
(mdifferentiableAt_fst I I').mfderiv_prod (mdifferentiabl... |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 79 | 81 | theorem einfsep_lt_iff {d} :
s.einfsep < d ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < d := by |
simp_rw [einfsep, iInf_lt_iff, exists_prop]
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 304 | 305 | theorem single_eq_same : (single a b : α →₀ M) a = b := by |
classical exact Pi.single_eq_same (f := fun _ ↦ M) a b
|
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 | 525 | 527 | theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s := by |
rw [Seminorm.ball_zero_eq]
exact gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs
|
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 368 | 376 | theorem Polynomial.bernoulli_three_eval_one_quarter :
(Polynomial.bernoulli 3).eval (1 / 4) = 3 / 64 := by |
simp_rw [Polynomial.bernoulli, Finset.sum_range_succ, Polynomial.eval_add,
Polynomial.eval_monomial]
rw [Finset.sum_range_zero, Polynomial.eval_zero, zero_add, bernoulli_one]
rw [bernoulli_eq_bernoulli'_of_ne_one zero_ne_one, bernoulli'_zero,
bernoulli_eq_bernoulli'_of_ne_one (by decide : 2 ≠ 1), bernoul... |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 629 | 633 | theorem sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependent {x y : M} :
SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] := by |
rw [← sameRay_or_sameRay_neg_iff_not_linearIndependent]
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0 <;> simp [hx, hy]
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 216 | 221 | theorem bernoulli_eq_bernoulli'_of_ne_one {n : ℕ} (hn : n ≠ 1) : bernoulli n = bernoulli' n := by |
by_cases h0 : n = 0; · simp [h0]
rw [bernoulli, neg_one_pow_eq_pow_mod_two]
cases' mod_two_eq_zero_or_one n with h h
· simp [h]
· simp [bernoulli'_odd_eq_zero (odd_iff.mpr h) (one_lt_iff_ne_zero_and_ne_one.mpr ⟨h0, hn⟩)]
|
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 | 1,110 | 1,119 | theorem lintegral_liminf_le' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ) :
∫⁻ a, liminf (fun n => f n a) atTop ∂μ ≤ liminf (fun n => ∫⁻ a, f n a ∂μ) atTop :=
calc
∫⁻ a, liminf (fun n => f n a) atTop ∂μ = ∫⁻ a, ⨆ n : ℕ, ⨅ i ≥ n, f i a ∂μ := by |
simp only [liminf_eq_iSup_iInf_of_nat]
_ = ⨆ n : ℕ, ∫⁻ a, ⨅ i ≥ n, f i a ∂μ :=
(lintegral_iSup' (fun n => aemeasurable_biInf _ (to_countable _) (fun i _ ↦ h_meas i))
(ae_of_all μ fun a n m hnm => iInf_le_iInf_of_subset fun i hi => le_trans hnm hi))
_ ≤ ⨆ n : ℕ, ⨅ i ≥ n, ∫⁻ a, f i a ∂μ := iS... |
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.angle.sphere from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open FiniteDimensional Complex
open scoped EuclideanGeomet... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 120 | 123 | theorem oangle_eq_pi_sub_two_zsmul_oangle_center_left {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂) : ∡ p₁ s.center p₂ = π - (2 : ℤ) • ∡ s.center p₂ p₁ := by |
rw [oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq h.symm
(dist_center_eq_dist_center_of_mem_sphere' hp₂ hp₁)]
|
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 | 496 | 497 | theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : (↑) ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a := by |
rw [← map_nhds_subtype_val, mem_map]
|
import Mathlib.Topology.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
#align_import topology.algebra.order.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
set_option autoImplicit true
open Filter Set TopologicalSpace
open scoped Classical
open Topology
theor... | Mathlib/Topology/Order/ExtendFrom.lean | 54 | 65 | theorem continuousOn_Ico_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ico... |
apply continuousOn_extendFrom
· rw [closure_Ioo hab.ne]
exact Ico_subset_Icc_self
· intro x x_in
rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)
· use la
simpa [hab]
· exact ⟨f x, hf x h⟩
|
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
import Mathlib.Data.Set.Lattice
#align_import group_theory.sylow from "leanprove... | Mathlib/GroupTheory/Sylow.lean | 698 | 702 | theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
(hdvd : p ∣ card G) : p ∣ card P := by |
rw [← pow_one p] at hdvd
have key := P.pow_dvd_card_of_pow_dvd_card hdvd
rwa [pow_one] at key
|
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.DiscreteSubset
import Mathlib.Tactic.Abel... | Mathlib/Topology/Algebra/UniformGroup.lean | 229 | 233 | theorem uniformGroup_inf {u₁ u₂ : UniformSpace β} (h₁ : @UniformGroup β u₁ _)
(h₂ : @UniformGroup β u₂ _) : @UniformGroup β (u₁ ⊓ u₂) _ := by |
rw [inf_eq_iInf]
refine uniformGroup_iInf fun b => ?_
cases b <;> assumption
|
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Algebra.Algebra.Defs
import Mathlib.LinearAlgebra.Projection
import Mat... | Mathlib/Topology/Algebra/Module/Basic.lean | 2,210 | 2,212 | theorem self_comp_symm (e : M₁ ≃SL[σ₁₂] M₂) : (e : M₁ → M₂) ∘ (e.symm : M₂ → M₁) = id := by |
ext x
exact apply_symm_apply e x
|
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 | 105 | 108 | theorem lmarginal_congr {x y : ∀ i, π i} (f : (∀ i, π i) → ℝ≥0∞)
(h : ∀ i ∉ s, x i = y i) :
(∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y := by |
dsimp [lmarginal, updateFinset_def]; rcongr; exact h _ ‹_›
|
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Star.Pointwise
import Mathlib.Algebra.Star.Module
import Mathlib.RingTheory.Adjoin.Basic
#align_import algebra.star.subalgebra from "leanprover-community/mathlib"@"160ef3e33... | Mathlib/Algebra/Star/Subalgebra.lean | 711 | 712 | theorem coe_iInf {ι : Sort*} {S : ι → StarSubalgebra R A} : (↑(⨅ i, S i) : Set A) = ⋂ i, S i := by |
simp [iInf]
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 166 | 166 | theorem cgf_zero' : cgf X μ 0 = log (μ Set.univ).toReal := by | simp only [cgf, mgf_zero']
|
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 | 288 | 291 | theorem map_fst_prod (f : Filter α) (g : Filter β) [NeBot g] : map Prod.fst (f ×ˢ g) = f := by |
ext s
simp only [mem_map, mem_prod_iff_left, mem_preimage, eventually_const, ← subset_def,
exists_mem_subset_iff]
|
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 394 | 400 | theorem single_one_eq_pow {R : Type _} [Ring R] (n : ℕ) :
single (n : ℤ) (1 : R) = single (1 : ℤ) 1 ^ n := by |
induction' n with n h_ind
· simp only [Nat.cast_zero, pow_zero]
rfl
· rw [← Int.ofNat_add_one_out, pow_succ', ← h_ind, HahnSeries.single_mul_single, one_mul,
add_comm]
|
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 | 727 | 728 | theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by | simp_rw [mul_comm, lintegral_const_mul'' r hf]
|
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 | 302 | 310 | theorem norm_eq_prod_automorphisms [FiniteDimensional K L] [IsGalois K L] (x : L) :
algebraMap K L (norm K x) = ∏ σ : L ≃ₐ[K] L, σ x := by |
apply NoZeroSMulDivisors.algebraMap_injective L (AlgebraicClosure L)
rw [map_prod (algebraMap L (AlgebraicClosure L))]
rw [← Fintype.prod_equiv (Normal.algHomEquivAut K (AlgebraicClosure L) L)]
· rw [← norm_eq_prod_embeddings _ _ x, ← IsScalarTower.algebraMap_apply]
· intro σ
simp only [Normal.algHomEqui... |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 132 | 134 | theorem continuous_iff (hg : Inducing g) :
Continuous f ↔ Continuous (g ∘ f) := by |
simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
|
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 | 150 | 161 | theorem exists_leftInverse_of_injective (f : V →ₗ[MonoidAlgebra k G] W)
(hf : LinearMap.ker f = ⊥) :
∃ g : W →ₗ[MonoidAlgebra k G] V, g.comp f = LinearMap.id := by |
let A := MonoidAlgebra k G
letI : Module k W := .compHom W (algebraMap k A)
letI : Module k V := .compHom V (algebraMap k A)
have := IsScalarTower.of_compHom k A W
have := IsScalarTower.of_compHom k A V
obtain ⟨φ, hφ⟩ := (f.restrictScalars k).exists_leftInverse_of_injective <| by
simp only [hf, Submodu... |
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Tactic.IntervalCases
namespace Polynomial
section IsDomain
variable {R : Type*} [CommRing R] [IsDomain R]
| Mathlib/Algebra/Polynomial/SpecificDegree.lean | 22 | 34 | theorem Monic.irreducible_iff_roots_eq_zero_of_degree_le_three {p : R[X]} (hp : p.Monic)
(hp2 : 2 ≤ p.natDegree) (hp3 : p.natDegree ≤ 3) : Irreducible p ↔ p.roots = 0 := by |
have hp0 : p ≠ 0 := hp.ne_zero
have hp1 : p ≠ 1 := by rintro rfl; rw [natDegree_one] at hp2; cases hp2
rw [hp.irreducible_iff_lt_natDegree_lt hp1]
simp_rw [show p.natDegree / 2 = 1 from
(Nat.div_le_div_right hp3).antisymm
(by apply Nat.div_le_div_right (c := 2) hp2),
show Finset.Ioc 0 1 = {1}... |
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.injective_resolution from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
noncomputable s... | Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean | 111 | 113 | theorem complex_d_comp (n : ℕ) :
I.cocomplex.d n (n + 1) ≫ I.cocomplex.d (n + 1) (n + 2) = 0 := by |
simp
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Rin... | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 114 | 132 | theorem exists_mulVec_eq_zero_iff_aux {K : Type*} [DecidableEq n] [Field K] {M : Matrix n n K} :
(∃ v ≠ 0, M *ᵥ v = 0) ↔ M.det = 0 := by |
constructor
· rintro ⟨v, hv, mul_eq⟩
contrapose! hv
exact eq_zero_of_mulVec_eq_zero hv mul_eq
· contrapose!
intro h
have : Function.Injective (Matrix.toLin' M) := by
simpa only [← LinearMap.ker_eq_bot, ker_toLin'_eq_bot_iff, not_imp_not] using h
have :
M *
LinearMap.toMa... |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 58 | 72 | theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : (0 : ℂ) ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by |
constructor
· intro hyp
simp only [cpow_def, eq_self_iff_true, if_true] at hyp
by_cases h : x = 0
· subst h
simp only [if_true, eq_self_iff_true] at hyp
right
exact ⟨rfl, hyp.symm⟩
· rw [if_neg h] at hyp
left
exact ⟨h, hyp.symm⟩
· rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)
·... |
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 337 | 338 | theorem degree_of_c_ne_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c ≠ 0) : P.toPoly.degree = 1 := by |
rw [of_b_eq_zero ha hb, degree_linear hc]
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 606 | 608 | theorem continuous_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
{f : X → E →L[𝕜] F} : Continuous f ↔ ∀ y, Continuous (f · y) := by |
simp_rw [continuous_iff_continuousOn_univ, continuousOn_clm_apply]
|
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {α β : Type*}
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
isCountablyGenerated : ∃ b... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 144 | 147 | theorem exists_measurableSet_of_ne [MeasurableSpace α] [SeparatesPoints α] {x y : α}
(h : x ≠ y) : ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s := by |
contrapose! h
exact separatesPoints_def h
|
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v... | Mathlib/Data/Set/NAry.lean | 325 | 329 | theorem image_image2_antidistrib {g : γ → δ} {f' : β' → α' → δ} {g₁ : β → β'} {g₂ : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g₁ b) (g₂ a)) :
(image2 f s t).image g = image2 f' (t.image g₁) (s.image g₂) := by |
rw [image2_swap f]
exact image_image2_distrib fun _ _ => h_antidistrib _ _
|
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Pseudoelements
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Images
#align_import category_theory.abelian.homology from "leanprover-community/mathlib"@"956af7c76589f444f... | Mathlib/CategoryTheory/Abelian/Homology.lean | 144 | 147 | theorem lift_ι {W : A} (e : W ⟶ cokernel f) (he : e ≫ cokernel.desc f g w = 0) :
lift f g w e he ≫ ι _ _ _ = e := by |
dsimp [ι, lift]
simp
|
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 | 33 | 41 | theorem mul_lt_mul (ac : a < c) (bd : b < d) : a * b < c * d := by |
rcases lt_iff_exists_nnreal_btwn.1 ac with ⟨a', aa', a'c⟩
lift a to ℝ≥0 using ne_top_of_lt aa'
rcases lt_iff_exists_nnreal_btwn.1 bd with ⟨b', bb', b'd⟩
lift b to ℝ≥0 using ne_top_of_lt bb'
norm_cast at *
calc
↑(a * b) < ↑(a' * b') := coe_lt_coe.2 (mul_lt_mul₀ aa' bb')
_ ≤ c * d := mul_le_mul' a'c.... |
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 185 | 187 | theorem realize_mul (x y : ring.Term α) (v : α → R) :
Term.realize v (x * y) = Term.realize v x * Term.realize v y := by |
simp [mul_def, funMap_mul]
|
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 | 1,081 | 1,106 | theorem lintegral_iInf_directed_of_measurable {mα : MeasurableSpace α} [Countable β]
{f : β → α → ℝ≥0∞} {μ : Measure α} (hμ : μ ≠ 0) (hf : ∀ b, Measurable (f b))
(hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : Directed (· ≥ ·) f) :
∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ := by |
cases nonempty_encodable β
cases isEmpty_or_nonempty β
· simp only [iInf_of_empty, lintegral_const,
ENNReal.top_mul (Measure.measure_univ_ne_zero.mpr hμ)]
inhabit β
have : ∀ a, ⨅ b, f b a = ⨅ n, f (h_directed.sequence f n) a := by
refine fun a =>
le_antisymm (le_iInf fun n => iInf_le _ _)
... |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 390 | 391 | theorem one_mem_nthRootsFinset (hn : 0 < n) : 1 ∈ nthRootsFinset n R := by |
rw [mem_nthRootsFinset hn, one_pow]
|
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
import Mathlib.RingTheory.Artinian
import Mathlib.LinearAlgebra.Trace
import Mathlib.LinearAlgebra.FreeModule.PID
#align_import algebra.lie.wei... | Mathlib/Algebra/Lie/Weights/Basic.lean | 316 | 319 | theorem exists_weightSpace_zero_le_ker_of_isNoetherian
[IsNoetherian R M] (x : L) :
∃ k : ℕ, weightSpace M (0 : L → R) ≤ LinearMap.ker (toEnd R L M x ^ k) := by |
simpa using exists_weightSpace_le_ker_of_isNoetherian M (0 : L → R) x
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 133 | 135 | theorem degrees_sum {ι : Type*} [DecidableEq σ] (s : Finset ι) (f : ι → MvPolynomial σ R) :
(∑ i ∈ s, f i).degrees ≤ s.sup fun i => (f i).degrees := by |
simp_rw [degrees_def]; exact supDegree_sum_le
|
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 | 85 | 90 | theorem card_Ioc : (Ioc a b).card = b - a := by |
rw [← Nat.card_Ioc]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Ioc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
import Mathlib.Order.Hom.Lattice
#align_import rin... | Mathlib/RingTheory/Filtration.lean | 371 | 403 | theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by |
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_iff_stable_ge]
constructor
· rintro H
refine H.stabilizes_of_iSup_eq
⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_
· intro n m e
rw [Submodule.span_le, Set.iUnion₂_subset_iff... |
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 267 | 271 | theorem uniformContinuous_dom₂ {f : SeparationQuotient α × SeparationQuotient β → γ} :
UniformContinuous f ↔ UniformContinuous fun p : α × β ↦ f (mk p.1, mk p.2) := by |
simp only [UniformContinuous, uniformity_prod_eq_prod, uniformity_eq, prod_map_map_eq,
tendsto_map'_iff]
rfl
|
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 181 | 184 | theorem congr_hom {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
stalkMap α x =
eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x) by rw [h]) ≫ stalkMap β x := by |
rw [← stalkMap.congr α β h x x rfl, eqToHom_refl, Category.comp_id]
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 298 | 299 | theorem sup_id_eq_sSup [CompleteLattice α] (s : Finset α) : s.sup id = sSup s := by |
simp [sSup_eq_iSup, sup_eq_iSup]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
namespace Matrix
variable {α β : Type*} {m n : Type*} {A : Matrix n n α}
open scoped Matrix
local ... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 281 | 282 | theorem IsHermitian.coe_re_apply_self {A : Matrix n n α} (h : A.IsHermitian) (i : n) :
(re (A i i) : α) = A i i := by | rw [← conj_eq_iff_re, ← star_def, ← conjTranspose_apply, h.eq]
|
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Basic
#align_import group_theory.group_action.conj_act from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable (α M G G₀ R K : Type*)
def ConjAct : Type _... | Mathlib/GroupTheory/GroupAction/ConjAct.lean | 306 | 307 | theorem mem_orbit_conjAct {g h : G} : g ∈ orbit (ConjAct G) h ↔ IsConj g h := by |
rw [isConj_comm, isConj_iff, mem_orbit_iff]; rfl
|
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
op... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 123 | 128 | theorem integrable_compProd_iff ⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
Integrable f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x))) ∧
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by |
simp only [Integrable, hasFiniteIntegral_compProd_iff' hf, hf.norm.integral_kernel_compProd,
hf, hf.compProd_mk_left, eventually_and, true_and_iff]
|
import Mathlib.Algebra.Polynomial.GroupRingAction
import Mathlib.Algebra.Ring.Action.Invariant
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Tower
#align_import field_theory.fixed from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
non... | Mathlib/FieldTheory/Fixed.lean | 225 | 241 | theorem irreducible_aux (f g : Polynomial (FixedPoints.subfield G F)) (hf : f.Monic) (hg : g.Monic)
(hfg : f * g = minpoly G F x) : f = 1 ∨ g = 1 := by |
have hf2 : f ∣ minpoly G F x := by rw [← hfg]; exact dvd_mul_right _ _
have hg2 : g ∣ minpoly G F x := by rw [← hfg]; exact dvd_mul_left _ _
have := eval₂ G F x
rw [← hfg, Polynomial.eval₂_mul, mul_eq_zero] at this
cases' this with this this
· right
have hf3 : f = minpoly G F x :=
Polynomial.eq_o... |
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import algebra.order.complete_field from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
variable {F α β γ : Type*}
noncomputable section
open Function ... | Mathlib/Algebra/Order/CompleteField.lean | 134 | 135 | theorem cutMap_coe (q : ℚ) : cutMap β (q : α) = Rat.cast '' {r : ℚ | (r : β) < q} := by |
simp_rw [cutMap, Rat.cast_lt]
|
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => ... | Mathlib/SetTheory/Ordinal/Principal.lean | 52 | 54 | theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :
Principal op o ↔ Principal (Function.swap op) o := by |
constructor <;> exact fun h a b ha hb => h hb ha
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def ImpartialAux : PGame → Prop
| G => (G ≈ -G) ∧ (∀ i... | Mathlib/SetTheory/Game/Impartial.lean | 226 | 230 | theorem exists_right_move_equiv_iff_fuzzy_zero : (∃ j, G.moveRight j ≈ 0) ↔ G ‖ 0 := by |
refine ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_lf G).2 (lf_of_moveRight_le hi.1), fun hn => ?_⟩
rw [fuzzy_zero_iff_lf G, lf_zero_le] at hn
cases' hn with i hi
exact ⟨i, (equiv_zero_iff_le _).2 hi⟩
|
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.BigOperators.RingEquiv
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Star.BigOperators... | Mathlib/Data/Matrix/Basic.lean | 2,061 | 2,066 | theorem transpose_one [DecidableEq n] [Zero α] [One α] : (1 : Matrix n n α)ᵀ = 1 := by |
ext i j
rw [transpose_apply, ← diagonal_one]
by_cases h : i = j
· simp only [h, diagonal_apply_eq]
· simp only [diagonal_apply_ne _ h, diagonal_apply_ne' _ h]
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.Common
#align_import data.set.pointwise.basic from "leanprover... | Mathlib/Data/Set/Pointwise/Basic.lean | 283 | 283 | theorem inv_singleton (a : α) : ({a} : Set α)⁻¹ = {a⁻¹} := by | rw [← image_inv, image_singleton]
|
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 622 | 630 | theorem iUnion_filter_not (π : Prepartition I) (p : Box ι → Prop) :
(π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion := by |
simp only [Prepartition.iUnion]
convert (@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm
· simp (config := { contextual := true })
· rw [Set.PairwiseDisjoint]
convert π.pairwiseDisjoint
rw [Set.union_eq_left, filter_boxes, coe_filter]
exact fun _ ⟨h, _⟩ => h
|
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 | 105 | 128 | theorem KaehlerDifferential.submodule_span_range_eq_ideal :
Submodule.span S (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
(KaehlerDifferential.ideal R S).restrictScalars S := by |
apply le_antisymm
· rw [Submodule.span_le]
rintro _ ⟨s, rfl⟩
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
· rintro x (hx : _ = _)
have : x - TensorProduct.lmul' (S := S) R x ⊗ₜ[R] (1 : S) = x := by
rw [hx, TensorProduct.zero_tmul, sub_zero]
rw [← this]
clear this hx
... |
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCompilation
#align_import linear_algebra.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_compilation
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 494 | 496 | theorem map₂_mk_top_top_eq_top : Submodule.map₂ (mk R M N) ⊤ ⊤ = ⊤ := by |
rw [← top_le_iff, ← span_tmul_eq_top, Submodule.map₂_eq_span_image2]
exact Submodule.span_mono fun _ ⟨m, n, h⟩ => ⟨m, trivial, n, trivial, h⟩
|
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 119 | 143 | theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by |
haveI := Nontrivial.of_polynomial_ne hp0
calc
degree (divX p) < (divX p * X + C (p.coeff 0)).degree :=
if h : degree p ≤ 0 then by
have h' : C (p.coeff 0) ≠ 0 := by rwa [← eq_C_of_degree_le_zero h]
rw [eq_C_of_degree_le_zero h, divX_C, degree_zero, zero_mul, zero_add]
exact lt_of_... |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 493 | 496 | theorem dNext_zero_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : dNext 0 f = 0 := by |
dsimp [dNext]
rw [P.shape, zero_comp]
rw [ChainComplex.next_nat_zero]; dsimp; decide
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (α : Type*) where... | Mathlib/Order/Filter/Basic.lean | 2,628 | 2,634 | theorem comap_eval_neBot_iff' {ι : Type*} {α : ι → Type*} {i : ι} {f : Filter (α i)} :
(comap (eval i) f).NeBot ↔ (∀ j, Nonempty (α j)) ∧ NeBot f := by |
cases' isEmpty_or_nonempty (∀ j, α j) with H H
· rw [filter_eq_bot_of_isEmpty (f.comap _), ← not_iff_not]
simp [← Classical.nonempty_pi]
· have : ∀ j, Nonempty (α j) := Classical.nonempty_pi.1 H
simp [comap_neBot_iff_frequently, *]
|
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 494 | 495 | theorem image₂_right_identity {f : γ → β → γ} {b : β} (h : ∀ a, f a b = a) (s : Finset γ) :
image₂ f s {b} = s := by | rw [image₂_singleton_right, funext h, image_id']
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
set_option autoImplicit true
universe u v
open Function Order
noncomputable section
def NatOrdinal : Type _ :=
... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 326 | 326 | theorem nadd_succ : a ♯ succ b = succ (a ♯ b) := by | rw [← nadd_one (a ♯ b), nadd_assoc, nadd_one]
|
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 | 369 | 370 | theorem associator_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
(f ≫ g) ◁ η ≫ (α_ f g h').hom = (α_ f g h).hom ≫ f ◁ g ◁ η := by | simp
|
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section InitsTails
@[simp]
theorem mem_inits : ∀ s t : List α, s ∈ in... | Mathlib/Data/List/Infix.lean | 440 | 442 | theorem map_reverse_inits (l : List α) : map reverse l.inits = (reverse <| tails <| reverse l) := by |
rw [inits_eq_tails l]
simp [reverse_involutive.comp_self, reverse_map]
|
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 867 | 870 | theorem total_comapDomain (f : α → α') (l : α' →₀ R) (hf : Set.InjOn f (f ⁻¹' ↑l.support)) :
Finsupp.total α M R v (Finsupp.comapDomain f l hf) =
(l.support.preimage f hf).sum fun i => l (f i) • v i := by |
rw [Finsupp.total_apply]; rfl
|
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 | 115 | 119 | theorem drop_sum_join' (L : List (List α)) (i : ℕ) :
L.join.drop (Nat.sum ((L.map length).take i)) = (L.drop i).join := by |
induction L generalizing i
· simp
· cases i <;> simp [drop_append, *]
|
import Mathlib.FieldTheory.Fixed
import Mathlib.FieldTheory.NormalClosure
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.GroupTheory.GroupAction.FixingSubgroup
#align_import field_theory.galois from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
open scoped Polynomial Interm... | Mathlib/FieldTheory/Galois.lean | 429 | 444 | theorem tfae [FiniteDimensional F E] :
List.TFAE [IsGalois F E, IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥,
Fintype.card (E ≃ₐ[F] E) = finrank F E, ∃ p: F[X], p.Separable ∧ p.IsSplittingField F E] := by |
tfae_have 1 → 2
· exact fun h => OrderIso.map_bot (@intermediateFieldEquivSubgroup F _ E _ _ _ h).symm
tfae_have 1 → 3
· intro; exact card_aut_eq_finrank F E
tfae_have 1 → 4
· intro; exact is_separable_splitting_field F E
tfae_have 2 → 1
· exact of_fixedField_eq_bot F E
tfae_have 3 → 1
· exact of_c... |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,274 | 1,278 | theorem restrict_trim (hle : m ≤ n) {i : Set α} (hi : MeasurableSet[m] i) :
@VectorMeasure.restrict α m M _ _ (v.trim hle) i = (v.restrict i).trim hle := by |
ext j hj
rw [@restrict_apply _ m, trim_measurableSet_eq hle hj, restrict_apply, trim_measurableSet_eq]
all_goals measurability
|
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 2,654 | 2,656 | theorem sub_filter_eq_filter_not [DecidableEq α] (p) [DecidablePred p] (s : Multiset α) :
s - s.filter p = s.filter (fun a ↦ ¬ p a) := by |
ext a; by_cases h : p a <;> simp [h]
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 689 | 695 | theorem cthickening_thickening_subset (hε : 0 ≤ ε) (δ : ℝ) (s : Set α) :
cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s := by |
obtain hδ | hδ := le_total δ 0
· simp only [thickening_of_nonpos hδ, cthickening_empty, empty_subset]
intro x
simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ]
exact fun hx => infEdist_le_infEdist_thickening_add.trans (add_le_add_right hx _)
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 992 | 999 | theorem inter_nonempty_of_nonempty_of_sup_direction_eq_top {s1 s2 : AffineSubspace k P}
(h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty)
(hd : s1.direction ⊔ s2.direction = ⊤) : ((s1 : Set P) ∩ s2).Nonempty := by |
by_contra h
rw [Set.not_nonempty_iff_eq_empty] at h
have hlt := sup_direction_lt_of_nonempty_of_inter_empty h1 h2 h
rw [hd] at hlt
exact not_top_lt hlt
|
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {α β γ ι ι' : Type*}
namespace Filter
theorem isCobounde... | Mathlib/Order/LiminfLimsup.lean | 1,277 | 1,284 | theorem eventually_lt_of_lt_liminf {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
{b : β} (h : b < liminf u f)
(hu : f.IsBoundedUnder (· ≥ ·) u := by | isBoundedDefault) :
∀ᶠ a in f, b < u a := by
obtain ⟨c, hc, hbc⟩ : ∃ (c : β) (_ : c ∈ { c : β | ∀ᶠ n : α in f, c ≤ u n }), b < c := by
simp_rw [exists_prop]
exact exists_lt_of_lt_csSup hu h
exact hc.mono fun x hx => lt_of_lt_of_le hbc hx
|
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 342 | 343 | theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv := by |
simp
|
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