Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | goals listlengths 0 224 | goals_before listlengths 0 220 |
|---|---|---|---|---|---|---|---|
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 | 131 | 134 | theorem δ_comp_δ_self {n} {i : Fin (n + 2)} :
X.δ (Fin.castSucc i) ≫ X.δ i = X.δ i.succ ≫ X.δ i := by |
dsimp [δ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ_self]
| [
" Category.{?u.61, max u v} (SimplicialObject C)",
" Category.{?u.61, max u v} (SimplexCategoryᵒᵖ ⥤ C)",
" HasLimitsOfShape J (SimplicialObject C)",
" HasLimitsOfShape J (SimplexCategoryᵒᵖ ⥤ C)",
" HasColimitsOfShape J (SimplicialObject C)",
" HasColimitsOfShape J (SimplexCategoryᵒᵖ ⥤ C)",
" f.app = g.a... | [
" Category.{?u.61, max u v} (SimplicialObject C)",
" Category.{?u.61, max u v} (SimplexCategoryᵒᵖ ⥤ C)",
" HasLimitsOfShape J (SimplicialObject C)",
" HasLimitsOfShape J (SimplexCategoryᵒᵖ ⥤ C)",
" HasColimitsOfShape J (SimplicialObject C)",
" HasColimitsOfShape J (SimplexCategoryᵒᵖ ⥤ C)",
" f.app = g.a... |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 213 | 215 | theorem diff_cylinder_same (s : Finset ι) (S T : Set (∀ i : s, α i)) :
cylinder s S \ cylinder s T = cylinder s (S \ T) := by |
ext1 f; simp only [mem_diff, mem_cylinder]
| [
" cylinder s ∅ = ∅",
" cylinder s univ = univ",
" cylinder s S = ∅ ↔ S = ∅",
" cylinder s S = ∅",
" S = ∅",
" False",
" f' ∈ cylinder s S",
" (fun i => f' ↑i) ∈ S",
" cylinder s₁ S₁ ∩ cylinder s₂ S₂ = cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∩ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)",
" f ∈ cylind... | [
" cylinder s ∅ = ∅",
" cylinder s univ = univ",
" cylinder s S = ∅ ↔ S = ∅",
" cylinder s S = ∅",
" S = ∅",
" False",
" f' ∈ cylinder s S",
" (fun i => f' ↑i) ∈ S",
" cylinder s₁ S₁ ∩ cylinder s₂ S₂ = cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∩ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)",
" f ∈ cylind... |
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Rat
#align_import data.rat.order from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
assert_not_exists Field
assert_not_exists Finset
assert_not_exists Set.Icc
assert_not_exists GaloisConnection
namespace Rat
variable {a... | Mathlib/Algebra/Order/Ring/Rat.lean | 50 | 59 | theorem ofScientific_nonneg (m : ℕ) (s : Bool) (e : ℕ) :
0 ≤ Rat.ofScientific m s e := by |
rw [Rat.ofScientific]
cases s
· rw [if_neg (by decide)]
refine num_nonneg.mp ?_
rw [num_natCast]
exact Int.natCast_nonneg _
· rw [if_pos rfl, normalize_eq_mkRat]
exact Rat.mkRat_nonneg (Int.natCast_nonneg _) _
| [
" 0 ≤ a /. b ↔ 0 ≤ a",
" 0 ≤ { num := n, den := d, den_nz := hd, reduced := hnd } ↔ 0 ≤ a",
" 0 ≤ a /. b",
" 0 ≤ a /. 0",
" 0 ≤ 0",
" 0 ≤ mkRat a b",
" 0 ≤ Rat.ofScientific m s e",
" 0 ≤ if s = true then normalize (↑m) (10 ^ e) ⋯ else ↑(m * 10 ^ e)",
" 0 ≤ if false = true then normalize (↑m) (10 ^ e... | [
" 0 ≤ a /. b ↔ 0 ≤ a",
" 0 ≤ { num := n, den := d, den_nz := hd, reduced := hnd } ↔ 0 ≤ a",
" 0 ≤ a /. b",
" 0 ≤ a /. 0",
" 0 ≤ 0",
" 0 ≤ mkRat a b"
] |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 195 | 199 | theorem ofFinsupp_pow (a) (n : ℕ) : (⟨a ^ n⟩ : R[X]) = ⟨a⟩ ^ n := by |
change _ = npowRec n _
induction n with
| zero => simp [npowRec]
| succ n n_ih => simp [npowRec, n_ih, pow_succ]
| [
" { toFinsupp := f.toFinsupp } = f",
" { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ }",
" { toFinsupp := a + b } = Polynomial.add { toFinsupp := a } { toFinsupp := b }",
" { toFinsupp := -a } = Polynomial.neg { toFinsupp := a }",
" { toFinsupp := a - b } = { toFinsupp :... | [
" { toFinsupp := f.toFinsupp } = f",
" { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ }",
" { toFinsupp := a + b } = Polynomial.add { toFinsupp := a } { toFinsupp := b }",
" { toFinsupp := -a } = Polynomial.neg { toFinsupp := a }",
" { toFinsupp := a - b } = { toFinsupp :... |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : ℕ) : Multiset (ℕ × ℕ) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 36 | 37 | theorem mem_antidiagonal {n : ℕ} {x : ℕ × ℕ} : x ∈ antidiagonal n ↔ x.1 + x.2 = n := by |
rw [antidiagonal, mem_coe, List.Nat.mem_antidiagonal]
| [
" x ∈ antidiagonal n ↔ x.1 + x.2 = n"
] | [] |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 106 | 113 | theorem Duplicate.mono_sublist {l' : List α} (hx : x ∈+ l) (h : l <+ l') : x ∈+ l' := by |
induction' h with l₁ l₂ y _ IH l₁ l₂ y h IH
· exact hx
· exact (IH hx).duplicate_cons _
· rw [duplicate_cons_iff] at hx ⊢
rcases hx with (⟨rfl, hx⟩ | hx)
· simp [h.subset hx]
· simp [IH hx]
| [
" x ∈ l",
" x ∈ x :: l'",
" x ∈ y :: l'",
" l ≠ [y]",
" x :: l' ≠ [y]",
" z :: l' ≠ [y]",
" x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l",
" y = x ∧ x ∈ l ∨ x ∈+ l",
" x = x ∧ x ∈ l ∨ x ∈+ l",
" x ∈+ y :: l",
" x ∈+ x :: l",
" x ∈+ l",
" x ∈+ y :: l ↔ x ∈+ l",
" x ∈+ l'",
" x ∈+ []",
" x ∈+ y ... | [
" x ∈ l",
" x ∈ x :: l'",
" x ∈ y :: l'",
" l ≠ [y]",
" x :: l' ≠ [y]",
" z :: l' ≠ [y]",
" x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l",
" y = x ∧ x ∈ l ∨ x ∈+ l",
" x = x ∧ x ∈ l ∨ x ∈+ l",
" x ∈+ y :: l",
" x ∈+ x :: l",
" x ∈+ l",
" x ∈+ y :: l ↔ x ∈+ l"
] |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic
#align_import algebra.algebraic_card from "leanprover-community/mathlib"@"40494fe75ecbd6d2ec61711baa630cf0a7b7d064"
universe u v
open Cardinal Polynomial Set
open Cardinal Polynomial
namespace Algebraic
theorem infinite_of_charZero... | Mathlib/Algebra/AlgebraicCard.lean | 45 | 54 | theorem cardinal_mk_lift_le_mul :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } ≤ Cardinal.lift.{v} #R[X] * ℵ₀ := by |
rw [← mk_uLift, ← mk_uLift]
choose g hg₁ hg₂ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => ?_
rw [lift_le_aleph0, le_aleph0_iff_set_countable]
suffices MapsTo (↑) (g ⁻¹' {f}) (f.rootSet A) from
this.countable_of_injOn Subtype.coe_i... | [
" lift.{u, v} #{ x // IsAlgebraic R x } ≤ lift.{v, u} #R[X] * ℵ₀",
" #(ULift.{u, v} { x // IsAlgebraic R x }) ≤ #(ULift.{v, u} R[X]) * ℵ₀",
" lift.{u, v} #↑(g ⁻¹' {f}) ≤ ℵ₀",
" (g ⁻¹' {f}).Countable",
" MapsTo Subtype.val (g ⁻¹' {f}) (f.rootSet A)",
" ↑x ∈ (g x).rootSet A"
] | [] |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 217 | 220 | theorem ker_mulVecLin_conjTranspose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aᴴ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, conjTranspose_mul_self_mulVec_eq_zero]
| [
" LinearMap.ker (Aᴴ * A).mulVecLin = LinearMap.ker A.mulVecLin",
" x ∈ LinearMap.ker (Aᴴ * A).mulVecLin ↔ x ∈ LinearMap.ker A.mulVecLin"
] | [] |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 63 | 66 | theorem Hom.cast_eq_iff_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) (e' : u' ⟶ v') :
e.cast hu hv = e' ↔ HEq e e' := by |
rw [Hom.cast_eq_cast]
exact _root_.cast_eq_iff_heq
| [
" (u ⟶ v) = (u' ⟶ v')",
" cast hu hv e = _root_.cast ⋯ e",
" cast ⋯ ⋯ e = _root_.cast ⋯ e",
" cast hu' hv' (cast hu hv e) = cast ⋯ ⋯ e",
" cast ⋯ ⋯ (cast ⋯ ⋯ e) = cast ⋯ ⋯ e",
" HEq (cast hu hv e) e",
" HEq (cast ⋯ ⋯ e) e",
" cast hu hv e = e' ↔ HEq e e'",
" _root_.cast ⋯ e = e' ↔ HEq e e'"
] | [
" (u ⟶ v) = (u' ⟶ v')",
" cast hu hv e = _root_.cast ⋯ e",
" cast ⋯ ⋯ e = _root_.cast ⋯ e",
" cast hu' hv' (cast hu hv e) = cast ⋯ ⋯ e",
" cast ⋯ ⋯ (cast ⋯ ⋯ e) = cast ⋯ ⋯ e",
" HEq (cast hu hv e) e",
" HEq (cast ⋯ ⋯ e) e"
] |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {K : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 107 | 109 | theorem disjoint_ker {f : F} {p : Submodule R M} :
Disjoint p (ker f) ↔ ∀ x ∈ p, f x = 0 → x = 0 := by |
simp [disjoint_def]
| [
" ker f ≤ ker (g.comp f)",
" ker f ≤ comap f (ker g)",
" ker f ⊔ ker g ≤ ker (f ∘ₗ g)",
" ker f ≤ ker (f ∘ₗ g)",
" ker f ≤ ker (g ∘ₗ f)",
" x ∈ comap f p",
" Disjoint p (ker f) ↔ ∀ x ∈ p, f x = 0 → x = 0"
] | [
" ker f ≤ ker (g.comp f)",
" ker f ≤ comap f (ker g)",
" ker f ⊔ ker g ≤ ker (f ∘ₗ g)",
" ker f ≤ ker (f ∘ₗ g)",
" ker f ≤ ker (g ∘ₗ f)",
" x ∈ comap f p"
] |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.Normed.Group.Completion
#align_import analysis.normed.group.hom_completion from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
noncomputable section
open Set NormedAddGroupHom UniformSpace
section Completion
variable {G... | Mathlib/Analysis/Normed/Group/HomCompletion.lean | 171 | 193 | theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
(h : f.SurjectiveOnWith f.range C) :
(f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range := by |
refine le_antisymm ?_ (closure_minimal f.ker_le_ker_completion f.completion.isClosed_ker)
rintro hatg (hatg_in : f.completion hatg = 0)
rw [SeminormedAddCommGroup.mem_closure_iff]
intro ε ε_pos
rcases h.exists_pos with ⟨C', C'_pos, hC'⟩
rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩
ob... | [
" (id G).completion = id (Completion G)",
" (id G).completion x = (id (Completion G)) x",
" _root_.id x = (id (Completion G)) x",
" g.completion.comp f.completion = (g.comp f).completion",
" (g.completion.comp f.completion) x = (g.comp f).completion x",
" Completion.map (⇑g ∘ ⇑f) x = Completion.map (⇑(g.c... | [
" (id G).completion = id (Completion G)",
" (id G).completion x = (id (Completion G)) x",
" _root_.id x = (id (Completion G)) x",
" g.completion.comp f.completion = (g.comp f).completion",
" (g.completion.comp f.completion) x = (g.comp f).completion x",
" Completion.map (⇑g ∘ ⇑f) x = Completion.map (⇑(g.c... |
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
#align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd"
namespace CategoryTheory
open Category Limits Preadditive
namespace Abelian
va... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 62 | 83 | theorem mono_of_epi_of_mono_of_mono' (hR₁ : R₁.map' 0 2 = 0)
(hR₁' : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
(hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact)
(h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) (h₃ : Mono (app' φ 3)) :
Mono (app' φ 2) := by |
apply mono_of_cancel_zero
intro A f₂ h₁
have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
rw [← cancel_mono (app' φ 3 _), assoc, NatTrans.naturality, reassoc_of% h₁,
zero_comp, zero_comp]
obtain ⟨A₁, π₁, _, f₁, hf₁⟩ := (hR₁'.exact 0).exact_up_to_refinements f₂ h₂
dsimp at hf₁
have h₃ : (f₁ ≫ app' φ 1) ≫ R₂.ma... | [
" Mono (app' φ 2 ⋯)",
" ∀ {P : C} (g : P ⟶ R₁.obj' 2 ⋯), g ≫ app' φ 2 ⋯ = 0 → g = 0",
" f₂ = 0",
" f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0",
" (f₁ ≫ app' φ 1 ⋯) ≫ R₂.map' 1 2 ⋯ ⋯ = 0",
" f₀ ≫ R₁.map' 0 1 ⋯ ⋯ = π₃ ≫ π₂ ≫ f₁",
" π₃ ≫ g₀ ≫ R₂.map (homOfLE ⋯) = π₃ ≫ g₀ ≫ ((mk₂ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).sc ⋯ 0 ⋯... | [] |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 100 | 102 | theorem iUnion_right_iff [Countable ι] {t : ι → Set α} :
AEDisjoint μ s (⋃ i, t i) ↔ ∀ i, AEDisjoint μ s (t i) := by |
simp only [AEDisjoint, inter_iUnion, measure_iUnion_null_iff]
| [
" ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \\ t i)",
" μ ((fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i) = 0",
" μ (⋃ i_1 ∈ {i}ᶜ, s i ∩ s i_1) = 0",
" Pairwise (Disjoint on fun i => s i \\ (fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i)",... | [
" ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \\ t i)",
" μ ((fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i) = 0",
" μ (⋃ i_1 ∈ {i}ᶜ, s i ∩ s i_1) = 0",
" Pairwise (Disjoint on fun i => s i \\ (fun i => toMeasurable μ (s i ∩ ⋃ j ∈ {i}ᶜ, s j)) i)",... |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 118 | 124 | theorem edist_congr_right {x y z : α} (h : edist x y = 0) : edist x z = edist y z := by |
apply le_antisymm
· rw [← zero_add (edist y z), ← h]
apply edist_triangle
· rw [edist_comm] at h
rw [← zero_add (edist x z), ← h]
apply edist_triangle
| [
" s ∈ U ↔ ∃ i > z, {p | D p.1 p.2 < i} ⊆ s",
" m = m'",
" mk edist_self✝ edist_comm✝ edist_triangle✝ U hU = m'",
" mk edist_self✝¹ edist_comm✝¹ edist_triangle✝¹ U hU = mk edist_self✝ edist_comm✝ edist_triangle✝ U' hU'",
" U = U'",
" edist x y ≤ edist z x + edist z y",
" edist x y ≤ edist x z + edist z y... | [
" s ∈ U ↔ ∃ i > z, {p | D p.1 p.2 < i} ⊆ s",
" m = m'",
" mk edist_self✝ edist_comm✝ edist_triangle✝ U hU = m'",
" mk edist_self✝¹ edist_comm✝¹ edist_triangle✝¹ U hU = mk edist_self✝ edist_comm✝ edist_triangle✝ U' hU'",
" U = U'",
" edist x y ≤ edist z x + edist z y",
" edist x y ≤ edist x z + edist z y... |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 66 | 68 | theorem mem_unitaryGroup_iff : A ∈ Matrix.unitaryGroup n α ↔ A * star A = 1 := by |
refine ⟨And.right, fun hA => ⟨?_, hA⟩⟩
simpa only [mul_eq_one_comm] using hA
| [
" A ∈ unitaryGroup n α ↔ A * star A = 1",
" star A * A = 1"
] | [] |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 154 | 156 | theorem mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
f ∈ idealOfSet R ({x}ᶜ : Set X) ↔ f x = 0 := by |
simp only [mem_idealOfSet, compl_compl, Set.mem_singleton_iff, forall_eq]
| [
" (f + g) x = 0",
" IsClosed ↑(idealOfSet R s)",
" IsClosed ↑{ carrier := ⋂ i ∈ sᶜ, {x | x i = 0}, add_mem' := ⋯, zero_mem' := ⋯ }",
" f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0",
" f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0",
" (¬∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0) ↔ ∃ x ∈ sᶜ, f x ≠ 0",
" (∃ x ∈ sᶜ, f x ≠ 0... | [
" (f + g) x = 0",
" IsClosed ↑(idealOfSet R s)",
" IsClosed ↑{ carrier := ⋂ i ∈ sᶜ, {x | x i = 0}, add_mem' := ⋯, zero_mem' := ⋯ }",
" f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0",
" f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0",
" (¬∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0) ↔ ∃ x ∈ sᶜ, f x ≠ 0",
" (∃ x ∈ sᶜ, f x ≠ 0... |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 117 | 119 | theorem nhds_basis_ball {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (ball x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_ball.comap _
| [
" closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r",
" IsCompact (closedBall x r)",
" IsCompact (⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r)",
" closure (ball x r) = closedBall x r",
" ∃ r ∈ Ioo 0 R, s ⊆ ball x r",
" (𝓝 x).HasBasis (fun r => 0 < r) (closedBall x)",
... | [
" closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r",
" IsCompact (closedBall x r)",
" IsCompact (⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r)",
" closure (ball x r) = closedBall x r",
" ∃ r ∈ Ioo 0 R, s ⊆ ball x r",
" (𝓝 x).HasBasis (fun r => 0 < r) (closedBall x)",
... |
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 | 156 | 157 | theorem add_one_mul [RightDistribClass α] (a b : α) : (a + 1) * b = a * b + b := by |
rw [add_mul, one_mul]
| [
" (a + b + c) * d = a * d + b * d + c * d",
" (a + 1) * b = a * b + b"
] | [
" (a + b + c) * d = a * d + b * d + c * d"
] |
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Directed
#align_import data.set.Union_lift from "leanprover-community/mathlib"@"5a4ea8453f128345f73cc656e80a49de2a54f481"
variable {α : Type*} {ι β : Sort _}
namespace Set
section UnionLift
@[nolint unusedArguments]
noncomputable def iUnionLift (S : ι → Set... | Mathlib/Data/Set/UnionLift.lean | 79 | 90 | theorem preimage_iUnionLift (t : Set β) :
iUnionLift S f hf T hT ⁻¹' t =
inclusion hT ⁻¹' (⋃ i, inclusion (subset_iUnion S i) '' (f i ⁻¹' t)) := by |
ext x
simp only [mem_preimage, mem_iUnion, mem_image]
constructor
· rcases mem_iUnion.1 (hT x.prop) with ⟨i, hi⟩
refine fun h => ⟨i, ⟨x, hi⟩, ?_, rfl⟩
rwa [iUnionLift_of_mem x hi] at h
· rintro ⟨i, ⟨y, hi⟩, h, hxy⟩
obtain rfl : y = x := congr_arg Subtype.val hxy
rwa [iUnionLift_of_mem x hi]
| [
" iUnionLift S f hf T hT x = f i ⟨↑x, hx⟩",
" iUnionLift S f hf T hT ⟨x, hx✝⟩ = f i ⟨↑⟨x, hx✝⟩, hx⟩",
" iUnionLift S f hf T hT ⁻¹' t = inclusion hT ⁻¹' ⋃ i, inclusion ⋯ '' (f i ⁻¹' t)",
" x ∈ iUnionLift S f hf T hT ⁻¹' t ↔ x ∈ inclusion hT ⁻¹' ⋃ i, inclusion ⋯ '' (f i ⁻¹' t)",
" iUnionLift S f hf T hT x ∈ t... | [
" iUnionLift S f hf T hT x = f i ⟨↑x, hx⟩",
" iUnionLift S f hf T hT ⟨x, hx✝⟩ = f i ⟨↑⟨x, hx✝⟩, hx⟩"
] |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 170 | 179 | theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)
(ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S)))
(S : Subalgebra R A) : P S := by |
classical
obtain ⟨t, rfl⟩ := S.fg_of_noetherian
refine Finset.induction_on t ?_ ?_
· simpa using base
intro x t _ h
rw [Finset.coe_insert]
simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
| [
" x ∈ span R ↑t",
" x ∈ toSubmodule S",
" toSubmodule (Algebra.adjoin R ↑s) = toSubmodule ⊤",
" ↑s ⊆ ↑(toSubmodule (Algebra.adjoin R ↑s))",
" (S.prod T).FG",
" ((Algebra.adjoin R s).prod (Algebra.adjoin R t)).FG",
" Algebra.adjoin R ↑(Finset.image (⇑f) s) = Subalgebra.map f S",
" map f (Algebra.adjoin... | [
" x ∈ span R ↑t",
" x ∈ toSubmodule S",
" toSubmodule (Algebra.adjoin R ↑s) = toSubmodule ⊤",
" ↑s ⊆ ↑(toSubmodule (Algebra.adjoin R ↑s))",
" (S.prod T).FG",
" ((Algebra.adjoin R s).prod (Algebra.adjoin R t)).FG",
" Algebra.adjoin R ↑(Finset.image (⇑f) s) = Subalgebra.map f S",
" map f (Algebra.adjoin... |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 92 | 102 | theorem condexpIndL1Fin_add (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x y : G) :
condexpIndL1Fin hm hs hμs (x + y) =
condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm hs hμs y := by |
ext1
refine (Memℒp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_add _ _).symm
refine EventuallyEq.trans ?_
(EventuallyEq.add (Memℒp.coeFn_toLp q).symm (Memℒp.coeFn_toLp q).symm)
rw [condexpIndSMul_add]
refine (Lp.coeFn_add _ _).trans (eventually_of_forall fun a => ?_)
rfl
| [
" Memℒp (↑↑(condexpIndSMul hm hs hμs x)) 1 μ",
" Integrable (↑↑(condexpIndSMul hm hs hμs x)) μ",
" condexpIndL1Fin hm hs hμs (x + y) = condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm hs hμs y",
" ↑↑(condexpIndL1Fin hm hs hμs (x + y)) =ᶠ[ae μ] ↑↑(condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm hs hμs y)",
... | [
" Memℒp (↑↑(condexpIndSMul hm hs hμs x)) 1 μ",
" Integrable (↑↑(condexpIndSMul hm hs hμs x)) μ"
] |
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
#align_import data.nat.size from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
namespace Nat
section
set_option linter.deprecated false
theorem shiftLeft_eq_mul_pow (m) : ∀ n, m <<< n = m * 2 ^ n := shiftLeft_eq _
#align nat.... | Mathlib/Data/Nat/Size.lean | 144 | 145 | theorem size_eq_zero {n : ℕ} : size n = 0 ↔ n = 0 := by |
simpa [Nat.pos_iff_ne_zero, not_iff_not] using size_pos
| [
" shiftLeft' true m 0 + 1 = (m + 1) * 2 ^ 0",
" shiftLeft' true m (k + 1) + 1 = (m + 1) * 2 ^ (k + 1)",
" bit1 (shiftLeft' true m k) + 1 = (m + 1) * (2 ^ k * 2)",
" 2 * shiftLeft' true m k + 1 + 1 = (m + 1) * (2 ^ k * 2)",
" 2 * (shiftLeft' true m k + 1) = (m + 1) * (2 ^ k * 2)",
" shiftLeft' b m n ≠ 0",
... | [
" shiftLeft' true m 0 + 1 = (m + 1) * 2 ^ 0",
" shiftLeft' true m (k + 1) + 1 = (m + 1) * 2 ^ (k + 1)",
" bit1 (shiftLeft' true m k) + 1 = (m + 1) * (2 ^ k * 2)",
" 2 * shiftLeft' true m k + 1 + 1 = (m + 1) * (2 ^ k * 2)",
" 2 * (shiftLeft' true m k + 1) = (m + 1) * (2 ^ k * 2)",
" shiftLeft' b m n ≠ 0",
... |
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.presented_group from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {α : Type*}
def PresentedGroup (rels : Set (FreeGroup α)) :=
FreeGroup α ⧸ Subgroup.normalClosu... | Mathlib/GroupTheory/PresentedGroup.lean | 101 | 104 | theorem ext {φ ψ : PresentedGroup rels →* G} (hx : ∀ (x : α), φ (.of x) = ψ (.of x)) : φ = ψ := by |
unfold PresentedGroup
ext
apply hx
| [
" Subgroup.closure (Set.range of) = ⊤",
" (QuotientGroup.mk' (Subgroup.normalClosure rels)).range = ⊤",
" ∀ {x : PresentedGroup rels}, g x = (toGroup h) x",
" g x = (toGroup h) x",
" ∀ (z : FreeGroup α), g ↑z = (toGroup h) ↑z",
" φ = ψ",
" (φ.comp (QuotientGroup.mk' (Subgroup.normalClosure rels))) (Free... | [
" Subgroup.closure (Set.range of) = ⊤",
" (QuotientGroup.mk' (Subgroup.normalClosure rels)).range = ⊤",
" ∀ {x : PresentedGroup rels}, g x = (toGroup h) x",
" g x = (toGroup h) x",
" ∀ (z : FreeGroup α), g ↑z = (toGroup h) ↑z"
] |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import data.polynomial.cardinal from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
universe u
open Cardinal Polynomial
open Cardinal
namespace Polynomial
@[simp]
theorem cardinal_mk_eq_max {R :... | Mathlib/Algebra/Polynomial/Cardinal.lean | 34 | 37 | theorem cardinal_mk_le_max {R : Type u} [Semiring R] : #(R[X]) ≤ max #R ℵ₀ := by |
cases subsingleton_or_nontrivial R
· exact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph0)
· exact cardinal_mk_eq_max.le
| [
" #(AddMonoidAlgebra R ℕ) = max #R ℵ₀",
" max (#R) (lift.{u, 0} #ℕ) = max #R ℵ₀",
" #R[X] ≤ max #R ℵ₀"
] | [
" #(AddMonoidAlgebra R ℕ) = max #R ℵ₀",
" max (#R) (lift.{u, 0} #ℕ) = max #R ℵ₀"
] |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 123 | 125 | theorem lcm_eq_zero_iff [Nontrivial α] : s.lcm f = 0 ↔ 0 ∈ f '' s := by |
simp only [Multiset.mem_map, lcm_def, Multiset.lcm_eq_zero_iff, Set.mem_image, mem_coe, ←
Finset.mem_def]
| [
" s.lcm f ∣ a ↔ ∀ b ∈ s, f b ∣ a",
" (∀ b ∈ Multiset.map f s.val, b ∣ a) ↔ ∀ b ∈ s, f b ∣ a",
" (∀ (b : α), ∀ x ∈ s.val, f x = b → b ∣ a) ↔ ∀ b ∈ s, f b ∣ a",
" (insert b s).lcm f = GCDMonoid.lcm (f b) (s.lcm f)",
" normalize (s.lcm f) = s.lcm f",
" (∅ ∪ s₂).lcm f = GCDMonoid.lcm (∅.lcm f) (s₂.lcm f)",
... | [
" s.lcm f ∣ a ↔ ∀ b ∈ s, f b ∣ a",
" (∀ b ∈ Multiset.map f s.val, b ∣ a) ↔ ∀ b ∈ s, f b ∣ a",
" (∀ (b : α), ∀ x ∈ s.val, f x = b → b ∣ a) ↔ ∀ b ∈ s, f b ∣ a",
" (insert b s).lcm f = GCDMonoid.lcm (f b) (s.lcm f)",
" normalize (s.lcm f) = s.lcm f",
" (∅ ∪ s₂).lcm f = GCDMonoid.lcm (∅.lcm f) (s₂.lcm f)",
... |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 52 | 55 | theorem invUnitsSub_mul_X (u : Rˣ) : invUnitsSub u * X = invUnitsSub u * C R u - 1 := by |
ext (_ | n)
· simp
· simp [n.succ_ne_zero, pow_succ']
| [
" (constantCoeff R) (invUnitsSub u) = 1 /ₚ u",
" invUnitsSub u * X = invUnitsSub u * (C R) ↑u - 1",
" (coeff R 0) (invUnitsSub u * X) = (coeff R 0) (invUnitsSub u * (C R) ↑u - 1)",
" (coeff R (n + 1)) (invUnitsSub u * X) = (coeff R (n + 1)) (invUnitsSub u * (C R) ↑u - 1)"
] | [
" (constantCoeff R) (invUnitsSub u) = 1 /ₚ u"
] |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 556 | 561 | theorem toNNReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toNNReal = ⨆ i, (f i).toNNReal := by |
lift f to ι → ℝ≥0 using hf
simp_rw [toNNReal_coe]
by_cases h : BddAbove (range f)
· rw [← coe_iSup h, toNNReal_coe]
· rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, top_toNNReal]
| [
" (iInf f).toNNReal = ⨅ i, (f i).toNNReal",
" (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i => ↑(f i)) i).toNNReal",
" (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)",
" (iSup f).toNNReal = ⨆ i, (f i).toNNReal",
" (⨆ i, ↑(f i)).toNNReal = ⨆ i, ((fun i => ↑(f i)) i).toNNReal",
" (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i... | [
" (iInf f).toNNReal = ⨅ i, (f i).toNNReal",
" (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i => ↑(f i)) i).toNNReal",
" (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)"
] |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 77 | 80 | theorem IsMatching.toEdge_eq_toEdge_of_adj {M : Subgraph G} {v w : V} (h : M.IsMatching)
(hv : v ∈ M.verts) (hw : w ∈ M.verts) (ha : M.Adj v w) :
h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩ := by |
rw [h.toEdge_eq_of_adj hv ha, h.toEdge_eq_of_adj hw (M.symm ha), Subtype.mk_eq_mk, Sym2.eq_swap]
| [
" h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩",
" s(v, Exists.choose ⋯) = s(v, w)",
" Exists.choose ⋯ = w",
" Function.Surjective h.toEdge",
" ∃ a, h.toEdge a = ⟨e, he⟩",
" ∃ a, h.toEdge a = ⟨s(x, y), he⟩",
" h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩"
] | [
" h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩",
" s(v, Exists.choose ⋯) = s(v, w)",
" Exists.choose ⋯ = w",
" Function.Surjective h.toEdge",
" ∃ a, h.toEdge a = ⟨e, he⟩",
" ∃ a, h.toEdge a = ⟨s(x, y), he⟩"
] |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 108 | 109 | theorem CNF_of_lt {b o : Ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩] := by |
simp only [CNF_ne_zero ho, log_eq_zero hb, opow_zero, div_one, mod_one, CNF_zero]
| [
" C o",
" C 0",
" (invImage (fun x => x) wellFoundedRelation).1 (o % b ^ b.log o) o",
" b.CNFRec H0 H 0 = H0",
" ⋯.mpr H0 = H0",
" b.CNFRec H0 H o = H o ho (b.CNFRec H0 H (o % b ^ b.log o))",
" CNF 0 o = [(0, o)]",
" CNF 1 o = [(0, o)]",
" b.CNF o = [(0, o)]"
] | [
" C o",
" C 0",
" (invImage (fun x => x) wellFoundedRelation).1 (o % b ^ b.log o) o",
" b.CNFRec H0 H 0 = H0",
" ⋯.mpr H0 = H0",
" b.CNFRec H0 H o = H o ho (b.CNFRec H0 H (o % b ^ b.log o))",
" CNF 0 o = [(0, o)]",
" CNF 1 o = [(0, o)]",
" b.CNF o = [(0, o)]"
] |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 123 | 127 | theorem LDL.lower_conj_diag : LDL.lower hS * LDL.diag hS * (LDL.lower hS)ᴴ = S := by |
rw [LDL.lower, conjTranspose_nonsing_inv, Matrix.mul_assoc,
Matrix.inv_mul_eq_iff_eq_mul_of_invertible (LDL.lowerInv hS),
Matrix.mul_inv_eq_iff_eq_mul_of_invertible]
exact LDL.diag_eq_lowerInv_conj hS
| [
" lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ",
" lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j",
" gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j",
" Invertible (lowerInv hS)",
" Inv... | [
" lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ",
" lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j",
" gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j",
" Invertible (lowerInv hS)",
" Inv... |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 57 | 66 | theorem LDL.lowerInv_eq_gramSchmidtBasis :
LDL.lowerInv hS =
((Pi.basisFun 𝕜 n).toMatrix
(@gramSchmidtBasis 𝕜 (n → 𝕜) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _
(Pi.basisFun 𝕜 n)))ᵀ := by |
letI := NormedAddCommGroup.ofMatrix hS.transpose
letI := InnerProductSpace.ofMatrix hS.transpose
ext i j
rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]
rfl
| [
" lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ",
" lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j",
" gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j"
] | [] |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 48 | 55 | theorem circleTransformDeriv_periodic (f : ℂ → E) :
Periodic (circleTransformDeriv R z w f) (2 * π) := by |
have := periodic_circleMap
simp_rw [Periodic] at *
intro x
simp_rw [circleTransformDeriv, this]
congr 2
simp [this]
| [
" Periodic (circleTransformDeriv R z w f) (2 * π)",
" ∀ (x : ℝ), circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x",
" circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x",
" (2 * ↑π * I)⁻¹ • deriv (circleMap z R) (x + 2 * π) • ((circleMap z R x - w) ^ 2)⁻¹ • f (c... | [] |
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 | 112 | 113 | theorem rpow_self_rpow_inv {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ (1 / y)) ^ y = x := by |
field_simp [← rpow_mul]
| [
" x ^ y = 0 ↔ x = 0 ∧ y ≠ 0",
" ↑x ^ y = ↑0 ↔ ↑x = 0 ∧ y ≠ 0",
" x ^ w = x ^ y * x ^ z",
" y + z ≠ 0",
" x ^ (-1) = x⁻¹",
" (x ^ y) ^ (1 / y) = x",
" (x ^ (1 / y)) ^ y = x"
] | [
" x ^ y = 0 ↔ x = 0 ∧ y ≠ 0",
" ↑x ^ y = ↑0 ↔ ↑x = 0 ∧ y ≠ 0",
" x ^ w = x ^ y * x ^ z",
" y + z ≠ 0",
" x ^ (-1) = x⁻¹",
" (x ^ y) ^ (1 / y) = x"
] |
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k G : Type*} [Semiring k]
namespace AddMonoidAlgebra
section
variable [AddCancelCommMonoid G]
noncomputable def divOf (x : k[G]) (g... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 105 | 109 | theorem of'_mul_divOf (a : G) (x : k[G]) : of' k G a * x /ᵒᶠ a = x := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
rw [AddMonoidAlgebra.divOf_apply, of'_apply, single_mul_apply_aux, one_mul]
intro c
exact add_right_inj _
| [
" x /ᵒᶠ 0 = x",
" (x /ᵒᶠ 0) x✝ = x x✝",
" x /ᵒᶠ (a + b) = x /ᵒᶠ a /ᵒᶠ b",
" (x /ᵒᶠ (a + b)) x✝ = (x /ᵒᶠ a /ᵒᶠ b) x✝",
" of' k G a * x /ᵒᶠ a = x",
" (of' k G a * x /ᵒᶠ a) x✝ = x x✝",
" ∀ (a_1 : G), a + a_1 = a + x✝ ↔ a_1 = x✝",
" a + c = a + x✝ ↔ c = x✝"
] | [
" x /ᵒᶠ 0 = x",
" (x /ᵒᶠ 0) x✝ = x x✝",
" x /ᵒᶠ (a + b) = x /ᵒᶠ a /ᵒᶠ b",
" (x /ᵒᶠ (a + b)) x✝ = (x /ᵒᶠ a /ᵒᶠ b) x✝"
] |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open sco... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 46 | 48 | theorem discr_ne_zero : discr K ≠ 0 := by |
rw [← (Int.cast_injective (α := ℚ)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis ℚ (integralBasis K)
| [
" discr K ≠ 0",
" Algebra.discr ℚ ⇑(integralBasis K) ≠ ↑0"
] | [] |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 119 | 121 | theorem add (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f + g) ℱ μ := by |
refine ⟨hf.adapted.add hg.adapted, fun i j hij => ?_⟩
exact (condexp_add (hf.integrable j) (hg.integrable j)).trans ((hf.2 i j hij).add (hg.2 i j hij))
| [
" μ[(fun x_1 x_2 => x) j|↑ℱ i] =ᶠ[ae μ] (fun x_1 x_2 => x) i",
" Martingale (fun x => f) ℱ μ",
" μ[(fun x => f) j|↑ℱ i] =ᶠ[ae μ] (fun x => f) i",
" μ[0 j|↑ℱ i] =ᶠ[ae μ] 0 i",
" 0 =ᶠ[ae μ] 0 i",
" ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ",
" ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (x : Ω) in s, (μ[f j|↑ℱ... | [
" μ[(fun x_1 x_2 => x) j|↑ℱ i] =ᶠ[ae μ] (fun x_1 x_2 => x) i",
" Martingale (fun x => f) ℱ μ",
" μ[(fun x => f) j|↑ℱ i] =ᶠ[ae μ] (fun x => f) i",
" μ[0 j|↑ℱ i] =ᶠ[ae μ] 0 i",
" 0 =ᶠ[ae μ] 0 i",
" ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ",
" ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (x : Ω) in s, (μ[f j|↑ℱ... |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}... | Mathlib/RingTheory/JacobsonIdeal.lean | 134 | 143 | theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by |
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exac... | [
" r * y * x + r - 1 ∈ I",
" -p ∈ I",
" z * -y * x + z ∈ M",
" z * i ∈ M",
" ∃ s, s * r - 1 ∈ I",
" s * r - 1 ∈ I",
" I.jacobson = I ↔ ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M",
" (∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M) → I.jacobson = I",
" I.jacobson = I",
" x ∈ I",
" ∀ ⦃I : I... | [
" r * y * x + r - 1 ∈ I",
" -p ∈ I",
" z * -y * x + z ∈ M",
" z * i ∈ M",
" ∃ s, s * r - 1 ∈ I",
" s * r - 1 ∈ I"
] |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 122 | 133 | theorem sym2_eq_image : s.sym2 = (s ×ˢ s).image Sym2.mk := by |
ext z
refine z.ind fun x y ↦ ?_
rw [mk_mem_sym2_iff, mem_image]
constructor
· intro h
use (x, y)
simp only [mem_product, h, and_self, true_and]
· rintro ⟨⟨a, b⟩, h⟩
simp only [mem_product, Sym2.eq_iff] at h
obtain ⟨h, (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)⟩ := h
<;> simp [h]
| [
" s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s",
" m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s",
" (∀ y ∈ m, y ∈ s.val) ↔ ∀ a ∈ m, a ∈ s",
" x ∈ univ.sym2",
" ∀ a ∈ x, a ∈ univ",
" univ.sym2 = univ",
" a✝ ∈ univ.sym2 ↔ a✝ ∈ univ",
" s.sym2 ⊆ t.sym2",
" s.val.sym2 ≤ t.val.sym2",
" s.val ≤ t.val",
" Function.Injective Finset... | [
" s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s",
" m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s",
" (∀ y ∈ m, y ∈ s.val) ↔ ∀ a ∈ m, a ∈ s",
" x ∈ univ.sym2",
" ∀ a ∈ x, a ∈ univ",
" univ.sym2 = univ",
" a✝ ∈ univ.sym2 ↔ a✝ ∈ univ",
" s.sym2 ⊆ t.sym2",
" s.val.sym2 ≤ t.val.sym2",
" s.val ≤ t.val",
" Function.Injective Finset... |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 28 | 30 | theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
| [
" l.prod.Coprime k ↔ ∀ n ∈ l, n.Coprime k",
" [].prod.Coprime k ↔ ∀ n ∈ [], n.Coprime k",
" (head✝ :: tail✝).prod.Coprime k ↔ ∀ n ∈ head✝ :: tail✝, n.Coprime k",
" k.Coprime l.prod ↔ ∀ n ∈ l, k.Coprime n",
" m.prod.Coprime k ↔ ∀ n ∈ m, n.Coprime k",
" (Multiset.prod ⟦a✝⟧).Coprime k ↔ ∀ n ∈ ⟦a✝⟧, n.Coprime... | [
" l.prod.Coprime k ↔ ∀ n ∈ l, n.Coprime k",
" [].prod.Coprime k ↔ ∀ n ∈ [], n.Coprime k",
" (head✝ :: tail✝).prod.Coprime k ↔ ∀ n ∈ head✝ :: tail✝, n.Coprime k",
" k.Coprime l.prod ↔ ∀ n ∈ l, k.Coprime n"
] |
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 138 | 141 | theorem contractRight_mul_ι (a : M) (b : CliffordAlgebra Q) :
b * ι Q a⌊d = d a • b - b⌊d * ι Q a := by |
rw [contractRight_eq, reverse.map_mul, reverse_ι, contractLeft_ι_mul, map_sub, map_smul,
reverse_reverse, reverse.map_mul, reverse_ι, contractRight_eq]
| [
" ((contractLeftAux Q d) v) ((ι Q) v * x, ((contractLeftAux Q d) v) (x, fx)) = Q v • fx",
" d v • ((ι Q) v * x) - (ι Q) v * (d v • x - (ι Q) v * fx) = Q v • fx",
" ((fun d => foldr' Q (contractLeftAux Q d) ⋯ 0) (d₁ + d₂)) x =\n ((fun d => foldr' Q (contractLeftAux Q d) ⋯ 0) d₁ + (fun d => foldr' Q (contractL... | [
" ((contractLeftAux Q d) v) ((ι Q) v * x, ((contractLeftAux Q d) v) (x, fx)) = Q v • fx",
" d v • ((ι Q) v * x) - (ι Q) v * (d v • x - (ι Q) v * fx) = Q v • fx",
" ((fun d => foldr' Q (contractLeftAux Q d) ⋯ 0) (d₁ + d₂)) x =\n ((fun d => foldr' Q (contractLeftAux Q d) ⋯ 0) d₁ + (fun d => foldr' Q (contractL... |
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.quotient_nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
theorem Ideal.isRadical_iff_quotient_reduced {R : Type*} [CommRing R] (I : Ideal R) :
I.IsRad... | Mathlib/RingTheory/QuotientNilpotent.lean | 26 | 51 | theorem Ideal.IsNilpotent.induction_on (hI : IsNilpotent I)
{P : ∀ ⦃S : Type _⦄ [CommRing S], Ideal S → Prop}
(h₁ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I : Ideal S, I ^ 2 = ⊥ → P I)
(h₂ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I J : Ideal S, I ≤ J → P I →
P (J.map (Ideal.Quotient.mk I)) → P J) :
P I := by |
obtain ⟨n, hI : I ^ n = ⊥⟩ := hI
induction' n using Nat.strong_induction_on with n H generalizing S
by_cases hI' : I = ⊥
· subst hI'
apply h₁
rw [← Ideal.zero_eq_bot, zero_pow two_ne_zero]
cases' n with n
· rw [pow_zero, Ideal.one_eq_top] at hI
haveI := subsingleton_of_bot_eq_top hI.symm
ex... | [
" I.IsRadical ↔ IsReduced (R ⧸ I)",
"R : Type u_1 inst✝ : CommRing R I : Ideal R | I.IsRadical",
" (RingHom.ker (Quotient.mk I)).IsRadical ↔ IsReduced (R ⧸ I)",
" P I",
" P ⊥",
" ⊥ ^ 2 = ⊥",
" P (I ^ 2)",
" (I ^ 2) ^ n.succ = ⊥",
" I ^ (2 * (n + 1)) ≤ I ^ (n + 1 + 1)",
" n + 1 + 1 ≤ 2 * (n + 1)",
... | [
" I.IsRadical ↔ IsReduced (R ⧸ I)",
"R : Type u_1 inst✝ : CommRing R I : Ideal R | I.IsRadical",
" (RingHom.ker (Quotient.mk I)).IsRadical ↔ IsReduced (R ⧸ I)"
] |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by
rw [← a... | Mathlib/Data/Int/Order/Lemmas.lean | 45 | 50 | theorem dvd_div_of_mul_dvd {a b c : ℤ} (h : a * b ∣ c) : b ∣ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [Int.ediv_zero, Int.dvd_zero]
rcases h with ⟨d, rfl⟩
refine ⟨d, ?_⟩
rw [mul_assoc, Int.mul_ediv_cancel_left _ ha]
| [
" a.natAbs = b.natAbs ↔ a * a = b * b",
" a.natAbs = b.natAbs ↔ ↑a.natAbs = ↑b.natAbs",
" a.natAbs < b.natAbs ↔ a * a < b * b",
" a.natAbs < b.natAbs ↔ ↑a.natAbs < ↑b.natAbs",
" a.natAbs ≤ b.natAbs ↔ a * a ≤ b * b",
" a.natAbs ≤ b.natAbs ↔ ↑a.natAbs ≤ ↑b.natAbs",
" b ∣ c / a",
" b ∣ c / 0",
" b ∣ a ... | [
" a.natAbs = b.natAbs ↔ a * a = b * b",
" a.natAbs = b.natAbs ↔ ↑a.natAbs = ↑b.natAbs",
" a.natAbs < b.natAbs ↔ a * a < b * b",
" a.natAbs < b.natAbs ↔ ↑a.natAbs < ↑b.natAbs",
" a.natAbs ≤ b.natAbs ↔ a * a ≤ b * b",
" a.natAbs ≤ b.natAbs ↔ ↑a.natAbs ≤ ↑b.natAbs"
] |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theory.constructions.pi from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Function Set MeasureTheory... | Mathlib/MeasureTheory/Constructions/Pi.lean | 182 | 184 | theorem piPremeasure_pi_eval {s : Set (∀ i, α i)} :
piPremeasure m (pi univ fun i => eval i '' s) = piPremeasure m s := by |
simp only [eval, piPremeasure_pi']; rfl
| [
" IsPiSystem (univ.pi '' univ.pi C)",
" univ.pi s₁ ∩ univ.pi s₂ ∈ univ.pi '' univ.pi C",
" (univ.pi fun i => s₁ i ∩ s₂ i) ∈ univ.pi '' univ.pi C",
" piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)",
" (m i) (s i) = 0",
" piPremeasure m (univ.pi fun i => eval i '' s) = piPremeasure m s",
" ∏ i : ι, (m ... | [
" IsPiSystem (univ.pi '' univ.pi C)",
" univ.pi s₁ ∩ univ.pi s₂ ∈ univ.pi '' univ.pi C",
" (univ.pi fun i => s₁ i ∩ s₂ i) ∈ univ.pi '' univ.pi C",
" piPremeasure m (univ.pi s) = ∏ i : ι, (m i) (s i)",
" (m i) (s i) = 0"
] |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 61 | 71 | theorem list_reverse_induction (p : List α → Prop) (base : p [])
(ind : ∀ (l : List α) (e : α), p l → p (l ++ [e])) : (∀ (l : List α), p l) := by |
let q := fun l ↦ p (reverse l)
have pq : ∀ l, p (reverse l) → q l := by simp only [q, reverse_reverse]; intro; exact id
have qp : ∀ l, q (reverse l) → p l := by simp only [q, reverse_reverse]; intro; exact id
intro l
apply qp
generalize (reverse l) = l
induction' l with head tail ih
· apply pq; simp on... | [
" List.oldMapIdxCore f n l = List.oldMapIdx (fun i a => f (i + n) a) l",
" List.oldMapIdxCore f n [] = List.oldMapIdx (fun i a => f (i + n) a) []",
" List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdx (fun i a => f (i + n) a) (hd :: tl)",
" List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdxCore (fun i a => f ... | [
" List.oldMapIdxCore f n l = List.oldMapIdx (fun i a => f (i + n) a) l",
" List.oldMapIdxCore f n [] = List.oldMapIdx (fun i a => f (i + n) a) []",
" List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdx (fun i a => f (i + n) a) (hd :: tl)",
" List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdxCore (fun i a => f ... |
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 | 159 | 175 | theorem le_degrees_add {p q : MvPolynomial σ R} (h : p.degrees.Disjoint q.degrees) :
p.degrees ≤ (p + q).degrees := by |
classical
apply Finset.sup_le
intro d hd
rw [Multiset.disjoint_iff_ne] at h
obtain rfl | h0 := eq_or_ne d 0
· rw [toMultiset_zero]; apply Multiset.zero_le
· refine Finset.le_sup_of_le (b := d) ?_ le_rfl
rw [mem_support_iff, coeff_add]
suffices q.coeff d = 0 by rwa [this, add_zero, coeff, ← Finsup... | [
" p.degrees = p.support.sup fun s => toMultiset s",
" (p.support.sup fun s => toMultiset s) = p.support.sup fun s => toMultiset s",
" ((monomial s) a).degrees ≤ toMultiset s",
" (if a = 0 then ⊥ else toMultiset s) ≤ toMultiset s",
" toMultiset s ≤ toMultiset s",
" ((monomial s) a).degrees = toMultiset s",... | [
" p.degrees = p.support.sup fun s => toMultiset s",
" (p.support.sup fun s => toMultiset s) = p.support.sup fun s => toMultiset s",
" ((monomial s) a).degrees ≤ toMultiset s",
" (if a = 0 then ⊥ else toMultiset s) ≤ toMultiset s",
" toMultiset s ≤ toMultiset s",
" ((monomial s) a).degrees = toMultiset s",... |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 105 | 114 | theorem rel_bot_eq_right_group_rel (H : Subgroup G) :
(setoid ↑H ↑(⊥ : Subgroup G)).Rel = (QuotientGroup.rightRel H).Rel := by |
ext a b
rw [rel_iff, Setoid.Rel, QuotientGroup.rightRel_apply]
constructor
· rintro ⟨b, hb, a, rfl : a = 1, rfl⟩
change b * a * 1 * a⁻¹ ∈ H
rwa [mul_one, mul_inv_cancel_right]
· rintro (h : b * a⁻¹ ∈ H)
exact ⟨b * a⁻¹, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]⟩
| [
" doset a s t = Set.image2 (fun x x_1 => x * a * x_1) s t",
" b ∈ doset a s t ↔ ∃ x ∈ s, ∃ y ∈ t, b = x * a * y",
" doset b ↑H ↑K = doset a ↑H ↑K",
" doset (h * a * k) ↑H ↑K = doset a ↑H ↑K",
" b ∈ doset a ↑H ↑K",
" ∃ x ∈ ↑H, ∃ y ∈ ↑K, b = x * a * y",
" b = y⁻¹ * l * a * (r * r'⁻¹)",
" doset a ↑H ↑K =... | [
" doset a s t = Set.image2 (fun x x_1 => x * a * x_1) s t",
" b ∈ doset a s t ↔ ∃ x ∈ s, ∃ y ∈ t, b = x * a * y",
" doset b ↑H ↑K = doset a ↑H ↑K",
" doset (h * a * k) ↑H ↑K = doset a ↑H ↑K",
" b ∈ doset a ↑H ↑K",
" ∃ x ∈ ↑H, ∃ y ∈ ↑K, b = x * a * y",
" b = y⁻¹ * l * a * (r * r'⁻¹)",
" doset a ↑H ↑K =... |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 125 | 134 | theorem sdiff_compl_neighborFinset_inter_eq {v w : V} (h : G.Adj v w) :
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) =
(G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ := by |
ext
simp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset,
mem_inter, mem_singleton]
rintro hnv hnw (rfl | rfl)
· exact hnv h
· apply hnw
rwa [adj_comm]
| [
" (fun v w => ¬⊥.Adj v w → Fintype.card ↑(⊥.commonNeighbors v w) = 0) v w",
" filter (fun x => x ∈ ⊥.commonNeighbors v w) univ = ∅",
" a✝ ∈ filter (fun x => x ∈ ⊥.commonNeighbors v w) univ ↔ a✝ ∈ ∅",
" Fintype.card ↑(⊤.commonNeighbors v w) = Fintype.card V - 2",
" v ≠ w",
" (G.neighborFinset v ∪ G.neighbo... | [
" (fun v w => ¬⊥.Adj v w → Fintype.card ↑(⊥.commonNeighbors v w) = 0) v w",
" filter (fun x => x ∈ ⊥.commonNeighbors v w) univ = ∅",
" a✝ ∈ filter (fun x => x ∈ ⊥.commonNeighbors v w) univ ↔ a✝ ∈ ∅",
" Fintype.card ↑(⊤.commonNeighbors v w) = Fintype.card V - 2",
" v ≠ w",
" (G.neighborFinset v ∪ G.neighbo... |
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 219 | 223 | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by |
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
| [
" upperCrossingTime a b f N (n + 1) ω =\n hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω",
" upperCrossingTime a b f N (n + 1) ω = hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω",
" hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N... | [
" upperCrossingTime a b f N (n + 1) ω =\n hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω",
" upperCrossingTime a b f N (n + 1) ω = hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω",
" hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N... |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.discrete_valuation_ring.tfae from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variab... | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | 92 | 150 | theorem maximalIdeal_isPrincipal_of_isDedekindDomain [LocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal := by |
classical
by_cases ne_bot : maximalIdeal R = ⊥
· rw [ne_bot]; infer_instance
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ maximalIdeal R, a ≠ (0 : R) := by
by_contra! h'; apply ne_bot; rwa [eq_bot_iff]
have hle : Ideal.span {a} ≤ maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]
have : (Ideal.span {a}... | [
" ∃ n, I = maximalIdeal R ^ n",
" I = maximalIdeal R ^ 0",
" x ≠ 0",
" False",
" maximalIdeal R = ⊥",
" ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r",
" ∃ n, Associated (x ^ n) r",
" ∃ n, Associated (x ^ n) f.prod",
" ∀ b ∈ f, Associated x b",
" Associated x b",
" ∃ n, Associate... | [
" ∃ n, I = maximalIdeal R ^ n",
" I = maximalIdeal R ^ 0",
" x ≠ 0",
" False",
" maximalIdeal R = ⊥",
" ∀ (r : R), r ≠ 0 → r ∈ nonunits R → ∃ n, Associated (x ^ n) r",
" ∃ n, Associated (x ^ n) r",
" ∃ n, Associated (x ^ n) f.prod",
" ∀ b ∈ f, Associated x b",
" Associated x b",
" ∃ n, Associate... |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 102 | 102 | theorem snorm_exponent_top {f : α → F} : snorm f ∞ μ = snormEssSup f μ := by | simp [snorm]
| [
" snorm f p μ = snorm' f p.toReal μ",
" snorm f p μ = (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ (1 / p.toReal)",
" snorm f 1 μ = ∫⁻ (x : α), ↑‖f x‖₊ ∂μ",
" snorm f ⊤ μ = snormEssSup f μ"
] | [
" snorm f p μ = snorm' f p.toReal μ",
" snorm f p μ = (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ (1 / p.toReal)",
" snorm f 1 μ = ∫⁻ (x : α), ↑‖f x‖₊ ∂μ"
] |
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 64 | 74 | theorem condexp_eq_zero_or_one_of_condIndepSet_self
[StandardBorelSpace Ω] [Nonempty Ω]
(hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
(h_indep : CondIndepSet m hm t t μ) :
∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 := by |
have h := ae_of_ae_trim hm (kernel.measure_eq_zero_or_one_of_indepSet_self h_indep)
filter_upwards [condexpKernel_ae_eq_condexp hm ht, h] with ω hω_eq hω
rw [← hω_eq, ENNReal.toReal_eq_zero_iff, ENNReal.toReal_eq_one_iff]
cases hω with
| inl h => exact Or.inl (Or.inl h)
| inr h => exact Or.inr h
| [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤",
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1",
" (κ a) t = 0 ∨ (κ a) t = 1",
" μ t = 0 ∨ μ t = 1",
" ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator ... | [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤",
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1",
" (κ a) t = 0 ∨ (κ a) t = 1",
" μ t = 0 ∨ μ t = 1"
] |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 138 | 158 | theorem multiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1 := by |
have hp' := hp.prime
have h0 : 2 ≤ p := hp.two_le
have h1 : 1 ≤ p * n + 1 := Nat.le_add_left _ _
have h2 : p * n + 1 ≤ p * (n + 1) := by linarith
have h3 : p * n + 1 ≤ p * (n + 1) + 1 := by omega
have hm : multiplicity p (p * n)! ≠ ⊤ := by
rw [Ne, eq_top_iff_not_finite, Classical.not_not, finite_nat_if... | [
" multiplicity m n = ↑(Ico 1 ((multiplicity m n).get ⋯ + 1)).card",
" i ∈ Ico 1 ((multiplicity m n).get ⋯ + 1) ↔ i ∈ filter (fun i => m ^ i ∣ n) (Ico 1 b)",
" 1 ≤ i ∧ m ^ i ∣ n ↔ (1 ≤ i ∧ m ^ i ∣ n) ∧ i < b",
" i ≤ m.log n",
" i ≤ log 0 n",
" i ≠ 0",
" i ≤ (m + 1).log n",
" multiplicity p 0! = ↑(∑ i ∈... | [
" multiplicity m n = ↑(Ico 1 ((multiplicity m n).get ⋯ + 1)).card",
" i ∈ Ico 1 ((multiplicity m n).get ⋯ + 1) ↔ i ∈ filter (fun i => m ^ i ∣ n) (Ico 1 b)",
" 1 ≤ i ∧ m ^ i ∣ n ↔ (1 ≤ i ∧ m ^ i ∣ n) ∧ i < b",
" i ≤ m.log n",
" i ≤ log 0 n",
" i ≠ 0",
" i ≤ (m + 1).log n",
" multiplicity p 0! = ↑(∑ i ∈... |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 140 | 146 | theorem homothety_norm [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
(hf : ∀ x, ‖f x‖ = a * ‖x‖) : ‖f‖ = a := by |
obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
rw [← norm_pos_iff] at hx
have ha : 0 ≤ a := by simpa only [hf, hx, mul_nonneg_iff_of_pos_right] using norm_nonneg (f x)
apply le_antisymm (f.opNorm_le_bound ha fun y => le_of_eq (hf y))
simpa only [hf, hx, mul_le_mul_right] using f.le_opNorm x
| [
" ‖f‖ * ‖x‖ = 0",
" f = 0 → ‖f‖ = 0",
" ‖0‖ = 0",
" ‖id 𝕜 E‖ = 1",
" ∃ x, ‖x‖ ≠ 0",
" ‖f‖ = a",
" 0 ≤ a",
" a ≤ ‖f‖"
] | [
" ‖f‖ * ‖x‖ = 0",
" f = 0 → ‖f‖ = 0",
" ‖0‖ = 0",
" ‖id 𝕜 E‖ = 1",
" ∃ x, ‖x‖ ≠ 0"
] |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 310 | 313 | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableWithinAt_iff, differentiableWithinAt_pi]
rfl
| [
" DifferentiableWithinAt 𝕜 f t y ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y",
" (∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) t y) ↔\n ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => f x i) t y"
] | [] |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 66 | 67 | theorem coe_injective ⦃f g : α →ₛ β⦄ (H : (f : α → β) = g) : f = g := by |
cases f; cases g; congr
| [
" f = g",
" { toFun := toFun✝, measurableSet_fiber' := measurableSet_fiber'✝, finite_range' := finite_range'✝ } = g",
" { toFun := toFun✝¹, measurableSet_fiber' := measurableSet_fiber'✝¹, finite_range' := finite_range'✝¹ } =\n { toFun := toFun✝, measurableSet_fiber' := measurableSet_fiber'✝, finite_range' :=... | [] |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 175 | 194 | theorem eq_zero_of_polynomial_mem_map_range (I : Ideal R[X]) (x : ((Quotient.mk I).comp C).range)
(hx : C x ∈ I.map (Polynomial.mapRingHom ((Quotient.mk I).comp C).rangeRestrict)) : x = 0 := by |
let i := ((Quotient.mk I).comp C).rangeRestrict
have hi' : RingHom.ker (Polynomial.mapRingHom i) ≤ I := by
refine fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => ?_
rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← RingHom.comp_apply]
rw [RingHom.mem_ker, coe_mapRingHom] at hf
replace h... | [
" ∀ a ∈ I, ((Quotient.mk (map C I)).comp C) a = 0",
" ((Quotient.mk (map C I)).comp C) a = 0",
" C a ∈ map C I",
" ∀ f ∈ map C I, (eval₂RingHom (C.comp (Quotient.mk I)) X) f = 0",
" (eval₂RingHom (C.comp (Quotient.mk I)) X) a = 0",
" (eval₂RingHom (C.comp (Quotient.mk I)) X) (a.sum fun n a => (monomial n)... | [
" ∀ a ∈ I, ((Quotient.mk (map C I)).comp C) a = 0",
" ((Quotient.mk (map C I)).comp C) a = 0",
" C a ∈ map C I",
" ∀ f ∈ map C I, (eval₂RingHom (C.comp (Quotient.mk I)) X) f = 0",
" (eval₂RingHom (C.comp (Quotient.mk I)) X) a = 0",
" (eval₂RingHom (C.comp (Quotient.mk I)) X) (a.sum fun n a => (monomial n)... |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 56 | 70 | theorem exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem {r : S}
(r_non_zero_divisor : ∀ {x}, x * r = 0 → x = 0) (hr : r ∈ I) {p : R[X]} :
p ≠ 0 → p.eval₂ f r = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ I.comap f := by |
refine p.recOnHorner ?_ ?_ ?_
· intro h
contradiction
· intro p a coeff_eq_zero a_ne_zero _ _ hp
refine ⟨0, ?_, coeff_zero_mem_comap_of_root_mem hr hp⟩
simp [coeff_eq_zero, a_ne_zero]
· intro p p_nonzero ih _ hp
rw [eval₂_mul, eval₂_X] at hp
obtain ⟨i, hi, mem⟩ := ih p_nonzero (r_non_zero_d... | [
" p.coeff 0 ∈ comap f I",
" eval₂ f r p.divX * r ∈ I",
" p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I",
" 0 ≠ 0 → eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I",
" eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I",
" ∀ (p : R[X]) (a : R),\n p.coeff 0 =... | [
" p.coeff 0 ∈ comap f I",
" eval₂ f r p.divX * r ∈ I"
] |
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.extend_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Filter Set Metric Contin... | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 37 | 106 | theorem has_fderiv_at_boundary_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
(f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
(f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
(h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
HasFDerivWithinAt f... |
classical
-- one can assume without loss of generality that `x` belongs to the closure of `s`, as the
-- statement is empty otherwise
by_cases hx : x ∉ closure s
· rw [← closure_closure] at hx; exact hasFDerivWithinAt_of_nmem_closure hx
push_neg at hx
rw [HasFDerivWithinAt, hasFDerivAtFilter_... | [
" HasFDerivWithinAt f f' (closure s) x",
" ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[closure s] x, ‖f x_1 - f x - f' (x_1 - x)‖ ≤ c * ‖x_1 - x‖",
" ∀ᶠ (x_1 : E) in 𝓝[closure s] x, ‖f x_1 - f x - f' (x_1 - x)‖ ≤ ε * ‖x_1 - x‖",
" ∃ δ > 0, ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε",
" y ∈ {x_1 | (fun x_2 =... | [] |
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 | 124 | 126 | theorem vars_mul [DecidableEq σ] (φ ψ : MvPolynomial σ R) : (φ * ψ).vars ⊆ φ.vars ∪ ψ.vars := by |
simp_rw [vars_def, ← Multiset.toFinset_add, Multiset.toFinset_subset]
exact Multiset.subset_of_le (degrees_mul φ ψ)
| [
" p.vars = p.degrees.toFinset",
" p.degrees.toFinset = p.degrees.toFinset",
" vars 0 = ∅",
" ((monomial s) r).vars = s.support",
" (C r).vars = ∅",
" (X n).vars = {n}",
" i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support",
" x v = 0",
" v ∈ f.vars",
" (p + q).vars ⊆ p.vars ∪ q.vars",
" x ∈ p.vars ∪ q.... | [
" p.vars = p.degrees.toFinset",
" p.degrees.toFinset = p.degrees.toFinset",
" vars 0 = ∅",
" ((monomial s) r).vars = s.support",
" (C r).vars = ∅",
" (X n).vars = {n}",
" i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support",
" x v = 0",
" v ∈ f.vars",
" (p + q).vars ⊆ p.vars ∪ q.vars",
" x ∈ p.vars ∪ q.... |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 164 | 164 | theorem kernelSubobjectMap_id : kernelSubobjectMap (𝟙 (Arrow.mk f)) = 𝟙 _ := by | aesop_cat
| [
" (kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow",
" (kernelSubobjectIso f).inv ≫ (kernelSubobject f).arrow = kernel.ι f",
" (kernelSubobject f).arrow ≫ f = 0",
" ((kernelSubobjectIso f).hom ≫ kernel.ι f) ≫ f = 0",
" kernel.lift f h w ≫ (MonoOver.mk' (kernel.ι f)).arrow = h",
" h ≫ f ... | [
" (kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow",
" (kernelSubobjectIso f).inv ≫ (kernelSubobject f).arrow = kernel.ι f",
" (kernelSubobject f).arrow ≫ f = 0",
" ((kernelSubobjectIso f).hom ≫ kernel.ι f) ≫ f = 0",
" kernel.lift f h w ≫ (MonoOver.mk' (kernel.ι f)).arrow = h",
" h ≫ f ... |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
open LinearMap (BilinForm)
universe u v w
variable {R : Type*} {M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
variable {R₁ : Type*} {M₁ : Type*} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁]
va... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 100 | 105 | theorem isOrtho_smul_left {x y : M₄} {a : R₄} (ha : a ≠ 0) :
IsOrtho G (a • x) y ↔ IsOrtho G x y := by |
dsimp only [IsOrtho]
rw [map_smul]
simp only [LinearMap.smul_apply, smul_eq_mul, mul_eq_zero, or_iff_right_iff_imp]
exact fun a ↦ (ha a).elim
| [
" G.IsOrtho (a • x) y ↔ G.IsOrtho x y",
" (G (a • x)) y = 0 ↔ (G x) y = 0",
" (a • G x) y = 0 ↔ (G x) y = 0",
" a = 0 → (G x) y = 0"
] | [] |
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Fintype.Basic
#align_import data.fintype.pi from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*}
open Finset
namespace Fintype
variable [DecidableEq α] [Fintype α] {γ δ : α → Type*} {s : ∀ a, Finset (γ a)}
def pi... | Mathlib/Data/Fintype/Pi.lean | 34 | 42 | theorem mem_piFinset {t : ∀ a, Finset (δ a)} {f : ∀ a, δ a} : f ∈ piFinset t ↔ ∀ a, f a ∈ t a := by |
constructor
· simp only [piFinset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ, exists_imp,
mem_pi]
rintro g hg hgf a
rw [← hgf]
exact hg a
· simp only [piFinset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi]
exact fun hf => ⟨fun a _ => f a, hf, rfl⟩
| [
" (fun f a => f a ⋯) x✝¹ = (fun f a => f a ⋯) x✝ → x✝¹ = x✝",
" f ∈ piFinset t ↔ ∀ (a : α), f a ∈ t a",
" f ∈ piFinset t → ∀ (a : α), f a ∈ t a",
" ∀ (x : (a : α) → a ∈ univ → δ a),\n (∀ (a : α), x a ⋯ ∈ t a) → { toFun := fun f a => f a ⋯, inj' := ⋯ } x = f → ∀ (a : α), f a ∈ t a",
" f a ∈ t a",
" { to... | [
" (fun f a => f a ⋯) x✝¹ = (fun f a => f a ⋯) x✝ → x✝¹ = x✝"
] |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 96 | 97 | theorem degrees_neg (p : MvPolynomial σ R) : (-p).degrees = p.degrees := by |
rw [degrees, support_neg]; rfl
| [
" (-p).degrees = p.degrees",
" (p.support.sup fun s => toMultiset s) = p.degrees"
] | [] |
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 | 75 | 78 | theorem prod_X_sub_C_eq [CommRing S] {x y z : S} :
(X - C x) * (X - C y) * (X - C z) =
toPoly ⟨1, -(x + y + z), x * y + x * z + y * z, -(x * y * z)⟩ := by |
rw [← one_mul <| X - C x, ← C_1, C_mul_prod_X_sub_C_eq, one_mul, one_mul, one_mul]
| [
" C w * (X - C x) * (X - C y) * (X - C z) =\n { a := w, b := w * -(x + y + z), c := w * (x * y + x * z + y * z), d := w * -(x * y * z) }.toPoly",
" C w * (X - C x) * (X - C y) * (X - C z) =\n C w * X ^ 3 + C w * -(C x + C y + C z) * X ^ 2 + C w * (C x * C y + C x * C z + C y * C z) * X +\n C w * -(C x ... | [
" C w * (X - C x) * (X - C y) * (X - C z) =\n { a := w, b := w * -(x + y + z), c := w * (x * y + x * z + y * z), d := w * -(x * y * z) }.toPoly",
" C w * (X - C x) * (X - C y) * (X - C z) =\n C w * X ^ 3 + C w * -(C x + C y + C z) * X ^ 2 + C w * (C x * C y + C x * C z + C y * C z) * X +\n C w * -(C x ... |
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.Analysis.RCLike.Lemmas
#align_import analysis.normed_space.hahn_banach.extension from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
univers... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 44 | 59 | theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ := by |
rcases exists_extension_of_le_sublinear ⟨p, f⟩ (fun x => ‖f‖ * ‖x‖)
(fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])
(fun x y => by -- Porting note: placeholder filled here
rw [← left_distrib]
exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@... | [
" ∃ g, (∀ (x : ↥p), g ↑x = f x) ∧ ‖g‖ = ‖f‖",
" (fun x => ‖f‖ * ‖x‖) (c • x) = c * (fun x => ‖f‖ * ‖x‖) x",
" (fun x => ‖f‖ * ‖x‖) (x + y) ≤ (fun x => ‖f‖ * ‖x‖) x + (fun x => ‖f‖ * ‖x‖) y",
" (fun x => ‖f‖ * ‖x‖) (x + y) ≤ ‖f‖ * (‖x‖ + ‖y‖)",
" ‖g'‖ = ‖f‖",
" ‖f‖ ≤ ‖g.mkContinuous ‖f‖ ⋯‖",
" ‖f x‖ ≤ ‖g... | [] |
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
universe v₁ v₂ u₁ u₂ u
open CategoryTheory MonoidalCategor... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 73 | 74 | theorem counit_comul_hom {Z : C} (f : M.X ⟶ Z) : M.comul ≫ (M.counit ⊗ f) = f ≫ (λ_ Z).inv := by |
rw [leftUnitor_inv_naturality, tensorHom_def, counit_comul_assoc]
| [
" (λ_ (𝟙_ C)).inv ≫ 𝟙 (𝟙_ C) ▷ 𝟙_ C = (λ_ (𝟙_ C)).inv",
" (λ_ (𝟙_ C)).inv ≫ 𝟙_ C ◁ 𝟙 (𝟙_ C) = (ρ_ (𝟙_ C)).inv",
" (λ_ (𝟙_ C)).inv ≫ 𝟙_ C ◁ (λ_ (𝟙_ C)).inv ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).inv = (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).inv ▷ 𝟙_ C",
" M.comul ≫ (M.counit ⊗ f) = f ≫ (λ_ Z).inv"
] | [
" (λ_ (𝟙_ C)).inv ≫ 𝟙 (𝟙_ C) ▷ 𝟙_ C = (λ_ (𝟙_ C)).inv",
" (λ_ (𝟙_ C)).inv ≫ 𝟙_ C ◁ 𝟙 (𝟙_ C) = (ρ_ (𝟙_ C)).inv",
" (λ_ (𝟙_ C)).inv ≫ 𝟙_ C ◁ (λ_ (𝟙_ C)).inv ≫ (α_ (𝟙_ C) (𝟙_ C) (𝟙_ C)).inv = (λ_ (𝟙_ C)).inv ≫ (λ_ (𝟙_ C)).inv ▷ 𝟙_ C"
] |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 141 | 143 | theorem derivative_zero (n : ℕ) :
Polynomial.derivative (bernsteinPolynomial R n 0) = -n * bernsteinPolynomial R (n - 1) 0 := by |
simp [bernsteinPolynomial, Polynomial.derivative_pow]
| [
" bernsteinPolynomial ℤ 3 2 = 3 * X ^ 2 - 3 * X ^ 3",
" 3 * X ^ 2 * (1 - X) = 3 * X ^ 2 - 3 * X ^ 3",
" bernsteinPolynomial R n ν = 0",
" Polynomial.map f (bernsteinPolynomial R n ν) = bernsteinPolynomial S n ν",
" (bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν)",
" bernsteinPol... | [
" bernsteinPolynomial ℤ 3 2 = 3 * X ^ 2 - 3 * X ^ 3",
" 3 * X ^ 2 * (1 - X) = 3 * X ^ 2 - 3 * X ^ 3",
" bernsteinPolynomial R n ν = 0",
" Polynomial.map f (bernsteinPolynomial R n ν) = bernsteinPolynomial S n ν",
" (bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν)",
" bernsteinPol... |
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 | 65 | 65 | theorem eval₂_zero : (0 : R[X]).eval₂ f x = 0 := by | simp [eval₂_eq_sum]
| [
" eval₂ f x p = p.sum fun e a => f a * x ^ e",
" f = g → s = t → φ = ψ → eval₂ f s φ = eval₂ g t ψ",
" eval₂ f s φ = eval₂ f s φ",
" eval₂ f 0 p = f (p.coeff 0)",
" eval₂ f x 0 = 0"
] | [
" eval₂ f x p = p.sum fun e a => f a * x ^ e",
" f = g → s = t → φ = ψ → eval₂ f s φ = eval₂ g t ψ",
" eval₂ f s φ = eval₂ f s φ",
" eval₂ f 0 p = f (p.coeff 0)"
] |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 119 | 121 | theorem rightUnitor_inv_hom {X : Action V G} : Hom.hom (ρ_ X).inv = (ρ_ X.V).inv := by |
dsimp
simp
| [
" (α_ X Y Z).hom.hom = (α_ X.V Y.V Z.V).hom",
" (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) ≫ (α_ X.V Y.V Z.V).hom ≫ (𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) = (α_ X.V Y.V Z.V).hom",
" (α_ X Y Z).inv.hom = (α_ X.V Y.V Z.V).inv",
" ((𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) ≫ (α_ X.V Y.V Z.V).inv) ≫ (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) = (α_ X.V Y.V Z.V).inv",
" (λ_ ... | [
" (α_ X Y Z).hom.hom = (α_ X.V Y.V Z.V).hom",
" (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) ≫ (α_ X.V Y.V Z.V).hom ≫ (𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) = (α_ X.V Y.V Z.V).hom",
" (α_ X Y Z).inv.hom = (α_ X.V Y.V Z.V).inv",
" ((𝟙 X.V ⊗ 𝟙 (Y.V ⊗ Z.V)) ≫ (α_ X.V Y.V Z.V).inv) ≫ (𝟙 (X.V ⊗ Y.V) ⊗ 𝟙 Z.V) = (α_ X.V Y.V Z.V).inv",
" (λ_ ... |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 120 | 122 | theorem PreservesPullback.iso_hom_fst :
(PreservesPullback.iso G f g).hom ≫ pullback.fst = G.map pullback.fst := by |
simp [PreservesPullback.iso]
| [
" G.map h ≫ G.map f = G.map k ≫ G.map g",
" ∀ (j : WalkingCospan),\n ((Cones.postcompose (diagramIsoCospan (cospan f g ⋙ G)).hom).obj (G.mapCone (PullbackCone.mk h k comm))).π.app j =\n (Iso.refl\n ((Cones.postcompose (diagramIsoCospan (cospan f g ⋙ G)).hom).obj\n (G.mapCone (Pul... | [
" G.map h ≫ G.map f = G.map k ≫ G.map g",
" ∀ (j : WalkingCospan),\n ((Cones.postcompose (diagramIsoCospan (cospan f g ⋙ G)).hom).obj (G.mapCone (PullbackCone.mk h k comm))).π.app j =\n (Iso.refl\n ((Cones.postcompose (diagramIsoCospan (cospan f g ⋙ G)).hom).obj\n (G.mapCone (Pul... |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 66 | 70 | theorem localization_localization_map_units [IsLocalization N T]
(y : localizationLocalizationSubmodule M N) : IsUnit (algebraMap R T y) := by |
obtain ⟨y', z, eq⟩ := mem_localizationLocalizationSubmodule.mp y.prop
rw [IsScalarTower.algebraMap_apply R S T, eq, RingHom.map_mul, IsUnit.mul_iff]
exact ⟨IsLocalization.map_units T y', (IsLocalization.map_units _ z).map (algebraMap S T)⟩
| [
" x ∈ localizationLocalizationSubmodule M N ↔ ∃ y z, (algebraMap R S) x = ↑y * (algebraMap R S) ↑z",
" (∃ y ∈ N, ∃ z ∈ Submonoid.map (algebraMap R S) M, y * z = (algebraMap R S) x) ↔\n ∃ y z, (algebraMap R S) x = ↑y * (algebraMap R S) ↑z",
" (∃ y ∈ N, ∃ z ∈ Submonoid.map (algebraMap R S) M, y * z = (algebraM... | [
" x ∈ localizationLocalizationSubmodule M N ↔ ∃ y z, (algebraMap R S) x = ↑y * (algebraMap R S) ↑z",
" (∃ y ∈ N, ∃ z ∈ Submonoid.map (algebraMap R S) M, y * z = (algebraMap R S) x) ↔\n ∃ y z, (algebraMap R S) x = ↑y * (algebraMap R S) ↑z",
" (∃ y ∈ N, ∃ z ∈ Submonoid.map (algebraMap R S) M, y * z = (algebraM... |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 514 | 525 | theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n := by |
intro ε hε
obtain ⟨k, hk⟩ : ∃ k : ℕ, (p : ℚ) ^ (-((k : ℕ) : ℤ)) < ε := exists_pow_neg_lt_rat p hε
use k
intro j hj
refine lt_of_le_of_lt ?_ hk
-- Need to do beta reduction first, as `norm_cast` doesn't.
-- Added to adapt to leanprover/lean4#2734.
beta_reduce
norm_cast
rw [← padicNorm.dvd_iff_norm_l... | [
" nthHom f 0 = 0",
" (fun n => 0) = 0",
" ↑p ^ i ∣ nthHom f r j - nthHom f r i",
" ↑(nthHom f r j) - ↑(nthHom f r i) = 0",
" ↑↑((f j) r).val - ↑↑((f i) r).val = 0",
" ↑↑((f j) r).val - ↑↑((ZMod.castHom ⋯ (ZMod (p ^ i))) ((f j) r)).val = 0",
" IsCauSeq (padicNorm p) fun n => ↑(nthHom f r n)",
" ∃ i, ∀ ... | [
" nthHom f 0 = 0",
" (fun n => 0) = 0",
" ↑p ^ i ∣ nthHom f r j - nthHom f r i",
" ↑(nthHom f r j) - ↑(nthHom f r i) = 0",
" ↑↑((f j) r).val - ↑↑((f i) r).val = 0",
" ↑↑((f j) r).val - ↑↑((ZMod.castHom ⋯ (ZMod (p ^ i))) ((f j) r)).val = 0"
] |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
#align_import group_theory.perm.option from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open Equiv
@[simp]
theo... | Mathlib/GroupTheory/Perm/Option.lean | 27 | 34 | theorem Equiv.optionCongr_swap {α : Type*} [DecidableEq α] (x y : α) :
optionCongr (swap x y) = swap (some x) (some y) := by |
ext (_ | i)
· simp [swap_apply_of_ne_of_ne]
· by_cases hx : i = x
· simp only [hx, optionCongr_apply, Option.map_some', swap_apply_left, Option.mem_def,
Option.some.injEq]
by_cases hy : i = y <;> simp [hx, hy, swap_apply_of_ne_of_ne]
| [
" optionCongr (swap x y) = swap (some x) (some y)",
" a✝ ∈ (optionCongr (swap x y)) none ↔ a✝ ∈ (swap (some x) (some y)) none",
" a✝ ∈ (optionCongr (swap x y)) (some i) ↔ a✝ ∈ (swap (some x) (some y)) (some i)"
] | [] |
import Mathlib.Order.Hom.Basic
import Mathlib.Order.BoundedOrder
#align_import order.hom.bounded from "leanprover-community/mathlib"@"f1a2caaf51ef593799107fe9a8d5e411599f3996"
open Function OrderDual
variable {F α β γ δ : Type*}
structure TopHom (α β : Type*) [Top α] [Top β] where
toFun : α → β
map_t... | Mathlib/Order/Hom/Bounded.lean | 146 | 149 | theorem map_eq_top_iff [LE α] [OrderTop α] [PartialOrder β] [OrderTop β] [OrderIsoClass F α β]
(f : F) {a : α} : f a = ⊤ ↔ a = ⊤ := by |
letI : TopHomClass F α β := OrderIsoClass.toTopHomClass
rw [← map_top f, (EquivLike.injective f).eq_iff]
| [
" f a = ⊤ ↔ a = ⊤"
] | [] |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 77 | 89 | theorem injective_quotient_le_comap_map (P : Ideal R[X]) :
Function.Injective <|
Ideal.quotientMap
(Ideal.map (Polynomial.mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)
(Polynomial.mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
le_comap_map := by |
refine quotientMap_injective' (le_of_eq ?_)
rw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
(map_surjective (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))) Ideal.Quotient.mk_surjective)]
refine le_antisymm (sup_le le_rfl ?_) (le_sup_of_le_left le_rfl)
refine fun p h... | [
" p.coeff 0 ∈ comap f I",
" eval₂ f r p.divX * r ∈ I",
" p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I",
" 0 ≠ 0 → eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I",
" eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I",
" ∀ (p : R[X]) (a : R),\n p.coeff 0 =... | [
" p.coeff 0 ∈ comap f I",
" eval₂ f r p.divX * r ∈ I",
" p ≠ 0 → eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ comap f I",
" 0 ≠ 0 → eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I",
" eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I",
" ∀ (p : R[X]) (a : R),\n p.coeff 0 =... |
import Mathlib.Topology.MetricSpace.PiNat
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Gluing
import Mathlib.Topology.Sets.Opens
import Mathlib.Analysis.Normed.Field.Basic
#align_import topology.metric_space.polish from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78... | Mathlib/Topology/MetricSpace/Polish.lean | 91 | 94 | theorem complete_polishSpaceMetric (α : Type*) [ht : TopologicalSpace α] [h : PolishSpace α] :
@CompleteSpace α (polishSpaceMetric α).toUniformSpace := by |
convert h.complete.choose_spec.2
exact MetricSpace.replaceTopology_eq _ _
| [
" CompleteSpace α",
" polishSpaceMetric α = ⋯.choose"
] | [] |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 150 | 154 | theorem Path.reverse_comp [HasReverse V] {a b c : V} (p : Path a b) (q : Path b c) :
(p.comp q).reverse = q.reverse.comp p.reverse := by |
induction' q with _ _ _ _ h
· simp
· simp [h]
| [
" reverse (reverse f) = f",
" reverse f = reverse g ↔ f = g",
" reverse f = reverse g → f = g",
" f = g",
" f = g → reverse f = reverse g",
" reverse f = reverse g",
" f = reverse g ↔ reverse f = g",
" (p.comp q).reverse = q.reverse.comp p.reverse",
" (p.comp nil).reverse = nil.reverse.comp p.revers... | [
" reverse (reverse f) = f",
" reverse f = reverse g ↔ f = g",
" reverse f = reverse g → f = g",
" f = g",
" f = g → reverse f = reverse g",
" reverse f = reverse g",
" f = reverse g ↔ reverse f = g"
] |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 447 | 451 | theorem HasStrictDerivAt.clm_comp (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x := by |
have := (hc.hasStrictFDerivAt.clm_comp hd.hasStrictFDerivAt).hasStrictDerivAt
rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| [
" HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x"
] | [] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 39 | 41 | theorem algebraMap_eval_T (x : R) (n : ℤ) :
algebraMap R A ((T R n).eval x) = (T A n).eval (algebraMap R A x) := by |
rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_T]
| [
" (aeval x) (T R n) = eval x (T A n)",
" (aeval x) (U R n) = eval x (U A n)",
" (algebraMap R A) (eval x (T R n)) = eval ((algebraMap R A) x) (T A n)"
] | [
" (aeval x) (T R n) = eval x (T A n)",
" (aeval x) (U R n) = eval x (U A n)"
] |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Localization.AsSubring
#align_import algebraic_geometry.prime_spectrum.maximal from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
noncomputable section
open scoped Classical
universe u v
variable (R : Typ... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean | 65 | 69 | theorem toPrimeSpectrum_range :
Set.range (@toPrimeSpectrum R _) = { x | IsClosed ({x} : Set <| PrimeSpectrum R) } := by |
simp only [isClosed_singleton_iff_isMaximal]
ext ⟨x, _⟩
exact ⟨fun ⟨y, hy⟩ => hy ▸ y.IsMaximal, fun hx => ⟨⟨x, hx⟩, rfl⟩⟩
| [
" { asIdeal := asIdeal✝¹, IsMaximal := IsMaximal✝¹ } = { asIdeal := asIdeal✝, IsMaximal := IsMaximal✝ }",
" range toPrimeSpectrum = {x | IsClosed {x}}",
" range toPrimeSpectrum = {x | x.asIdeal.IsMaximal}",
" { asIdeal := x, IsPrime := IsPrime✝ } ∈ range toPrimeSpectrum ↔\n { asIdeal := x, IsPrime := IsPri... | [
" { asIdeal := asIdeal✝¹, IsMaximal := IsMaximal✝¹ } = { asIdeal := asIdeal✝, IsMaximal := IsMaximal✝ }"
] |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 232 | 232 | theorem inner_im_symm (x y : F) : im ⟪x, y⟫ = -im ⟪y, x⟫ := by | rw [← inner_conj_symm, conj_im]
| [
" 0 ≤ re ⟪x, x⟫_𝕜",
" 0 ≤ ‖x‖ ^ 2",
" ‖x‖ ^ 2 = 0",
" im ⟪x, x⟫_𝕜 = 0",
" I * ((starRingEnd 𝕜) ⟪x, x⟫_𝕜 - ⟪x, x⟫_𝕜) / 2 = ↑0",
" ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜",
" (starRingEnd 𝕜) ⟪y, x⟫_𝕜 + (starRingEnd 𝕜) ⟪z, x⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜",
" ↑(normSq x) = ⟪x, x⟫_𝕜",
" re ↑(normSq ... | [
" 0 ≤ re ⟪x, x⟫_𝕜",
" 0 ≤ ‖x‖ ^ 2",
" ‖x‖ ^ 2 = 0",
" im ⟪x, x⟫_𝕜 = 0",
" I * ((starRingEnd 𝕜) ⟪x, x⟫_𝕜 - ⟪x, x⟫_𝕜) / 2 = ↑0",
" ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜",
" (starRingEnd 𝕜) ⟪y, x⟫_𝕜 + (starRingEnd 𝕜) ⟪z, x⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜",
" ↑(normSq x) = ⟪x, x⟫_𝕜",
" re ↑(normSq ... |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
noncomputable section
namespace Finsupp
variable {n : ℕ} (i : Fin n) {M : Type*} [Zero M] (y : M) (t : Fin (n + 1) →₀ M) (s : Fin n →₀ M)
def tail (s : Fin (n + 1) →₀ ... | Mathlib/Data/Finsupp/Fin.lean | 60 | 64 | theorem cons_tail : cons (t 0) (tail t) = t := by |
ext a
by_cases c_a : a = 0
· rw [c_a, cons_zero]
· rw [← Fin.succ_pred a c_a, cons_succ, ← tail_apply]
| [
" (cons y s).tail k = s k",
" cons (t 0) t.tail = t",
" (cons (t 0) t.tail) a = t a"
] | [
" (cons y s).tail k = s k"
] |
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Norm
import Mathlib.Topology.Instances.Complex
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.embeddings from "leanprov... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 73 | 77 | theorem range_eval_eq_rootSet_minpoly :
(range fun φ : K →+* A => φ x) = (minpoly ℚ x).rootSet A := by |
convert (NumberField.isAlgebraic K).range_eval_eq_rootSet_minpoly A x using 1
ext a
exact ⟨fun ⟨φ, hφ⟩ => ⟨φ.toRatAlgHom, hφ⟩, fun ⟨φ, hφ⟩ => ⟨φ.toRingHom, hφ⟩⟩
| [
" (range fun φ => φ x) = (minpoly ℚ x).rootSet A",
" (range fun φ => φ x) = range fun ψ => ψ x",
" (a ∈ range fun φ => φ x) ↔ a ∈ range fun ψ => ψ x"
] | [] |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 171 | 176 | theorem finprod_eq_prod_plift_of_mulSupport_toFinset_subset {f : α → M}
(hf : (mulSupport (f ∘ PLift.down)).Finite) {s : Finset (PLift α)} (hs : hf.toFinset ⊆ s) :
∏ᶠ i, f i = ∏ i ∈ s, f i.down := by |
rw [finprod, dif_pos]
refine Finset.prod_subset hs fun x _ hxf => ?_
rwa [hf.mem_toFinset, nmem_mulSupport] at hxf
| [
" ∏ᶠ (i : α), f i = ∏ i ∈ s, f i.down",
" (mulSupport ((fun i => f i) ∘ PLift.down)).Finite",
" f x.down = 1"
] | [] |
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Tactic.Monotonicity
#align_import algebra.continued_fractions.computation.approximations from "leanprover-commu... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 115 | 127 | theorem succ_nth_stream_b_le_nth_stream_fr_inv {ifp_n ifp_succ_n : IntFractPair K}
(nth_stream_eq : IntFractPair.stream v n = some ifp_n)
(succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n) :
(ifp_succ_n.b : K) ≤ ifp_n.fr⁻¹ := by |
suffices (⌊ifp_n.fr⁻¹⌋ : K) ≤ ifp_n.fr⁻¹ by
cases' ifp_n with _ ifp_n_fr
have : ifp_n_fr ≠ 0 := by
intro h
simp [h, IntFractPair.stream, nth_stream_eq] at succ_nth_stream_eq
have : IntFractPair.of ifp_n_fr⁻¹ = ifp_succ_n := by
simpa [this, IntFractPair.stream, nth_stream_eq, Option.coe_... | [
" 0 ≤ ifp_n.fr ∧ ifp_n.fr < 1",
" IntFractPair.of v = ifp_n",
" 0 ≤ { b := ⌊v⌋, fr := fract v }.fr ∧ { b := ⌊v⌋, fr := fract v }.fr < 1",
" 0 ≤ { b := ⌊w✝.fr⁻¹⌋, fr := fract w✝.fr⁻¹ }.fr ∧ { b := ⌊w✝.fr⁻¹⌋, fr := fract w✝.fr⁻¹ }.fr < 1",
" 1 ≤ ifp_succ_n.b",
" 1 ≤ (IntFractPair.of ifp_n.fr⁻¹).b",
" 1 ≤ ... | [
" 0 ≤ ifp_n.fr ∧ ifp_n.fr < 1",
" IntFractPair.of v = ifp_n",
" 0 ≤ { b := ⌊v⌋, fr := fract v }.fr ∧ { b := ⌊v⌋, fr := fract v }.fr < 1",
" 0 ≤ { b := ⌊w✝.fr⁻¹⌋, fr := fract w✝.fr⁻¹ }.fr ∧ { b := ⌊w✝.fr⁻¹⌋, fr := fract w✝.fr⁻¹ }.fr < 1",
" 1 ≤ ifp_succ_n.b",
" 1 ≤ (IntFractPair.of ifp_n.fr⁻¹).b",
" 1 ≤ ... |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.asymptotics.superpolynomial_decay from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 116 | 120 | theorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :
SuperpolynomialDecay l k (k ^ n * f) := by |
induction' n with n hn
· simpa only [Nat.zero_eq, one_mul, pow_zero] using hf
· simpa only [pow_succ', mul_assoc] using hn.param_mul
| [
" Tendsto (fun a => k a ^ z * 0 a) l (𝓝 0)",
" Tendsto (fun a => k a ^ z * (f + g) a) l (𝓝 0)",
" Tendsto (fun a => k a ^ z * (f * g) a) l (𝓝 0)",
" Tendsto (fun a => k a ^ z * (fun n => f n * c) a) l (𝓝 0)",
" x ∈ (fun a => k a ^ z * (k * f) a) ⁻¹' s",
" SuperpolynomialDecay l k (k ^ n * f)",
" Sup... | [
" Tendsto (fun a => k a ^ z * 0 a) l (𝓝 0)",
" Tendsto (fun a => k a ^ z * (f + g) a) l (𝓝 0)",
" Tendsto (fun a => k a ^ z * (f * g) a) l (𝓝 0)",
" Tendsto (fun a => k a ^ z * (fun n => f n * c) a) l (𝓝 0)",
" x ∈ (fun a => k a ^ z * (k * f) a) ⁻¹' s"
] |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
#align_import analysis.special_functions.sqrt from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set
open scoped Topology
namespace Real
noncomputable def sqPartialHomeomorph : PartialHo... | Mathlib/Analysis/SpecialFunctions/Sqrt.lean | 46 | 58 | theorem deriv_sqrt_aux {x : ℝ} (hx : x ≠ 0) :
HasStrictDerivAt (√·) (1 / (2 * √x)) x ∧ ∀ n, ContDiffAt ℝ n (√·) x := by |
cases' hx.lt_or_lt with hx hx
· rw [sqrt_eq_zero_of_nonpos hx.le, mul_zero, div_zero]
have : (√·) =ᶠ[𝓝 x] fun _ => 0 := (gt_mem_nhds hx).mono fun x hx => sqrt_eq_zero_of_nonpos hx.le
exact
⟨(hasStrictDerivAt_const x (0 : ℝ)).congr_of_eventuallyEq this.symm, fun n =>
contDiffAt_const.congr_of... | [
" HasStrictDerivAt (fun x => √x) (1 / (2 * √x)) x ∧ ∀ (n : ℕ∞), ContDiffAt ℝ n (fun x => √x) x",
" HasStrictDerivAt (fun x => √x) 0 x ∧ ∀ (n : ℕ∞), ContDiffAt ℝ n (fun x => √x) x",
" 2 * √x ^ (2 - 1) ≠ 0",
" HasStrictDerivAt (fun x => √x) (1 / (2 * √x)) x",
" ∀ (n : ℕ∞), ContDiffAt ℝ n (fun x => √x) x"
] | [] |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 353 | 356 | theorem cons_vecMulVec (x : α) (v : Fin m → α) (w : n' → α) :
vecMulVec (vecCons x v) w = vecCons (x • w) (vecMulVec v w) := by |
ext i
refine Fin.cases ?_ ?_ i <;> simp [vecMulVec]
| [
" vecCons v B i j = vecCons (v j) (fun i => B i j) i",
" vecCons v B 0 j = vecCons (v j) (fun i => B i j) 0",
" ∀ (i : Fin m), vecCons v B i.succ j = vecCons (v j) (fun i => B i j) i.succ",
" vecMulVec (vecCons x v) w = vecCons (x • w) (vecMulVec v w)",
" vecMulVec (vecCons x v) w i j✝ = vecCons (x • w) (ve... | [
" vecCons v B i j = vecCons (v j) (fun i => B i j) i",
" vecCons v B 0 j = vecCons (v j) (fun i => B i j) 0",
" ∀ (i : Fin m), vecCons v B i.succ j = vecCons (v j) (fun i => B i j) i.succ"
] |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 88 | 90 | theorem reduceOption_concat_of_some (l : List (Option α)) (x : α) :
(l.concat (some x)).reduceOption = l.reduceOption.concat x := by |
simp only [reduceOption_nil, concat_eq_append, reduceOption_append, reduceOption_cons_of_some]
| [
" (some x :: l).reduceOption = x :: l.reduceOption",
" (none :: l).reduceOption = l.reduceOption",
" (map (Option.map f) l).reduceOption = map f l.reduceOption",
" (map (Option.map f) []).reduceOption = map f [].reduceOption",
" (map (Option.map f) (hd :: tl)).reduceOption = map f (hd :: tl).reduceOption",
... | [
" (some x :: l).reduceOption = x :: l.reduceOption",
" (none :: l).reduceOption = l.reduceOption",
" (map (Option.map f) l).reduceOption = map f l.reduceOption",
" (map (Option.map f) []).reduceOption = map f [].reduceOption",
" (map (Option.map f) (hd :: tl)).reduceOption = map f (hd :: tl).reduceOption",
... |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 90 | 99 | theorem IsComplement'.symm (h : IsComplement' H K) : IsComplement' K H := by |
let ϕ : H × K ≃ K × H :=
Equiv.mk (fun x => ⟨x.2⁻¹, x.1⁻¹⟩) (fun x => ⟨x.2⁻¹, x.1⁻¹⟩)
(fun x => Prod.ext (inv_inv _) (inv_inv _)) fun x => Prod.ext (inv_inv _) (inv_inv _)
let ψ : G ≃ G := Equiv.mk (fun g : G => g⁻¹) (fun g : G => g⁻¹) inv_inv inv_inv
suffices hf : (ψ ∘ fun x : H × K => x.1.1 * x.2.1) ... | [
" K.IsComplement' H",
" Function.Bijective ((fun x => ↑x.1 * ↑x.2) ∘ ⇑ϕ)",
" Function.Bijective (⇑ψ ∘ fun x => ↑x.1 * ↑x.2)",
" (⇑ψ ∘ fun x => ↑x.1 * ↑x.2) = (fun x => ↑x.1 * ↑x.2) ∘ ⇑ϕ"
] | [] |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q +... | Mathlib/MeasureTheory/Integral/Gamma.lean | 65 | 69 | theorem integral_exp_neg_mul_rpow {p b : ℝ} (hp : 0 < p) (hb : 0 < b) :
∫ x in Ioi (0:ℝ), exp (- b * x ^ p) = b ^ (- 1 / p) * Gamma (1 / p + 1) := by |
convert (integral_rpow_mul_exp_neg_mul_rpow hp neg_one_lt_zero hb) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp)), mul_assoc]
| [
" ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * ((q + 1) / p).Gamma",
" ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) =\n ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x))",
" ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-(x ^ (1 / p)) ^ p)) =\n ... | [
" ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * ((q + 1) / p).Gamma",
" ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) =\n ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x))",
" ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-(x ^ (1 / p)) ^ p)) =\n ... |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputab... | Mathlib/Probability/Kernel/Basic.lean | 113 | 114 | theorem finset_sum_apply (I : Finset ι) (κ : ι → kernel α β) (a : α) :
(∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a := by | rw [coe_finset_sum, Finset.sum_apply]
| [
" ↑⊥ a ≤ ↑κ a",
" (∑ i ∈ I, κ i) a = ∑ i ∈ I, (κ i) a"
] | [
" ↑⊥ a ≤ ↑κ a"
] |
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
set_option autoImplicit true
variable {C : Levenshtein.Cost α β δ} [CanonicallyLinearOrderedAddCommMonoid δ]
theorem suffixLevenshtein_minimum_le_levenshtein... | Mathlib/Data/List/EditDistance/Bounds.lean | 89 | 92 | theorem le_levenshtein_cons (xs : List α) (y ys) :
∃ xs', xs' <:+ xs ∧ levenshtein C xs' ys ≤ levenshtein C xs (y :: ys) := by |
simpa [suffixLevenshtein_eq_tails_map, List.minimum_le_coe_iff] using
suffixLevenshtein_minimum_le_levenshtein_cons (δ := δ) xs y ys
| [
" (↑(suffixLevenshtein C xs ys)).minimum ≤ ↑(levenshtein C xs (y :: ys))",
" (↑(suffixLevenshtein C [] ys)).minimum ≤ ↑(levenshtein C [] (y :: ys))",
" levenshtein C [] ys ≤ C.insert y + levenshtein C [] ys",
" 0 ≤ C.insert y",
" (↑(suffixLevenshtein C (x :: xs) ys)).minimum ≤ ↑(levenshtein C (x :: xs) (y :... | [
" (↑(suffixLevenshtein C xs ys)).minimum ≤ ↑(levenshtein C xs (y :: ys))",
" (↑(suffixLevenshtein C [] ys)).minimum ≤ ↑(levenshtein C [] (y :: ys))",
" levenshtein C [] ys ≤ C.insert y + levenshtein C [] ys",
" 0 ≤ C.insert y",
" (↑(suffixLevenshtein C (x :: xs) ys)).minimum ≤ ↑(levenshtein C (x :: xs) (y :... |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 65 | 68 | theorem quadraticChar_neg_two [DecidableEq F] (hF : ringChar F ≠ 2) :
quadraticChar F (-2) = χ₈' (Fintype.card F) := by |
rw [(by norm_num : (-2 : F) = -1 * 2), map_mul, χ₈'_eq_χ₄_mul_χ₈, quadraticChar_neg_one hF,
quadraticChar_two hF, @cast_natCast _ (ZMod 4) _ _ _ (by decide : 4 ∣ 8)]
| [
" IsSquare 2 ↔ Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5",
" Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5",
" (if Fintype.card F % 2 = 0 then 0 else if Fintype.card F % 8 = 1 ∨ Fintype.card F % 8 = 7 then 1 else -1) = 1 ↔\n Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5",
" -1 ≠ 1",
" Fintype.c... | [
" IsSquare 2 ↔ Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5",
" Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5",
" (if Fintype.card F % 2 = 0 then 0 else if Fintype.card F % 8 = 1 ∨ Fintype.card F % 8 = 7 then 1 else -1) = 1 ↔\n Fintype.card F % 8 ≠ 3 ∧ Fintype.card F % 8 ≠ 5",
" -1 ≠ 1",
" Fintype.c... |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 102 | 103 | theorem iUnion_Ico_left (b : α) : ⋃ a, Ico a b = Iio b := by |
simp only [← Ici_inter_Iio, ← iUnion_inter, iUnion_Ici, univ_inter]
| [
" Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b",
" ⋃ b, Icc a b = Ici a",
" ⋃ b, Ioc a b = Ioi a",
" ⋃ a, Icc a b = Iic b",
" ⋃ a, Ico a b = Iio b"
] | [
" Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b",
" ⋃ b, Icc a b = Ici a",
" ⋃ b, Ioc a b = Ioi a",
" ⋃ a, Icc a b = Iic b"
] |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 90 | 93 | theorem existsUnique_sub_zsmul_mem_Ico {a : α} (ha : 0 < a) (b c : α) :
∃! m : ℤ, b - m • a ∈ Set.Ico c (c + a) := by |
simpa only [mem_Ico, le_sub_iff_add_le, zero_add, add_comm c, sub_lt_iff_lt_add', add_assoc] using
existsUnique_zsmul_near_of_pos' ha (b - c)
| [
" x ≤ n • y",
" ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
" -↑k ∈ s",
" ∀ n ∈ s, n ≤ ↑k",
" n ≤ ↑k",
" n • a ≤ ↑k • a",
" g < (m + 1) • a",
" ∃ z ∈ s, m < z",
" m < n + 1",
" m • a < (n + 1) • a",
" ∃! k, 0 ≤ g - k • a ∧ g - k • a < a",
" ∃! m, b - m • a ∈ Ico c (c + a)"
] | [
" x ≤ n • y",
" ∃! k, k • a ≤ g ∧ g < (k + 1) • a",
" -↑k ∈ s",
" ∀ n ∈ s, n ≤ ↑k",
" n ≤ ↑k",
" n • a ≤ ↑k • a",
" g < (m + 1) • a",
" ∃ z ∈ s, m < z",
" m < n + 1",
" m • a < (n + 1) • a",
" ∃! k, 0 ≤ g - k • a ∧ g - k • a < a"
] |
import Mathlib.Order.Filter.Ultrafilter
import Mathlib.Order.Filter.Germ
#align_import order.filter.filter_product from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {α : Type u} {β : Type v} {φ : Ultrafilter α}
open scoped Classical
namespace Filter
local not... | Mathlib/Order/Filter/FilterProduct.lean | 65 | 66 | theorem coe_lt [Preorder β] {f g : α → β} : (f : β*) < g ↔ ∀* x, f x < g x := by |
simp only [lt_iff_le_not_le, eventually_and, coe_le, eventually_not, EventuallyLE]
| [
" ((fun x => Inv.inv ∘ x) fun x => 0) =ᶠ[↑φ] fun x => 0",
" ↑f < ↑g ↔ ∀* (x : α), f x < g x"
] | [
" ((fun x => Inv.inv ∘ x) fun x => 0) =ᶠ[↑φ] fun x => 0"
] |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 106 | 110 | theorem strongMono_comp [StrongMono f] [StrongMono g] : StrongMono (f ≫ g) :=
{ mono := mono_comp _ _
rlp := by |
intros
infer_instance }
| [
" ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [inst : Mono z], HasLiftingProperty (f ≫ g) z",
" HasLiftingProperty (f ≫ g) z✝",
" ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [inst : Epi z], HasLiftingProperty z (f ≫ g)",
" HasLiftingProperty z✝ (f ≫ g)"
] | [
" ∀ ⦃X Y : C⦄ (z : X ⟶ Y) [inst : Mono z], HasLiftingProperty (f ≫ g) z",
" HasLiftingProperty (f ≫ g) z✝"
] |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 | theorem mem_space_iff : x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ convexHull 𝕜 (s : Set E) := by |
simp [space]
| [
" x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ (convexHull 𝕜) ↑s"
] | [] |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 152 | 154 | theorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
(h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ zpowers g := by |
simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]
| [
" x ∈ zpowers 1",
" 1 ∈ zpowers 1",
" Nontrivial α",
" IsCyclic α",
" ∃ m, ∀ (g : G), σ g = g ^ m",
" σ g = g ^ m",
" σ ((fun x => h ^ x) n) = (fun x => h ^ x) n ^ m",
" ∀ (x_1 : α), x_1 ∈ zpowers x",
" ↑(zpowers x) = Set.univ",
" H = ⊥ ∨ H = ⊤",
" zpowers g = ⊤",
" g' ∈ zpowers g"
] | [
" x ∈ zpowers 1",
" 1 ∈ zpowers 1",
" Nontrivial α",
" IsCyclic α",
" ∃ m, ∀ (g : G), σ g = g ^ m",
" σ g = g ^ m",
" σ ((fun x => h ^ x) n) = (fun x => h ^ x) n ^ m",
" ∀ (x_1 : α), x_1 ∈ zpowers x",
" ↑(zpowers x) = Set.univ",
" H = ⊥ ∨ H = ⊤",
" zpowers g = ⊤"
] |
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