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.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int
#align_import data.int.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Int
open Nat
variable {α : Type*}
@[norm_cast]
theorem cast_neg_natCast {R} [DivisionRing R] (n : ℕ) : ((-n : ℤ) : R) = -... | Mathlib/Data/Int/Cast/Field.lean | 38 | 42 | theorem cast_div [DivisionRing α] {m n : ℤ} (n_dvd : n ∣ m) (hn : (n : α) ≠ 0) :
((m / n : ℤ) : α) = m / n := by |
rcases n_dvd with ⟨k, rfl⟩
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Int.mul_ediv_cancel_left _ this, mul_comm n, Int.cast_mul, mul_div_cancel_right₀ _ hn]
| [
" ↑(-↑n) = -↑n",
" ↑(m / n) = ↑m / ↑n",
" ↑(n * k / n) = ↑(n * k) / ↑n",
" n ≠ 0",
" False"
] | [
" ↑(-↑n) = -↑n"
] |
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Topology
namespa... | Mathlib/Probability/Process/Adapted.lean | 99 | 105 | theorem Filtration.adapted_natural [MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β]
{u : ι → Ω → β} (hum : ∀ i, StronglyMeasurable[m] (u i)) :
Adapted (Filtration.natural u hum) u := by |
intro i
refine StronglyMeasurable.mono ?_ (le_iSup₂_of_le i (le_refl i) le_rfl)
rw [stronglyMeasurable_iff_measurable_separable]
exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).isSeparable_range⟩
| [
" Adapted (natural u hum) u",
" StronglyMeasurable (u i)",
" Measurable (u i) ∧ IsSeparable (Set.range (u i))"
] | [] |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
#align_import group_theory.subgroup.saturated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
namespace Subgroup
variable {G : Type*} [Group G]
@[to_additive
"An additive subgroup `H` of `G` is *... | Mathlib/GroupTheory/Subgroup/Saturated.lean | 42 | 56 | theorem saturated_iff_zpow {H : Subgroup G} :
Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := by |
constructor
· intros hH n g hgn
induction' n with n n
· simp only [Int.natCast_eq_zero, Int.ofNat_eq_coe, zpow_natCast] at hgn ⊢
exact hH hgn
· suffices g ^ (n + 1) ∈ H by
refine (hH this).imp ?_ id
simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]
simpa only [inv_mem_iff, zp... | [
" H.Saturated ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H",
" H.Saturated → ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H",
" n = 0 ∨ g ∈ H",
" Int.ofNat n = 0 ∨ g ∈ H",
" Int.negSucc n = 0 ∨ g ∈ H",
" n + 1 = 0 → Int.negSucc n = 0",
" g ^ (n + 1) ∈ H",
" (∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ ... | [] |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Order Set
variable {α β γ ι ι' : Type*} {r p q : α → α → Prop}
section Pairwise
variabl... | Mathlib/Data/Set/Pairwise/Basic.lean | 41 | 42 | theorem pairwise_on_bool (hr : Symmetric r) {a b : α} :
Pairwise (r on fun c => cond c a b) ↔ r a b := by | simpa [Pairwise, Function.onFun] using @hr a b
| [
" Pairwise (r on fun c => bif c then a else b) ↔ r a b"
] | [] |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 122 | 123 | theorem hammingDist_lt_one {x y : ∀ i, β i} : hammingDist x y < 1 ↔ x = y := by |
rw [Nat.lt_one_iff, hammingDist_eq_zero]
| [
" hammingDist x x = 0",
" ∀ ⦃x_1 : ι⦄, x_1 ∈ univ → ¬x x_1 ≠ x x_1",
" hammingDist x y = hammingDist y x",
" hammingDist x z ≤ hammingDist x y + hammingDist y z",
" (filter (fun i => x i ≠ z i) univ).card ≤\n (filter (fun i => x i ≠ y i) univ).card + (filter (fun i => y i ≠ z i) univ).card",
" filter (... | [
" hammingDist x x = 0",
" ∀ ⦃x_1 : ι⦄, x_1 ∈ univ → ¬x x_1 ≠ x x_1",
" hammingDist x y = hammingDist y x",
" hammingDist x z ≤ hammingDist x y + hammingDist y z",
" (filter (fun i => x i ≠ z i) univ).card ≤\n (filter (fun i => x i ≠ y i) univ).card + (filter (fun i => y i ≠ z i) univ).card",
" filter (... |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 316 | 316 | theorem support_one : (1 : Perm α).support = ∅ := by | rw [support_eq_empty_iff]
| [
" x ∈ f.support ↔ f x ≠ x",
" x ∉ f.support ↔ f x = x",
" ↑f.support = {x | f x ≠ x}",
" x✝ ∈ ↑f.support ↔ x✝ ∈ {x | f x ≠ x}",
" σ.support = ∅ ↔ σ = 1",
" support 1 = ∅"
] | [
" x ∈ f.support ↔ f x ≠ x",
" x ∉ f.support ↔ f x = x",
" ↑f.support = {x | f x ≠ x}",
" x✝ ∈ ↑f.support ↔ x✝ ∈ {x | f x ≠ x}",
" σ.support = ∅ ↔ σ = 1"
] |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 123 | 163 | theorem Finset.all_card_le_biUnion_card_iff_exists_injective {ι : Type u} {α : Type v}
[DecidableEq α] (t : ι → Finset α) :
(∀ s : Finset ι, s.card ≤ (s.biUnion t).card) ↔
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by |
constructor
· intro h
-- Set up the functor
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Nonempty ((hallMatchingsFunctor t).obj ι') := fun ι' =>
hallMatchingsOn.nonempty t h ι'.unop
classical
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
intro ι'
rw [hallMatchi... | [
" ↑(hallMatchingsOn t ι')",
" (fun i => ↑f ⟨↑i, ⋯⟩) ∈ hallMatchingsOn t ι'",
" Function.Injective fun i => ↑f ⟨↑i, ⋯⟩",
" ⟨i, hi⟩ = ⟨j, hj⟩",
" Nonempty ↑(hallMatchingsOn t ι')",
" ∃ x, x ∈ hallMatchingsOn t ι'",
" ∀ (s : Finset { x // x ∈ ι' }), s.card ≤ (s.biUnion fun i => t ↑i).card",
" s'.card ≤ (... | [
" ↑(hallMatchingsOn t ι')",
" (fun i => ↑f ⟨↑i, ⋯⟩) ∈ hallMatchingsOn t ι'",
" Function.Injective fun i => ↑f ⟨↑i, ⋯⟩",
" ⟨i, hi⟩ = ⟨j, hj⟩",
" Nonempty ↑(hallMatchingsOn t ι')",
" ∃ x, x ∈ hallMatchingsOn t ι'",
" ∀ (s : Finset { x // x ∈ ι' }), s.card ≤ (s.biUnion fun i => t ↑i).card",
" s'.card ≤ (... |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 123 | 123 | theorem zsmul (z : ℤ) (x : WittVector p R) : mapFun f (z • x) = z • mapFun f x := by | map_fun_tac
| [
" Injective (mapFun f)",
" a₁✝ = a₂✝",
" a₁✝.coeff p = a₂✝.coeff p",
" mapFun f (mk p fun n => Classical.choose ⋯) = x",
" (mapFun f (mk p fun n => Classical.choose ⋯)).coeff n = x.coeff n",
" mapFun (⇑f) 0 = 0",
" mapFun (⇑f) 1 = 1",
" mapFun (⇑f) (x + y) = mapFun (⇑f) x + mapFun (⇑f) y",
" mapFun ... | [
" Injective (mapFun f)",
" a₁✝ = a₂✝",
" a₁✝.coeff p = a₂✝.coeff p",
" mapFun f (mk p fun n => Classical.choose ⋯) = x",
" (mapFun f (mk p fun n => Classical.choose ⋯)).coeff n = x.coeff n",
" mapFun (⇑f) 0 = 0",
" mapFun (⇑f) 1 = 1",
" mapFun (⇑f) (x + y) = mapFun (⇑f) x + mapFun (⇑f) y",
" mapFun ... |
import Mathlib.Combinatorics.Young.YoungDiagram
#align_import combinatorics.young.semistandard_tableau from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
structure SemistandardYoungTableau (μ : YoungDiagram) where
entry : ℕ → ℕ → ℕ
row_weak' : ∀ {i j1 j2 : ℕ}, j1 < j2 → (i, ... | Mathlib/Combinatorics/Young/SemistandardTableau.lean | 129 | 133 | theorem row_weak_of_le {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ}
(hj : j1 ≤ j2) (cell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 := by |
cases' eq_or_lt_of_le hj with h h
· rw [h]
· exact T.row_weak h cell
| [
" T = T'",
" { entry := entry✝, row_weak' := row_weak'✝, col_strict' := col_strict'✝, zeros' := zeros'✝ } = T'",
" { entry := entry✝¹, row_weak' := row_weak'✝¹, col_strict' := col_strict'✝¹, zeros' := zeros'✝¹ } =\n { entry := entry✝, row_weak' := row_weak'✝, col_strict' := col_strict'✝, zeros' := zeros'✝ }"... | [
" T = T'",
" { entry := entry✝, row_weak' := row_weak'✝, col_strict' := col_strict'✝, zeros' := zeros'✝ } = T'",
" { entry := entry✝¹, row_weak' := row_weak'✝¹, col_strict' := col_strict'✝¹, zeros' := zeros'✝¹ } =\n { entry := entry✝, row_weak' := row_weak'✝, col_strict' := col_strict'✝, zeros' := zeros'✝ }"... |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 185 | 185 | theorem coe_eq_zero {f : α →₀ M} : (f : α → M) = 0 ↔ f = 0 := by | rw [← coe_zero, DFunLike.coe_fn_eq]
| [
" Injective toFun",
" { support := s, toFun := f, mem_support_toFun := hf } = { support := t, toFun := f, mem_support_toFun := hg }",
" s = t",
" a ∈ s ↔ a ∈ t",
" ⇑f = 0 ↔ f = 0"
] | [
" Injective toFun",
" { support := s, toFun := f, mem_support_toFun := hf } = { support := t, toFun := f, mem_support_toFun := hg }",
" s = t",
" a ∈ s ↔ a ∈ t"
] |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 42 | 44 | theorem Ico_filter_dvd_card : ((Ico a b).filter (r ∣ ·)).card =
max (⌈b / (r : ℚ)⌉ - ⌈a / (r : ℚ)⌉) 0 := by |
rw [Ico_filter_dvd_eq _ _ hr, card_map, card_Ico, toNat_eq_max]
| [
" filter (fun x => r ∣ x) (Ico a b) = map { toFun := fun x => x * r, inj' := ⋯ } (Ico ⌈↑a / ↑r⌉ ⌈↑b / ↑r⌉)",
" x ∈ filter (fun x => r ∣ x) (Ico a b) ↔ x ∈ map { toFun := fun x => x * r, inj' := ⋯ } (Ico ⌈↑a / ↑r⌉ ⌈↑b / ↑r⌉)",
" ((a ≤ x ∧ x < b) ∧ ∃ c, x = c * r) ↔\n ∃ a_1, (a ≤ a_1 * r ∧ a_1 * r < b) ∧ { toF... | [
" filter (fun x => r ∣ x) (Ico a b) = map { toFun := fun x => x * r, inj' := ⋯ } (Ico ⌈↑a / ↑r⌉ ⌈↑b / ↑r⌉)",
" x ∈ filter (fun x => r ∣ x) (Ico a b) ↔ x ∈ map { toFun := fun x => x * r, inj' := ⋯ } (Ico ⌈↑a / ↑r⌉ ⌈↑b / ↑r⌉)",
" ((a ≤ x ∧ x < b) ∧ ∃ c, x = c * r) ↔\n ∃ a_1, (a ≤ a_1 * r ∧ a_1 * r < b) ∧ { toF... |
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf1682e59106d6c942434a32"
universe u ... | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | 137 | 212 | theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
(hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 := by |
-- First define some abbreviations.
letI := B.finite
let P := minpoly R B.gen
obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
have finrank_K_L : FiniteDimensional.finrank K L = B.dim := B.finrank
have deg_K_P : (minpoly K B.gen).natDegree = B.dim := B.natDegree_minpoly
have deg_R_P : P.natD... | [
" p ∣ Q.coeff 0",
" P.natDegree = B.dim",
" p ^ n.succ ∣ Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n",
" Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n =\n Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) * ↑((-1) ^ (n.succ * n))",
" Q.coeff 0 ^ n.succ * (minpoly R B.... | [] |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749fc4661828cba18cd0f4e3c5eb66a8e80598"
open scoped Pointwise
universe u₁ u₂ u₃
namespace MonoidAlgebra
open Finset Finsupp
variable {k : Type u₁} ... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 25 | 30 | theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
(a * b).support ⊆ a.support * b.support := by |
rw [MonoidAlgebra.mul_def]
exact support_sum.trans <| biUnion_subset.2 fun _x hx ↦
support_sum.trans <| biUnion_subset.2 fun _y hy ↦
support_single_subset.trans <| singleton_subset_iff.2 <| mem_image₂_of_mem hx hy
| [
" (a * b).support ⊆ a.support * b.support",
" (sum a fun a₁ b₁ => sum b fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)).support ⊆ a.support * b.support"
] | [] |
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6"
namespace MvPolynomial
variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S]
noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno... | Mathlib/Algebra/MvPolynomial/Expand.lean | 59 | 61 | theorem expand_one : expand 1 = AlgHom.id R (MvPolynomial σ R) := by |
ext1 f
rw [expand_one_apply, AlgHom.id_apply]
| [
" (expand 1) f = f",
" expand 1 = AlgHom.id R (MvPolynomial σ R)",
" (expand 1) (X f) = (AlgHom.id R (MvPolynomial σ R)) (X f)"
] | [
" (expand 1) f = f"
] |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 104 | 104 | theorem prod_univ {s : Set α} : s ×ˢ (univ : Set β) = Prod.fst ⁻¹' s := by | simp [prod_eq]
| [
" (∃ x ∈ s ×ˢ t, p x) ↔ ∃ x ∈ s, ∃ y ∈ t, p (x, y)",
" s ×ˢ ∅ = ∅",
" x✝ ∈ s ×ˢ ∅ ↔ x✝ ∈ ∅",
" ∅ ×ˢ t = ∅",
" x✝ ∈ ∅ ×ˢ t ↔ x✝ ∈ ∅",
" univ ×ˢ univ = univ",
" x✝ ∈ univ ×ˢ univ ↔ x✝ ∈ univ",
" univ ×ˢ t = Prod.snd ⁻¹' t",
" s ×ˢ univ = Prod.fst ⁻¹' s"
] | [
" (∃ x ∈ s ×ˢ t, p x) ↔ ∃ x ∈ s, ∃ y ∈ t, p (x, y)",
" s ×ˢ ∅ = ∅",
" x✝ ∈ s ×ˢ ∅ ↔ x✝ ∈ ∅",
" ∅ ×ˢ t = ∅",
" x✝ ∈ ∅ ×ˢ t ↔ x✝ ∈ ∅",
" univ ×ˢ univ = univ",
" x✝ ∈ univ ×ˢ univ ↔ x✝ ∈ univ",
" univ ×ˢ t = Prod.snd ⁻¹' t"
] |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 285 | 286 | theorem mem_row_iff {μ : YoungDiagram} {i : ℕ} {c : ℕ × ℕ} : c ∈ μ.row i ↔ c ∈ μ ∧ c.fst = i := by |
simp [row]
| [
" μ = ν",
" c ∈ μ.row i ↔ c ∈ μ ∧ c.1 = i"
] | [
" μ = ν"
] |
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial In... | Mathlib/FieldTheory/PrimitiveElement.lean | 282 | 292 | theorem FiniteDimensional.of_finite_intermediateField
[Finite (IntermediateField F E)] : FiniteDimensional F E := by |
let IF := { K : IntermediateField F E // ∃ x, K = F⟮x⟯ }
have := isAlgebraic_of_finite_intermediateField F E
haveI : ∀ K : IF, FiniteDimensional F K.1 := fun ⟨_, x, rfl⟩ ↦ adjoin.finiteDimensional
(Algebra.IsIntegral.isIntegral _)
have hfin := finiteDimensional_iSup_of_finite (t := fun K : IF ↦ K.1)
have... | [
" IsAlgebraic F α",
" (aeval α) (X ^ n - C y) = 0",
" s ≠ 0",
" False",
" f ≠ 0",
" f.coeff (n * s.natDegree + m) ≠ 0",
" 0 < n",
" ¬n ∣ n * s.natDegree + m",
" ¬n ∣ m",
" s.leadingCoeff ≠ 0",
" (aeval α) f = 0",
" FiniteDimensional F E"
] | [
" IsAlgebraic F α",
" (aeval α) (X ^ n - C y) = 0",
" s ≠ 0",
" False",
" f ≠ 0",
" f.coeff (n * s.natDegree + m) ≠ 0",
" 0 < n",
" ¬n ∣ n * s.natDegree + m",
" ¬n ∣ m",
" s.leadingCoeff ≠ 0",
" (aeval α) f = 0"
] |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 45 | 56 | theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
(A / C).card * B.card ≤ (A / B).card * (B / C).card := by |
rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
refine card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ ?_)
fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
((mem_bipartiteBelow _).1 hu).2.symm.trans ?_
obtain ⟨a, ha, c, hc, rfl⟩ := mem_div.1 hx
refine card_le_card_of_inj_on (fun... | [
" (A / C).card * B.card ≤ (A / B).card * (B / C).card",
" (A / C).card * B.card ≤ ((A / B) ×ˢ (B / C)).card * 1",
" x.1 * x.2 = b✝",
" (fun b => (a / b, b / c)) b ∈ bipartiteAbove (fun b ac => ac.1 * ac.2 = b) ((A / B) ×ˢ (B / C)) (a / c)",
" (fun b => (a / b, b / c)) b ∈ (A / B) ×ˢ (B / C) ∧\n ((fun b =... | [] |
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Preadditive
Opposite Simplicial
noncomputable section
namespace Alge... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 137 | 139 | theorem postComp_φ : (f.postComp h).φ = f.φ ≫ h := by |
unfold φ postComp
simp only [add_comp, sum_comp, assoc]
| [
" (Q q).f (n + 1) = ∑ i ∈ Finset.filter (fun i => ↑i < q) Finset.univ, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev",
" (Q 0).f (n + 1) = ∑ i ∈ Finset.filter (fun i => ↑i < 0) Finset.univ, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev",
" (Q (q + 1)).f (n + 1) =\n ∑ i ∈ Finset.filter (fun i => ↑i < q + 1) F... | [
" (Q q).f (n + 1) = ∑ i ∈ Finset.filter (fun i => ↑i < q) Finset.univ, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev",
" (Q 0).f (n + 1) = ∑ i ∈ Finset.filter (fun i => ↑i < 0) Finset.univ, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev",
" (Q (q + 1)).f (n + 1) =\n ∑ i ∈ Finset.filter (fun i => ↑i < q + 1) F... |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 78 | 79 | theorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a := by |
rw [← succ_zero, opow_succ]; simp only [opow_zero, one_mul]
| [
" 0 ^ a = 1 - a",
" 0 ^ a = 0",
" a ^ 0 = 1",
" a ^ succ b = a ^ b * a",
" 0 ^ succ b = 0 ^ b * 0",
" a ^ b = b.bsup fun c x => a ^ c",
" (b.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup) =\n b.bsup fun c x => c.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup",
" a ^ b ≤ c ↔ ∀ b' < b, a ^... | [
" 0 ^ a = 1 - a",
" 0 ^ a = 0",
" a ^ 0 = 1",
" a ^ succ b = a ^ b * a",
" 0 ^ succ b = 0 ^ b * 0",
" a ^ b = b.bsup fun c x => a ^ c",
" (b.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup) =\n b.bsup fun c x => c.limitRecOn 1 (fun x IH => IH * a) fun b x => b.bsup",
" a ^ b ≤ c ↔ ∀ b' < b, a ^... |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.IntervalCases
#align_import number_the... | Mathlib/NumberTheory/LucasLehmer.lean | 145 | 145 | theorem sMod_mod (p i : ℕ) : sMod p i % (2 ^ p - 1) = sMod p i := by | cases i <;> simp [sMod]
| [
" 2 ^ m < 2 ^ n",
" 1 < 2",
" mersenne k + 1 = 2 ^ k",
" 1 ≤ 2 ^ k",
" 1 ≤ 2",
" 0 ≤ sMod p i",
" 0 ≤ sMod p 0",
" 0 ≤ sMod p (n✝ + 1)",
" 0 ≤ 4 % (2 ^ p - 1)",
" 0 ≤ (sMod p n✝ ^ 2 - 2) % (2 ^ p - 1)",
" 2 ^ p - 1 ≠ 0",
" sMod p i % (2 ^ p - 1) = sMod p i",
" sMod p 0 % (2 ^ p - 1) = sMod p... | [
" 2 ^ m < 2 ^ n",
" 1 < 2",
" mersenne k + 1 = 2 ^ k",
" 1 ≤ 2 ^ k",
" 1 ≤ 2",
" 0 ≤ sMod p i",
" 0 ≤ sMod p 0",
" 0 ≤ sMod p (n✝ + 1)",
" 0 ≤ 4 % (2 ^ p - 1)",
" 0 ≤ (sMod p n✝ ^ 2 - 2) % (2 ^ p - 1)",
" 2 ^ p - 1 ≠ 0"
] |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs)... | Mathlib/Data/List/Sym.lean | 46 | 47 | theorem sym2_eq_nil_iff {xs : List α} : xs.sym2 = [] ↔ xs = [] := by |
cases xs <;> simp [List.sym2]
| [
" z ∈ (x :: xs).sym2 ↔ z = s(x, x) ∨ (∃ y ∈ xs, z = s(x, y)) ∨ z ∈ xs.sym2",
" z = s(x, x) ∨ (∃ a ∈ xs, s(x, a) = z) ∨ z ∈ xs.sym2 ↔ z = s(x, x) ∨ (∃ y ∈ xs, z = s(x, y)) ∨ z ∈ xs.sym2",
" xs.sym2 = [] ↔ xs = []",
" [].sym2 = [] ↔ [] = []",
" (head✝ :: tail✝).sym2 = [] ↔ head✝ :: tail✝ = []"
] | [
" z ∈ (x :: xs).sym2 ↔ z = s(x, x) ∨ (∃ y ∈ xs, z = s(x, y)) ∨ z ∈ xs.sym2",
" z = s(x, x) ∨ (∃ a ∈ xs, s(x, a) = z) ∨ z ∈ xs.sym2 ↔ z = s(x, x) ∨ (∃ y ∈ xs, z = s(x, y)) ∨ z ∈ xs.sym2"
] |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 35 | 41 | theorem Ioi_add_bij : BijOn (· + d) (Ioi a) (Ioi (a + d)) := by |
refine
⟨fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h =>
?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ioi.mp h).le
rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm]⟩
| [
" BijOn (fun x => x + d) (Ici a) (Ici (a + d))",
" x✝ ∈ (fun x => x + d) '' Ici a",
" a + d + c ∈ (fun x => x + d) '' Ici a",
" (fun x => x + d) (a + c) = a + d + c",
" BijOn (fun x => x + d) (Ioi a) (Ioi (a + d))",
" x✝ ∈ (fun x => x + d) '' Ioi a",
" a + d + c ∈ (fun x => x + d) '' Ioi a"
] | [
" BijOn (fun x => x + d) (Ici a) (Ici (a + d))",
" x✝ ∈ (fun x => x + d) '' Ici a",
" a + d + c ∈ (fun x => x + d) '' Ici a",
" (fun x => x + d) (a + c) = a + d + c"
] |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 86 | 89 | theorem single_mem_gradeBy {R} [CommSemiring R] (f : M → ι) (m : M) (r : R) :
Finsupp.single m r ∈ gradeBy R f (f m) := by |
intro x hx
rw [Finset.mem_singleton.mp (Finsupp.support_single_subset hx)]
| [
" f m = i",
" a ∈ gradeBy R f i ↔ ↑a.support ⊆ f ⁻¹' {i}",
" a ∈ grade R m ↔ a.support ⊆ {m}",
" a ∈ grade R m ↔ ↑a.support ⊆ {m}",
" a ∈ grade R m ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
" (∃ b, a = Finsupp.single m b) ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
" ∀ (a_1 : R), a = Finsupp.single m ... | [
" f m = i",
" a ∈ gradeBy R f i ↔ ↑a.support ⊆ f ⁻¹' {i}",
" a ∈ grade R m ↔ a.support ⊆ {m}",
" a ∈ grade R m ↔ ↑a.support ⊆ {m}",
" a ∈ grade R m ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
" (∃ b, a = Finsupp.single m b) ↔ a ∈ LinearMap.range (Finsupp.lsingle m)",
" ∀ (a_1 : R), a = Finsupp.single m ... |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 90 | 92 | theorem multiplicity_eq_factorization {n p : ℕ} (pp : p.Prime) (hn : n ≠ 0) :
multiplicity p n = n.factorization p := by |
simp [factorization, pp, padicValNat_def' pp.ne_one hn.bot_lt]
| [
" ∀ (a : ℕ), a ∈ n.primeFactors ↔ (fun p => if p.Prime then padicValNat p n else 0) a ≠ 0",
" ∀ (a : ℕ), a.Prime → (a ∣ n ∧ ¬n = 0 ↔ ¬a = 1 ∧ ¬n = 0 ∧ a ∣ n)",
" n.factorization p = padicValNat p n",
" count p n.factors = n.factorization p",
" count p (factors 0) = (factorization 0) p",
" 0 = n.factorizat... | [
" ∀ (a : ℕ), a ∈ n.primeFactors ↔ (fun p => if p.Prime then padicValNat p n else 0) a ≠ 0",
" ∀ (a : ℕ), a.Prime → (a ∣ n ∧ ¬n = 0 ↔ ¬a = 1 ∧ ¬n = 0 ∧ a ∣ n)",
" n.factorization p = padicValNat p n",
" count p n.factors = n.factorization p",
" count p (factors 0) = (factorization 0) p",
" 0 = n.factorizat... |
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
noncomputable section
namespace CategoryTheor... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 124 | 126 | theorem kernel.ι_unop :
(kernel.ι g.unop).op = eqToHom (Opposite.op_unop _) ≫ cokernel.π g ≫ (kernelUnopOp g).inv := by |
simp
| [
" Abelian Cᵒᵖ",
" (cokernel.π f).op ≫ f.op = 0",
" f ≫ (kernel.ι f.op).unop = 0",
" (kernel.ι f.op ≫ f.op).unop = 0",
" (kernel.lift f.op (cokernel.π f).op ⋯).unop ≫ cokernel.desc f (kernel.ι f.op).unop ⋯ = 𝟙 (kernel f.op).unop",
" ((cokernel.desc f (kernel.ι f.op).unop ⋯).op ≫ kernel.lift f.op (cokernel... | [
" Abelian Cᵒᵖ",
" (cokernel.π f).op ≫ f.op = 0",
" f ≫ (kernel.ι f.op).unop = 0",
" (kernel.ι f.op ≫ f.op).unop = 0",
" (kernel.lift f.op (cokernel.π f).op ⋯).unop ≫ cokernel.desc f (kernel.ι f.op).unop ⋯ = 𝟙 (kernel f.op).unop",
" ((cokernel.desc f (kernel.ι f.op).unop ⋯).op ≫ kernel.lift f.op (cokernel... |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 109 | 112 | theorem hasBasis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
HasBasis (𝓝 x) (fun _ : Unit => True) fun _ => {x} := by |
rw [nhds_of_ne_zero h]
exact hasBasis_pure _
| [
" 𝓝 = update pure 0 (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ))",
" pure 0 ≤ ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)",
" 𝓝 0 = ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)",
" (𝓝 0).HasBasis (fun γ => γ ≠ 0) Iio",
" (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)).HasBasis (fun γ => γ ≠ 0) Iio",
" DirectedOn ((fun γ => Iio γ) ⁻¹'o fun x x_1 => x ≥ x_1... | [
" 𝓝 = update pure 0 (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ))",
" pure 0 ≤ ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)",
" 𝓝 0 = ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)",
" (𝓝 0).HasBasis (fun γ => γ ≠ 0) Iio",
" (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)).HasBasis (fun γ => γ ≠ 0) Iio",
" DirectedOn ((fun γ => Iio γ) ⁻¹'o fun x x_1 => x ≥ x_1... |
import Mathlib.Analysis.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 119 | 121 | theorem Sphere.ne_iff {s₁ s₂ : Sphere P} :
s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius := by |
rw [← not_and_or, ← Sphere.ext_iff]
| [
" { center := s.center, radius := s.radius } = s",
" { center := s.center, radius := s.radius }.center = s.center",
" { center := s.center, radius := s.radius }.radius = s.radius",
" s₁ ≠ s₂ ↔ s₁.center ≠ s₂.center ∨ s₁.radius ≠ s₂.radius"
] | [
" { center := s.center, radius := s.radius } = s",
" { center := s.center, radius := s.radius }.center = s.center",
" { center := s.center, radius := s.radius }.radius = s.radius"
] |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 43 | 46 | theorem braiding_naturality_left {X Y : ModuleCat R} (f : X ⟶ Y) (Z : ModuleCat R) :
f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
| [
" (f ⊗ g) ≫ (Y₁.braiding Y₂).hom = (X₁.braiding X₂).hom ≫ (g ⊗ f)",
" ∀ (x : ↑X₁) (y : ↑X₂), ((f ⊗ g) ≫ (Y₁.braiding Y₂).hom) (x ⊗ₜ[R] y) = ((X₁.braiding X₂).hom ≫ (g ⊗ f)) (x ⊗ₜ[R] y)",
" ((f ⊗ g) ≫ (Y₁.braiding Y₂).hom) (x ⊗ₜ[R] y) = ((X₁.braiding X₂).hom ≫ (g ⊗ f)) (x ⊗ₜ[R] y)",
" f ▷ Z ≫ (Y.braiding Z).ho... | [
" (f ⊗ g) ≫ (Y₁.braiding Y₂).hom = (X₁.braiding X₂).hom ≫ (g ⊗ f)",
" ∀ (x : ↑X₁) (y : ↑X₂), ((f ⊗ g) ≫ (Y₁.braiding Y₂).hom) (x ⊗ₜ[R] y) = ((X₁.braiding X₂).hom ≫ (g ⊗ f)) (x ⊗ₜ[R] y)",
" ((f ⊗ g) ≫ (Y₁.braiding Y₂).hom) (x ⊗ₜ[R] y) = ((X₁.braiding X₂).hom ≫ (g ⊗ f)) (x ⊗ₜ[R] y)"
] |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ... | Mathlib/RingTheory/Int/Basic.lean | 99 | 102 | theorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :
p ∣ n.natAbs := by |
rw [Int.natCast_dvd, Int.natAbs_pow] at h
exact hp.dvd_of_dvd_pow h
| [
" p ∣ m.natAbs ∨ p ∣ n.natAbs",
" ↑p ∣ m ∨ ↑p ∣ n",
" p ∣ n.natAbs"
] | [
" p ∣ m.natAbs ∨ p ∣ n.natAbs",
" ↑p ∣ m ∨ ↑p ∣ n"
] |
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
#align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 120 | 164 | theorem intervalIntegrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [[a, b]]) :
IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) μ a b := by |
by_cases h2 : (0 : ℝ) ∉ [[a, b]]
· -- Easy case #1: 0 ∉ [a, b] -- use continuity.
refine (ContinuousAt.continuousOn fun x hx => ?_).intervalIntegrable
exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2)
rw [eq_false h2, or_false_iff] at h
rcases lt_or_eq_of_le h with... | [
" IntervalIntegrable (fun x => x ^ r) volume a b",
" ∀ (c : ℝ), IntervalIntegrable (fun x => x ^ r) volume 0 c",
" ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c",
" IntervalIntegrable (fun x => x ^ r) volume 0 c",
" IntegrableOn (fun x => x ^ r) (Set.Ioc 0 c) volume",
" ∀ x ∈ Set.Ioo 0... | [
" IntervalIntegrable (fun x => x ^ r) volume a b",
" ∀ (c : ℝ), IntervalIntegrable (fun x => x ^ r) volume 0 c",
" ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c",
" IntervalIntegrable (fun x => x ^ r) volume 0 c",
" IntegrableOn (fun x => x ^ r) (Set.Ioc 0 c) volume",
" ∀ x ∈ Set.Ioo 0... |
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 188 | 192 | theorem congr_point {X Y : PresheafedSpace.{_, _, v} C}
(α : X ⟶ Y) (x x' : X) (h : x = x') :
stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h]) =
eqToHom (show Y.stalk (α.base x) = Y.stalk (α.base x') by rw [h]) ≫ stalkMap α x' := by |
rw [stalkMap.congr α α rfl x x' h]
| [] | [] |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 204 | 206 | theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by |
ext1 x; simp_rw [weightedSMul_apply]; congr 2
| [
" (weightedSMul μ s) x = (μ s).toReal • x",
" weightedSMul 0 = 0",
" weightedSMul 0 x✝ = 0 x✝",
" weightedSMul μ ∅ = 0",
" (weightedSMul μ ∅) x = 0 x",
" (μ ∅).toReal • x = 0 x",
" weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
" (weightedSMul (μ + ν) s) x = (weightedSMul μ s + weighte... | [
" (weightedSMul μ s) x = (μ s).toReal • x",
" weightedSMul 0 = 0",
" weightedSMul 0 x✝ = 0 x✝",
" weightedSMul μ ∅ = 0",
" (weightedSMul μ ∅) x = 0 x",
" (μ ∅).toReal • x = 0 x",
" weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
" (weightedSMul (μ + ν) s) x = (weightedSMul μ s + weighte... |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 50 | 50 | theorem Disjoint.symm : Disjoint f g → Disjoint g f := by | simp only [Disjoint, or_comm, imp_self]
| [
" f.Disjoint g → g.Disjoint f"
] | [] |
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 | 153 | 154 | theorem of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.toPoly = C P.d := by |
rw [of_b_eq_zero ha hb, hc, C_0, zero_mul, zero_add]
| [
" 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.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 88 | 90 | theorem t'_jii (i j : D.J) : D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd := by |
rw [← Category.assoc, ← D.t_fac]
simp
| [
" D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom",
" D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd",
" D.t' j i i = (D.t' j i i ≫ pullback.snd) ≫ inv pullback.snd"
] | [
" D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom"
] |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 106 | 108 | theorem conjTranspose_col [Star α] (v : m → α) : (col v)ᴴ = row (star v) := by |
ext
rfl
| [
" col (v + w) = col v + col w",
" col (v + w) i✝ j✝ = (col v + col w) i✝ j✝",
" col (x • v) = x • col v",
" col (x • v) i✝ j✝ = (x • col v) i✝ j✝",
" row (v + w) = row v + row w",
" row (v + w) i✝ j✝ = (row v + row w) i✝ j✝",
" row (x • v) = x • row v",
" row (x • v) i✝ j✝ = (x • row v) i✝ j✝",
" (c... | [
" col (v + w) = col v + col w",
" col (v + w) i✝ j✝ = (col v + col w) i✝ j✝",
" col (x • v) = x • col v",
" col (x • v) i✝ j✝ = (x • col v) i✝ j✝",
" row (v + w) = row v + row w",
" row (v + w) i✝ j✝ = (row v + row w) i✝ j✝",
" row (x • v) = x • row v",
" row (x • v) i✝ j✝ = (x • row v) i✝ j✝",
" (c... |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 84 | 93 | theorem matrix_mul_apply [Fintype m] [Semiring α] [DecidableEq n] (M : Matrix l m α) (f : m ≃. n)
(i j) : (M * f.toMatrix :) i j = Option.casesOn (f.symm j) 0 fun fj => M i fj := by |
dsimp [toMatrix, Matrix.mul_apply]
cases' h : f.symm j with fj
· simp [h, ← f.eq_some_iff]
· rw [Finset.sum_eq_single fj]
· simp [h, ← f.eq_some_iff]
· rintro b - n
simp [h, ← f.eq_some_iff, n.symm]
· simp
| [
" (f.toMatrix * M) i j = Option.casesOn (f i) 0 fun fi => M fi j",
" ∑ j_1 : m, (if j_1 ∈ f i then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) (f i)",
" ∑ j_1 : m, (if j_1 ∈ none then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) none",
" ∑ j_1 : m, (if j_1 ∈ some fi then 1 else 0) * M j... | [
" (f.toMatrix * M) i j = Option.casesOn (f i) 0 fun fi => M fi j",
" ∑ j_1 : m, (if j_1 ∈ f i then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) (f i)",
" ∑ j_1 : m, (if j_1 ∈ none then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) none",
" ∑ j_1 : m, (if j_1 ∈ some fi then 1 else 0) * M j... |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_limits_deriv from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Filter
open scoped uniformity Filter Topology
section L... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 176 | 220 | theorem uniformCauchySeqOn_ball_of_fderiv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
(hf : ∀ n : ι, ∀ y : E, y ∈ Metric.ball x r → HasFDerivAt (f n) (f' n y) y)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) := by |
letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
have : NeBot l := (cauchy_map_iff.1 hfg).1
rcases le_or_lt r 0 with (hr | hr)
· simp only [Metric.ball_eq_empty.2 hr, UniformCauchySeqOn, Set.mem_empty_iff_false,
IsEmpty.forall_iff, eventually_const, imp_true_iff]
rw [SeminormedAddGroup.unif... | [
" UniformCauchySeqOnFilter f l (𝓝 x)",
" TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) (𝓝 x)",
" ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,\n ((fun n z => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) + fun n x_1 => f n.1 x - f n.2 x) n.1 n.2 =\n f n.1.1 n.2 - f n.1.2 n.2",
" TendstoU... | [
" UniformCauchySeqOnFilter f l (𝓝 x)",
" TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) (𝓝 x)",
" ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,\n ((fun n z => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) + fun n x_1 => f n.1 x - f n.2 x) n.1 n.2 =\n f n.1.1 n.2 - f n.1.2 n.2",
" TendstoU... |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 109 | 113 | theorem areaForm_apply_self (x : E) : ω x x = 0 := by |
rw [areaForm_to_volumeForm]
refine o.volumeForm.map_eq_zero_of_eq ![x, x] ?_ (?_ : (0 : Fin 2) ≠ 1)
· simp
· norm_num
| [
" E →ₗ[ℝ] E →ₗ[ℝ] ℝ",
" (o.areaForm x) y = o.volumeForm ![x, y]",
" (o.areaForm x) x = 0",
" o.volumeForm ![x, x] = 0",
" ![x, x] 0 = ![x, x] 1",
" 0 ≠ 1"
] | [
" E →ₗ[ℝ] E →ₗ[ℝ] ℝ",
" (o.areaForm x) y = o.volumeForm ![x, y]"
] |
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u → Type u) [Functor F] where
P : PFunctor.{u}
abs : ∀ {α}, P α → F α
repr : ∀ {α}, F α → P α
abs_repr : ∀ {α} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 117 | 131 | theorem liftp_iff' {α : Type u} (p : α → Prop) (x : F α) :
Liftp p x ↔ ∃ u : q.P α, abs u = x ∧ ∀ i, p (u.snd i) := by |
constructor
· rintro ⟨y, hy⟩
cases' h : repr y with a f
use ⟨a, fun i => (f i).val⟩
dsimp
constructor
· rw [← hy, ← abs_repr y, h, ← abs_map]
rfl
intro i
apply (f i).property
rintro ⟨⟨a, f⟩, h₀, h₁⟩; dsimp at *
use abs ⟨a, fun i => ⟨f i, h₁ i⟩⟩
rw [← abs_map, ← h₀]; rfl
| [
" id <$> x = x",
" id <$> abs (repr x) = abs (repr x)",
" id <$> abs ⟨a, f⟩ = abs ⟨a, f⟩",
" abs ((P F).map id ⟨a, f⟩) = abs ⟨a, f⟩",
" (g ∘ f) <$> x = g <$> f <$> x",
" (g ∘ f) <$> abs (repr x) = g <$> f <$> abs (repr x)",
" (g ∘ f✝) <$> abs ⟨a, f⟩ = g <$> f✝ <$> abs ⟨a, f⟩",
" abs ((P F).map (g ∘ f✝... | [
" id <$> x = x",
" id <$> abs (repr x) = abs (repr x)",
" id <$> abs ⟨a, f⟩ = abs ⟨a, f⟩",
" abs ((P F).map id ⟨a, f⟩) = abs ⟨a, f⟩",
" (g ∘ f) <$> x = g <$> f <$> x",
" (g ∘ f) <$> abs (repr x) = g <$> f <$> abs (repr x)",
" (g ∘ f✝) <$> abs ⟨a, f⟩ = g <$> f✝ <$> abs ⟨a, f⟩",
" abs ((P F).map (g ∘ f✝... |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 128 | 138 | theorem parallelepiped_eq_sum_segment (v : ι → E) : parallelepiped v = ∑ i, segment ℝ 0 (v i) := by |
ext
simp only [mem_parallelepiped_iff, Set.mem_finset_sum, Finset.mem_univ, forall_true_left,
segment_eq_image, smul_zero, zero_add, ← Set.pi_univ_Icc, Set.mem_univ_pi]
constructor
· rintro ⟨t, ht, rfl⟩
exact ⟨t • v, fun {i} => ⟨t i, ht _, by simp⟩, rfl⟩
rintro ⟨g, hg, rfl⟩
choose t ht hg using @hg... | [
" x ∈ parallelepiped v ↔ ∃ t ∈ Icc 0 1, x = ∑ i : ι, t i • v i",
" parallelepiped ⇑b = {x | ∀ (i : ι), (b.repr x) i ∈ Icc 0 1}",
" x ∈ parallelepiped ⇑b ↔ x ∈ {x | ∀ (i : ι), (b.repr x) i ∈ Icc 0 1}",
" (∃ t, ∀ (x_1 : ι), (0 ≤ t x_1 ∧ t x_1 ≤ 1) ∧ (b.repr x) x_1 = t x_1) ↔ ∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.rep... | [
" x ∈ parallelepiped v ↔ ∃ t ∈ Icc 0 1, x = ∑ i : ι, t i • v i",
" parallelepiped ⇑b = {x | ∀ (i : ι), (b.repr x) i ∈ Icc 0 1}",
" x ∈ parallelepiped ⇑b ↔ x ∈ {x | ∀ (i : ι), (b.repr x) i ∈ Icc 0 1}",
" (∃ t, ∀ (x_1 : ι), (0 ≤ t x_1 ∧ t x_1 ≤ 1) ∧ (b.repr x) x_1 = t x_1) ↔ ∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.rep... |
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 180 | 181 | theorem mem_restrict {f : α →. β} {s : Set α} (h : s ⊆ f.Dom) (a : α) (b : β) :
b ∈ f.restrict h a ↔ a ∈ s ∧ b ∈ f a := by | simp [restrict]
| [
" x ∈ f.Dom ↔ ∃ y, y ∈ f x",
" (fun f => ⟨fun a => (f a).Dom, f.asSubtype⟩) ((fun f x => { Dom := f.fst x, get := fun h => f.snd ⟨x, h⟩ }) ⟨p, f⟩) =\n ⟨p, f⟩",
" ⟨fun a => p a, asSubtype fun x => { Dom := p x, get := fun h => f ⟨x, h⟩ }⟩ = ⟨p, f⟩",
" b ∈ f.restrict h a ↔ a ∈ s ∧ b ∈ f a"
] | [
" x ∈ f.Dom ↔ ∃ y, y ∈ f x",
" (fun f => ⟨fun a => (f a).Dom, f.asSubtype⟩) ((fun f x => { Dom := f.fst x, get := fun h => f.snd ⟨x, h⟩ }) ⟨p, f⟩) =\n ⟨p, f⟩",
" ⟨fun a => p a, asSubtype fun x => { Dom := p x, get := fun h => f ⟨x, h⟩ }⟩ = ⟨p, f⟩"
] |
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 | 586 | 597 | theorem limNthHom_spec (r : R) :
∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ n ≥ N, ‖limNthHom f_compat r - nthHom f r n‖ < ε := by |
intro ε hε
obtain ⟨ε', hε'0, hε'⟩ : ∃ v : ℚ, (0 : ℝ) < v ∧ ↑v < ε := exists_rat_btwn hε
norm_cast at hε'0
obtain ⟨N, hN⟩ := padicNormE.defn (nthHomSeq f_compat r) hε'0
use N
intro n hn
apply _root_.lt_trans _ hε'
change (padicNormE _ : ℝ) < _
norm_cast
exact hN _ hn
| [
" 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",
" IsCauSeq (padicNorm p) fun n => ↑(nthHom f r n)",
" ∃ i, ∀ ... |
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.SingleObj
import Mathlib.Logic.Equiv.TransferInstance
universe u₁ u₂ v₁ v₂ v w
namespace CategoryTheory
namespace PreGaloisCat... | Mathlib/CategoryTheory/Galois/GaloisObjects.lean | 81 | 84 | theorem isGalois_iff_pretransitive (X : C) [IsConnected X] :
IsGalois X ↔ MulAction.IsPretransitive (Aut X) (F.obj X) := by |
rw [isGalois_iff_aux, Equiv.nonempty_congr <| quotientByAutTerminalEquivUniqueQuotient F X]
exact (MulAction.pretransitive_iff_unique_quotient_of_nonempty (Aut X) (F.obj X)).symm
| [
" PreservesColimitsOfShape (SingleObj G) FintypeCat.incl",
" 1 • a = a",
" F.map (𝟙 X) a = a",
" (g * h) • a = g • h • a",
" F.map (h.hom ≫ g.hom) a = (F.map h.hom ≫ F.map g.hom) a",
" IsTerminal (colimit (SingleObj.functor (Aut.toEnd X))) ≃ Unique (MulAction.orbitRel.Quotient (Aut X) ↑(F.obj X))",
" I... | [
" PreservesColimitsOfShape (SingleObj G) FintypeCat.incl",
" 1 • a = a",
" F.map (𝟙 X) a = a",
" (g * h) • a = g • h • a",
" F.map (h.hom ≫ g.hom) a = (F.map h.hom ≫ F.map g.hom) a",
" IsTerminal (colimit (SingleObj.functor (Aut.toEnd X))) ≃ Unique (MulAction.orbitRel.Quotient (Aut X) ↑(F.obj X))",
" I... |
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
open Polynomial Algebra FiniteD... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 107 | 108 | theorem empty [h : IsCyclotomicExtension ∅ A B] : (⊥ : Subalgebra A B) = ⊤ := by |
simpa [Algebra.eq_top_iff, isCyclotomicExtension_iff] using h
| [
" IsCyclotomicExtension {n} A B ↔ (∃ r, IsPrimitiveRoot r ↑n) ∧ ∀ (x : B), x ∈ adjoin A {b | b ^ ↑n = 1}",
" ⊥ = ⊤"
] | [
" IsCyclotomicExtension {n} A B ↔ (∃ r, IsPrimitiveRoot r ↑n) ∧ ∀ (x : B), x ∈ adjoin A {b | b ^ ↑n = 1}"
] |
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 130 | 134 | theorem basisSets_neg (U) (hU' : U ∈ p.basisSets) :
∃ V ∈ p.basisSets, V ⊆ (fun x : E => -x) ⁻¹' U := by |
rcases p.basisSets_iff.mp hU' with ⟨s, r, _, hU⟩
rw [hU, neg_preimage, neg_ball (s.sup p), neg_zero]
exact ⟨U, hU', Eq.subset hU⟩
| [
" U ∈ p.basisSets ↔ ∃ i r, 0 < r ∧ U = (i.sup p).ball 0 r",
" (p i).ball 0 r = ({i}.sup p).ball 0 r",
" p.basisSets.Nonempty",
" (p i).ball 0 1 ∈ p.basisSets",
" ∃ z ∈ p.basisSets, z ⊆ U ∩ V",
" ((s ∪ t).sup p).ball 0 (min r₁ r₂) ∈ p.basisSets ∧ ((s ∪ t).sup p).ball 0 (min r₁ r₂) ⊆ U ∩ V",
" ((s ∪ t).su... | [
" U ∈ p.basisSets ↔ ∃ i r, 0 < r ∧ U = (i.sup p).ball 0 r",
" (p i).ball 0 r = ({i}.sup p).ball 0 r",
" p.basisSets.Nonempty",
" (p i).ball 0 1 ∈ p.basisSets",
" ∃ z ∈ p.basisSets, z ⊆ U ∩ V",
" ((s ∪ t).sup p).ball 0 (min r₁ r₂) ∈ p.basisSets ∧ ((s ∪ t).sup p).ball 0 (min r₁ r₂) ⊆ U ∩ V",
" ((s ∪ t).su... |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 51 | 55 | theorem Ioo_add_bij : BijOn (· + d) (Ioo a b) (Ioo (a + d) (b + d)) := by |
rw [← Ioi_inter_Iio, ← Ioi_inter_Iio]
exact
(Ioi_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx =>
lt_of_add_lt_add_right hx.2
| [
" BijOn (fun x => x + d) (Ici a) (Ici (a + d))",
" x✝ ∈ (fun x => x + d) '' Ici a",
" a + d + c ∈ (fun x => x + d) '' Ici a",
" (fun x => x + d) (a + c) = a + d + c",
" BijOn (fun x => x + d) (Ioi a) (Ioi (a + d))",
" x✝ ∈ (fun x => x + d) '' Ioi a",
" a + d + c ∈ (fun x => x + d) '' Ioi a",
" BijOn (... | [
" BijOn (fun x => x + d) (Ici a) (Ici (a + d))",
" x✝ ∈ (fun x => x + d) '' Ici a",
" a + d + c ∈ (fun x => x + d) '' Ici a",
" (fun x => x + d) (a + c) = a + d + c",
" BijOn (fun x => x + d) (Ioi a) (Ioi (a + d))",
" x✝ ∈ (fun x => x + d) '' Ioi a",
" a + d + c ∈ (fun x => x + d) '' Ioi a",
" BijOn (... |
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 57 | 57 | theorem toNat_apply_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : toNat c = 0 := by | simp [h]
| [
" toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c",
" toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀",
" ↑(toNat c) = c",
" ↑(toNat ↑c) = ↑c",
" ↑(toNat c) = ↑(Classical.choose ⋯)",
" toNat c = 0"
] | [
" toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c",
" toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀",
" ↑(toNat c) = c",
" ↑(toNat ↑c) = ↑c",
" ↑(toNat c) = ↑(Classical.choose ⋯)"
] |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 158 | 177 | theorem Perfect.splitting [T25Space α] (hC : Perfect C) (hnonempty : C.Nonempty) :
∃ C₀ C₁ : Set α,
(Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C) ∧ Disjoint C₀ C₁ := by |
cases' hnonempty with y yC
obtain ⟨x, xC, hxy⟩ : ∃ x ∈ C, x ≠ y := by
have := hC.acc _ yC
rw [accPt_iff_nhds] at this
rcases this univ univ_mem with ⟨x, xC, hxy⟩
exact ⟨x, xC.2, hxy⟩
obtain ⟨U, xU, Uop, V, yV, Vop, hUV⟩ := exists_open_nhds_disjoint_closure hxy
use closure (U ∩ C), closure (V ∩ ... | [
" AccPt x (𝓟 (U ∩ C))",
" 𝓝[≠] x ≤ 𝓟 U",
" U ∈ 𝓝[≠] x",
" (𝓝[≠] x ⊓ 𝓟 C).NeBot",
" Preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x",
" Preperfect (U ∩ C)",
" U ∈ 𝓝 x",
" Perfect (closure C)",
" IsClosed (closure C)",
" Preperfect (closure C)",
" AccPt x (𝓟 (closure C))",
" AccPt... | [
" AccPt x (𝓟 (U ∩ C))",
" 𝓝[≠] x ≤ 𝓟 U",
" U ∈ 𝓝[≠] x",
" (𝓝[≠] x ⊓ 𝓟 C).NeBot",
" Preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x",
" Preperfect (U ∩ C)",
" U ∈ 𝓝 x",
" Perfect (closure C)",
" IsClosed (closure C)",
" Preperfect (closure C)",
" AccPt x (𝓟 (closure C))",
" AccPt... |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 142 | 145 | theorem subsingleton (h : 0 ∈ S) : Subsingleton (LocalizedModule S M) := by |
refine ⟨fun a b ↦ ?_⟩
induction a,b using LocalizedModule.induction_on₂
exact mk_eq.mpr ⟨⟨0, h⟩, by simp only [Submonoid.mk_smul, zero_smul]⟩
| [
" 1 • (m, s).2 • (m, s).1 = 1 • (m, s).2 • (m, s).1",
" r S M (m1, s1) (m3, s3)",
" (u1 * u2 * s2) • (m3, s3).2 • (m1, s1).1 = (u1 * u2 * s2) • (m1, s1).2 • (m3, s3).1",
" (s2 * (s3 * (u1 * u2))) • m1 = (s1 * (s2 * (u1 * u2))) • m3",
" ∀ (x : LocalizedModule S M), β x",
" β (Quot.mk Setoid.r (m, s))",
"... | [
" 1 • (m, s).2 • (m, s).1 = 1 • (m, s).2 • (m, s).1",
" r S M (m1, s1) (m3, s3)",
" (u1 * u2 * s2) • (m3, s3).2 • (m1, s1).1 = (u1 * u2 * s2) • (m1, s1).2 • (m3, s3).1",
" (s2 * (s3 * (u1 * u2))) • m1 = (s1 * (s2 * (u1 * u2))) • m3",
" ∀ (x : LocalizedModule S M), β x",
" β (Quot.mk Setoid.r (m, s))",
"... |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 80 | 80 | theorem volume_Ico {a b : ℝ} : volume (Ico a b) = ofReal (b - a) := by | simp [volume_val]
| [
" volume = StieltjesFunction.id.measure",
" StieltjesFunction.id.measure (Ioo ↑p ↑q) = (Measure.map (fun x => a + x) StieltjesFunction.id.measure) (Ioo ↑p ↑q)",
" StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1",
" StieltjesFunction.id.measure (parallelepiped ⇑(stdOrthonorma... | [
" volume = StieltjesFunction.id.measure",
" StieltjesFunction.id.measure (Ioo ↑p ↑q) = (Measure.map (fun x => a + x) StieltjesFunction.id.measure) (Ioo ↑p ↑q)",
" StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1",
" StieltjesFunction.id.measure (parallelepiped ⇑(stdOrthonorma... |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 140 | 142 | theorem term_add_constg {α} [AddCommGroup α] (n x a k a') (h : a + k = a') :
@termg α _ n x a + k = termg n x a' := by |
simp [h.symm, termg, add_assoc]
| [
" k + term n x a = term n x a'",
" k + termg n x a = termg n x a'",
" term n x a + k = term n x a'",
" termg n x a + k = termg n x a'"
] | [
" k + term n x a = term n x a'",
" k + termg n x a = termg n x a'",
" term n x a + k = term n x a'"
] |
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
universe u₁ u₂ w
namespace CategoryTheory
open Limits Functor
variable {C : Type u₁} [Category.{u₂} C]
namespace PreGaloisCategory
variable [GaloisCategory C]
section Decomposition
private lemma... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 118 | 121 | theorem has_decomp_connected_components' (X : C) :
∃ (ι : Type) (_ : Finite ι) (f : ι → C) (_ : ∐ f ≅ X), ∀ i, IsConnected (f i) := by |
obtain ⟨ι, f, g, hl, hc, hf⟩ := has_decomp_connected_components X
exact ⟨ι, hf, f, colimit.isoColimitCocone ⟨Cofan.mk X g, hl⟩, hc⟩
| [
" ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι",
" (∀ (i : Unit), IsConnected ((fun x => X) i)) ∧ Finite Unit",
" ∃ x, (∀ (i : Empty), IsConnected ((fun x => X) i)) ∧ Finite Empty",
" ∀ (j : Empty), (Cofan.mk X fun x => 𝟙 X).inj j ≫ (fun s => h.to s.pt) s = s.inj j",
" (∀ (i : Empty), IsConnected (... | [
" ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι",
" (∀ (i : Unit), IsConnected ((fun x => X) i)) ∧ Finite Unit",
" ∃ x, (∀ (i : Empty), IsConnected ((fun x => X) i)) ∧ Finite Empty",
" ∀ (j : Empty), (Cofan.mk X fun x => 𝟙 X).inj j ≫ (fun s => h.to s.pt) s = s.inj j",
" (∀ (i : Empty), IsConnected (... |
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
open Set Function Metric MeasurableSpace intervalIntegral
open s... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 49 | 53 | theorem toSphere_apply_aux (s : Set (sphere (0 : E) 1)) (r : Ioi (0 : ℝ)) :
μ ((↑) '' (homeomorphUnitSphereProd E ⁻¹' s ×ˢ Iio r)) = μ (Ioo (0 : ℝ) r • ((↑) '' s)) := by |
rw [← image2_smul, image2_image_right, ← Homeomorph.image_symm, image_image,
← image_subtype_val_Ioi_Iio, image2_image_left, image2_swap, ← image_prod]
rfl
| [
" μ (Subtype.val '' (⇑(homeomorphUnitSphereProd E) ⁻¹' s ×ˢ Iio r)) = μ (Ioo 0 ↑r • Subtype.val '' s)",
" μ ((fun x => ↑((homeomorphUnitSphereProd E).symm x)) '' s ×ˢ Iio r) = μ ((fun x => SMul.smul ↑x.2 ↑x.1) '' s ×ˢ Iio r)"
] | [] |
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 217 | 219 | theorem flip_isReduced : (flip u).IsReduced ↔ u.IsReduced := by |
dsimp [IsReduced, flip]
constructor <;> intro h <;> exact h.symm
| [
" u.v = succ₂ u.vp",
" u.v.1 = (succ₂ u.vp).1",
" u.v.2 = (succ₂ u.vp).2",
" (u.wp + 1) * (u.ap + 1) + u.x * (u.bp + 1) = u.wp + u.x + u.ap + u.wp * u.ap + u.x * u.bp + 1",
" u.y * (u.ap + 1) + (u.zp + 1) * (u.bp + 1) = u.y + u.zp + u.bp + u.y * u.ap + u.zp * u.bp + 1",
" u.IsSpecial ↔ u.IsSpecial'",
" ... | [
" u.v = succ₂ u.vp",
" u.v.1 = (succ₂ u.vp).1",
" u.v.2 = (succ₂ u.vp).2",
" (u.wp + 1) * (u.ap + 1) + u.x * (u.bp + 1) = u.wp + u.x + u.ap + u.wp * u.ap + u.x * u.bp + 1",
" u.y * (u.ap + 1) + (u.zp + 1) * (u.bp + 1) = u.y + u.zp + u.bp + u.y * u.ap + u.zp * u.bp + 1",
" u.IsSpecial ↔ u.IsSpecial'",
" ... |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 45 | 46 | theorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by |
ext X Y f; rw [universallyClosed_iff]
| [
" @UniversallyClosed = (topologically @IsClosedMap).universally",
" UniversallyClosed f ↔ (topologically @IsClosedMap).universally f"
] | [] |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Int.Cast.Field
import Mathlib.Data.Int.Cast.Lemmas
#align_import data.int.char_zero from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Nat Set
variable {α β : Type*}
namespace Int
@[simp, norm_cast]
| Mathlib/Data/Int/CharZero.lean | 24 | 28 | theorem cast_div_charZero {k : Type*} [DivisionRing k] [CharZero k] {m n : ℤ} (n_dvd : n ∣ m) :
((m / n : ℤ) : k) = m / n := by |
rcases eq_or_ne n 0 with (rfl | hn)
· simp [Int.ediv_zero]
· exact cast_div n_dvd (cast_ne_zero.mpr hn)
| [
" ↑(m / n) = ↑m / ↑n",
" ↑(m / 0) = ↑m / ↑0"
] | [] |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 140 | 141 | theorem moebius_mul_log_eq_vonMangoldt : (μ : ArithmeticFunction ℝ) * log = Λ := by |
rw [mul_comm]; simp
| [
" (fun n => (↑n).log) 0 = 0",
" Λ 1 = 0",
" 0 ≤ Λ n",
" 0 ≤ if IsPrimePow n then (↑n.minFac).log else 0",
" 0 ≤ (↑n.minFac).log",
" 0 ≤ 0",
" Λ (n ^ k) = Λ n",
" Λ p = (↑p).log",
" Λ n ≠ 0 ↔ IsPrimePow n",
" Λ 1 ≠ 0 ↔ IsPrimePow 1",
" ∑ i ∈ n.divisors, Λ i = (↑n).log",
" ∑ i ∈ divisors 0, Λ i ... | [
" (fun n => (↑n).log) 0 = 0",
" Λ 1 = 0",
" 0 ≤ Λ n",
" 0 ≤ if IsPrimePow n then (↑n.minFac).log else 0",
" 0 ≤ (↑n.minFac).log",
" 0 ≤ 0",
" Λ (n ^ k) = Λ n",
" Λ p = (↑p).log",
" Λ n ≠ 0 ↔ IsPrimePow n",
" Λ 1 ≠ 0 ↔ IsPrimePow 1",
" ∑ i ∈ n.divisors, Λ i = (↑n).log",
" ∑ i ∈ divisors 0, Λ i ... |
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
open MeasureTheory ProbabilityTheory
open scoped MeasureTheory ENNReal NNReal
namesp... | Mathlib/Probability/Kernel/WithDensity.lean | 125 | 132 | theorem withDensity_add_left (κ η : kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η]
(f : α → β → ℝ≥0∞) : withDensity (κ + η) f = withDensity κ f + withDensity η f := by |
by_cases hf : Measurable (Function.uncurry f)
· ext a s
simp only [kernel.withDensity_apply _ hf, coeFn_add, Pi.add_apply, withDensity_add_measure,
Measure.add_apply]
· simp_rw [withDensity_of_not_measurable _ hf]
rw [zero_add]
| [
" (fun a => (κ a).withDensity (f a)) ∈ kernel α β",
" Measurable fun b => ((κ b).withDensity (f b)) s",
" Measurable fun b => ∫⁻ (a : β) in s, f b a ∂κ b",
" withDensity κ f = 0",
" (withDensity κ f) a = (κ a).withDensity (f a)",
" ⟨fun a => (κ a).withDensity (f a), ⋯⟩ a = (κ a).withDensity (f a)",
" ((... | [
" (fun a => (κ a).withDensity (f a)) ∈ kernel α β",
" Measurable fun b => ((κ b).withDensity (f b)) s",
" Measurable fun b => ∫⁻ (a : β) in s, f b a ∂κ b",
" withDensity κ f = 0",
" (withDensity κ f) a = (κ a).withDensity (f a)",
" ⟨fun a => (κ a).withDensity (f a), ⋯⟩ a = (κ a).withDensity (f a)",
" ((... |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 189 | 191 | theorem orderOf_r [NeZero n] (i : ZMod n) : orderOf (r i) = n / Nat.gcd n i.val := by |
conv_lhs => rw [← ZMod.natCast_zmod_val i]
rw [← r_one_pow, orderOf_pow, orderOf_r_one]
| [
" ∀ (a b c : DihedralGroup n), a * b * c = a * (b * c)",
" r a * r b * r c = r a * (r b * r c)",
" r a * r b * sr c = r a * (r b * sr c)",
" r a * sr b * r c = r a * (sr b * r c)",
" r a * sr b * sr c = r a * (sr b * sr c)",
" sr a * r b * r c = sr a * (r b * r c)",
" sr a * r b * sr c = sr a * (r b * s... | [
" ∀ (a b c : DihedralGroup n), a * b * c = a * (b * c)",
" r a * r b * r c = r a * (r b * r c)",
" r a * r b * sr c = r a * (r b * sr c)",
" r a * sr b * r c = r a * (sr b * r c)",
" r a * sr b * sr c = r a * (sr b * sr c)",
" sr a * r b * r c = sr a * (r b * r c)",
" sr a * r b * sr c = sr a * (r b * s... |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 130 | 131 | theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by |
rw [← card_Icc, Fintype.card_ofFinset]
| [
" map valEmbedding (Icc a b) = Icc ↑a ↑b",
" map valEmbedding (Ico a b) = Ico ↑a ↑b",
" map valEmbedding (Ioc a b) = Ioc ↑a ↑b",
" map valEmbedding (Ioo a b) = Ioo ↑a ↑b",
" (Icc a b).card = ↑b + 1 - ↑a",
" (Ico a b).card = ↑b - ↑a",
" (Ioc a b).card = ↑b - ↑a",
" (Ioo a b).card = ↑b - ↑a - 1",
" (u... | [
" map valEmbedding (Icc a b) = Icc ↑a ↑b",
" map valEmbedding (Ico a b) = Ico ↑a ↑b",
" map valEmbedding (Ioc a b) = Ioc ↑a ↑b",
" map valEmbedding (Ioo a b) = Ioo ↑a ↑b",
" (Icc a b).card = ↑b + 1 - ↑a",
" (Ico a b).card = ↑b - ↑a",
" (Ioc a b).card = ↑b - ↑a",
" (Ioo a b).card = ↑b - ↑a - 1",
" (u... |
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_import ring_theory.witt_vector.compare from "leanprover-community/mathlib"@"168ad7fc5d8173ad38be9767a22d50b8ecf1cd00"
noncomputable section
variable {p : ℕ} [hp : Fact... | Mathlib/RingTheory/WittVector/Compare.lean | 43 | 53 | theorem eq_of_le_of_cast_pow_eq_zero [CharP R p] (i : ℕ) (hin : i ≤ n)
(hpi : (p : TruncatedWittVector p n R) ^ i = 0) : i = n := by |
contrapose! hpi
replace hin := lt_of_le_of_ne hin hpi; clear hpi
have : (p : TruncatedWittVector p n R) ^ i = WittVector.truncate n ((p : 𝕎 R) ^ i) := by
rw [RingHom.map_pow, map_natCast]
rw [this, ne_eq, ext_iff, not_forall]; clear this
use ⟨i, hin⟩
rw [WittVector.coeff_truncate, coeff_zero, Fin.val_... | [
" i = n",
" ↑p ^ i ≠ 0",
" ↑p ^ i = (WittVector.truncate n) (↑p ^ i)",
" ∃ x, ¬coeff x ((WittVector.truncate n) (↑p ^ i)) = coeff x 0",
" ¬coeff ⟨i, hin⟩ ((WittVector.truncate n) (↑p ^ i)) = coeff ⟨i, hin⟩ 0",
" ¬1 = 0"
] | [] |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e"
section
open Finset Polynomial Function Nat
variable {R : Type*} {G : Ty... | Mathlib/RingTheory/IntegralDomain.lean | 122 | 133 | theorem card_nthRoots_subgroup_units [Fintype G] [DecidableEq G] (f : G →* R) (hf : Injective f)
{n : ℕ} (hn : 0 < n) (g₀ : G) :
Finset.card (Finset.univ.filter (fun g ↦ g^n = g₀)) ≤ Multiset.card (nthRoots n (f g₀)) := by |
haveI : DecidableEq R := Classical.decEq _
refine le_trans ?_ (nthRoots n (f g₀)).toFinset_card_le
apply card_le_card_of_inj_on f
· intro g hg
rw [mem_filter] at hg
rw [Multiset.mem_toFinset, mem_nthRoots hn, ← f.map_pow, hg.2]
· intros
apply hf
assumption
| [
" (filter (fun g => g ^ n = g₀) univ).card ≤ Multiset.card (nthRoots n (f g₀))",
" (filter (fun g => g ^ n = g₀) univ).card ≤ (nthRoots n (f g₀)).toFinset.card",
" ∀ a ∈ filter (fun g => g ^ n = g₀) univ, f a ∈ (nthRoots n (f g₀)).toFinset",
" f g ∈ (nthRoots n (f g₀)).toFinset",
" ∀ a₁ ∈ filter (fun g => g... | [] |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 144 | 147 | theorem prod_eq_pow_single [DecidableEq α] (a : α) (h : ∀ a' ≠ a, a' ∈ s → a' = 1) :
s.prod = a ^ s.count a := by |
induction' s using Quotient.inductionOn with l
simp [List.prod_eq_pow_single a h]
| [
" (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
" (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y",
" foldl (fun x y => y * x) ⋯ 1 s = foldl (fun x x_1 => x * x_1) ⋯ 1 s",
" s.toList... | [
" (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
" (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y",
" foldl (fun x y => y * x) ⋯ 1 s = foldl (fun x x_1 => x * x_1) ⋯ 1 s",
" s.toList... |
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open Set
open Computability
-- "ε_NFA"
set_option linter.uppercaseLean3 false
universe u v
structure εNFA (α : Type u) (σ : Type v) where
step : σ → Opt... | Mathlib/Computability/EpsilonNFA.lean | 82 | 83 | theorem mem_stepSet_iff : s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.εClosure (M.step t a) := by |
simp_rw [stepSet, mem_iUnion₂, exists_prop]
| [
" False",
" s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.εClosure (M.step t (some a))"
] | [
" False"
] |
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {α β γ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 96 | 98 | theorem tendsto_floor_left_pure_sub_one (n : ℤ) :
Tendsto (floor : α → ℤ) (𝓝[<] n) (pure (n - 1)) := by |
simpa only [ceil_intCast] using tendsto_floor_left_pure_ceil_sub_one (n : α)
| [
" b ≤ ⌊↑(b + 1)⌋",
" b ≤ b + 1",
" ⌈↑(b - 1)⌉ ≤ b",
" b - 1 ≤ b",
" Tendsto floor (𝓝[≥] ↑n) (pure n)",
" Tendsto ceil (𝓝[≤] ↑n) (pure n)",
" ↑(⌈x⌉ - 1) < x",
" ↑⌈x⌉ < x + 1",
" x ≤ ↑(⌈x⌉ - 1) + 1",
" x ≤ ↑⌈x⌉",
" Tendsto floor (𝓝[<] ↑n) (pure (n - 1))"
] | [
" b ≤ ⌊↑(b + 1)⌋",
" b ≤ b + 1",
" ⌈↑(b - 1)⌉ ≤ b",
" b - 1 ≤ b",
" Tendsto floor (𝓝[≥] ↑n) (pure n)",
" Tendsto ceil (𝓝[≤] ↑n) (pure n)",
" ↑(⌈x⌉ - 1) < x",
" ↑⌈x⌉ < x + 1",
" x ≤ ↑(⌈x⌉ - 1) + 1",
" x ≤ ↑⌈x⌉"
] |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finsupp.Basic
import Mathlib.LinearAlgebra.Finsupp
#align_import algebra.monoid_algebra.basic from "leanprover-community/mathlib"@... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 174 | 176 | theorem mul_def {f g : MonoidAlgebra k G} :
f * g = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂) := by |
with_unfolding_all rfl
| [
" f * g = sum f fun a₁ b₁ => sum g fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)"
] | [] |
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 237 | 238 | theorem SymmetricRel.inter {U V : Set (α × α)} (hU : SymmetricRel U) (hV : SymmetricRel V) :
SymmetricRel (U ∩ V) := by | rw [SymmetricRel, preimage_inter, hU.eq, hV.eq]
| [
" idRel ⊆ s ↔ ∀ (a : α), (a, a) ∈ s",
" (a, b) ∈ Prod.swap '' idRel ↔ (a, b) ∈ idRel",
" (a, b) ∈ idRel ○ r ↔ (a, b) ∈ r",
" r ○ s ○ t = r ○ (s ○ t)",
" (a, b) ∈ r ○ s ○ t ↔ (a, b) ∈ r ○ (s ○ t)",
" (∃ z, (∃ z_1, (a, z_1) ∈ r ∧ (z_1, z) ∈ s) ∧ (z, b) ∈ t) ↔ ∃ z, (a, z) ∈ r ∧ ∃ z_1, (z, z_1) ∈ s ∧ (z_1, b)... | [
" idRel ⊆ s ↔ ∀ (a : α), (a, a) ∈ s",
" (a, b) ∈ Prod.swap '' idRel ↔ (a, b) ∈ idRel",
" (a, b) ∈ idRel ○ r ↔ (a, b) ∈ r",
" r ○ s ○ t = r ○ (s ○ t)",
" (a, b) ∈ r ○ s ○ t ↔ (a, b) ∈ r ○ (s ○ t)",
" (∃ z, (∃ z_1, (a, z_1) ∈ r ∧ (z_1, z) ∈ s) ∧ (z, b) ∈ t) ↔ ∃ z, (a, z) ∈ r ∧ ∃ z_1, (z, z_1) ∈ s ∧ (z_1, b)... |
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.Topology.Algebra.ValuedField
#align_import number_theory.function_field from "leanprover-community/mathlib"@"70fd9563... | Mathlib/NumberTheory/FunctionField.lean | 168 | 176 | theorem InftyValuation.map_mul' (x y : RatFunc Fq) :
inftyValuationDef Fq (x * y) = inftyValuationDef Fq x * inftyValuationDef Fq y := by |
rw [inftyValuationDef, inftyValuationDef, inftyValuationDef]
by_cases hx : x = 0
· rw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]
· by_cases hy : y = 0
· rw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithZero.coe_mul, WithZero.coe_inj,
... | [
" FunctionField Fq F ↔ FiniteDimensional Fqt F",
" ∀ (c : RatFunc Fq) (x : F), e c • x = c • x",
" e c • x = c • x",
" (algebraMap Fqt F) (e c) * x = (algebraMap (RatFunc Fq) F) c * x",
" (algebraMap Fqt F) (e c) = (algebraMap (RatFunc Fq) F) c",
" (fun c => (algebraMap Fqt F) (e c)) = ⇑(algebraMap (RatFu... | [
" FunctionField Fq F ↔ FiniteDimensional Fqt F",
" ∀ (c : RatFunc Fq) (x : F), e c • x = c • x",
" e c • x = c • x",
" (algebraMap Fqt F) (e c) * x = (algebraMap (RatFunc Fq) F) c * x",
" (algebraMap Fqt F) (e c) = (algebraMap (RatFunc Fq) F) c",
" (fun c => (algebraMap Fqt F) (e c)) = ⇑(algebraMap (RatFu... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 117 | 126 | theorem _root_.ZMod.card_units_eq_totient (n : ℕ) [NeZero n] [Fintype (ZMod n)ˣ] :
Fintype.card (ZMod n)ˣ = φ n :=
calc
Fintype.card (ZMod n)ˣ = Fintype.card { x : ZMod n // x.val.Coprime n } :=
Fintype.card_congr ZMod.unitsEquivCoprime
_ = φ n := by |
obtain ⟨m, rfl⟩ : ∃ m, n = m + 1 := exists_eq_succ_of_ne_zero NeZero.out
simp only [totient, Finset.card_eq_sum_ones, Fintype.card_subtype, Finset.sum_filter, ←
Fin.sum_univ_eq_sum_range, @Nat.coprime_comm (m + 1)]
rfl
| [
" φ n = Nat.card ↑{m | m < n ∧ n.Coprime m}",
" ↑m ∈ filter n.Coprime (range n)",
" ↑m ∈ {m | m < n ∧ n.Coprime m}",
" (fun m => ⟨↑m, ⋯⟩) ((fun m => ⟨↑m, ⋯⟩) m) = m",
" 0 ∈ range n ∧ ¬n.Coprime 0",
" φ 0 = 0 ↔ 0 = 0",
" (n + 1).gcd (1 % (n + 1)) = 1",
" φ (n + 1) = 0 ↔ n + 1 = 0",
" 0 < φ n ↔ 0 < n"... | [
" φ n = Nat.card ↑{m | m < n ∧ n.Coprime m}",
" ↑m ∈ filter n.Coprime (range n)",
" ↑m ∈ {m | m < n ∧ n.Coprime m}",
" (fun m => ⟨↑m, ⋯⟩) ((fun m => ⟨↑m, ⋯⟩) m) = m",
" 0 ∈ range n ∧ ¬n.Coprime 0",
" φ 0 = 0 ↔ 0 = 0",
" (n + 1).gcd (1 % (n + 1)) = 1",
" φ (n + 1) = 0 ↔ n + 1 = 0",
" 0 < φ n ↔ 0 < n"... |
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 | 137 | 138 | theorem of_a_eq_zero (ha : P.a = 0) : P.toPoly = C P.b * X ^ 2 + C P.c * X + C P.d := by |
rw [toPoly, ha, C_0, zero_mul, zero_add]
| [
" 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.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 140 | 146 | theorem gold_irrational : Irrational φ := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
convert this
norm_num
field_simp
| [
" φ⁻¹ = -ψ",
" 0 < 1",
" 0 < 5",
" 2 * 2 = 5 - 1",
" ψ⁻¹ = -φ",
" -ψ = φ⁻¹",
" φ * ψ = -1",
" (1 + √5) * (1 - √5) = -(2 * 2)",
" 1 ^ 2 - √5 ^ 2 = -(2 * 2)",
" ψ * φ = -1",
" φ + ψ = 1",
" (1 + √5) / 2 + (1 - √5) / 2 = 1",
" 1 - φ = ψ",
" 1 - ψ = φ",
" φ - ψ = √5",
" φ ^ (n + 2) - φ ^ (... | [
" φ⁻¹ = -ψ",
" 0 < 1",
" 0 < 5",
" 2 * 2 = 5 - 1",
" ψ⁻¹ = -φ",
" -ψ = φ⁻¹",
" φ * ψ = -1",
" (1 + √5) * (1 - √5) = -(2 * 2)",
" 1 ^ 2 - √5 ^ 2 = -(2 * 2)",
" ψ * φ = -1",
" φ + ψ = 1",
" (1 + √5) / 2 + (1 - √5) / 2 = 1",
" 1 - φ = ψ",
" 1 - ψ = φ",
" φ - ψ = √5",
" φ ^ (n + 2) - φ ^ (... |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 170 | 171 | theorem coe_of_rat_eq : ((IntFractPair.of q).mapFr (↑) : IntFractPair K) = IntFractPair.of v := by |
simp [IntFractPair.of, v_eq_q]
| [
" mapFr Rat.cast (IntFractPair.of q) = IntFractPair.of v"
] | [] |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 629 | 634 | theorem oangle_left_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arctan (dist p₃ p₂ / dist p₁ p₂) := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(left_ne_of_oangle_eq_pi_div_two h)]
| [
" ∡ p₂ p₃ p₁ = ↑(dist p₃ p₂ / dist p₁ p₃).arccos",
" (∡ p₂ p₃ p₁).sign = 1",
" ∡ p₃ p₁ p₂ = ↑(dist p₁ p₂ / dist p₁ p₃).arccos",
" (∡ p₃ p₁ p₂).sign = 1",
" ∡ p₂ p₃ p₁ = ↑(dist p₁ p₂ / dist p₁ p₃).arcsin",
" ∡ p₃ p₁ p₂ = ↑(dist p₃ p₂ / dist p₁ p₃).arcsin",
" ∡ p₂ p₃ p₁ = ↑(dist p₁ p₂ / dist p₃ p₂).arctan... | [
" ∡ p₂ p₃ p₁ = ↑(dist p₃ p₂ / dist p₁ p₃).arccos",
" (∡ p₂ p₃ p₁).sign = 1",
" ∡ p₃ p₁ p₂ = ↑(dist p₁ p₂ / dist p₁ p₃).arccos",
" (∡ p₃ p₁ p₂).sign = 1",
" ∡ p₂ p₃ p₁ = ↑(dist p₁ p₂ / dist p₁ p₃).arcsin",
" ∡ p₃ p₁ p₂ = ↑(dist p₃ p₂ / dist p₁ p₃).arcsin",
" ∡ p₂ p₃ p₁ = ↑(dist p₁ p₂ / dist p₃ p₂).arctan... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 137 | 139 | theorem map_nsmul_add [AddCommMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (n : ℕ) (x : G) : f (n • a + x) = f x + n • b := by |
rw [add_comm, map_add_nsmul]
| [
" f (x + n • a) = f x + n • b",
" f (x + ↑n) = f x + n • b",
" f (x + ↑n) = f x + ↑n",
" f a = f 0 + b",
" f (n • a) = f 0 + n • b",
" f ↑n = f 0 + n • b",
" f ↑n = f 0 + ↑n",
" f (a + x) = f x + b",
" f (n • a + x) = f x + n • b"
] | [
" f (x + n • a) = f x + n • b",
" f (x + ↑n) = f x + n • b",
" f (x + ↑n) = f x + ↑n",
" f a = f 0 + b",
" f (n • a) = f 0 + n • b",
" f ↑n = f 0 + n • b",
" f ↑n = f 0 + ↑n",
" f (a + x) = f x + b"
] |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.complex_deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
namespace Complex
open Set Filter
open scoped Real
theorem hasStrictDerivAt_t... | Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 37 | 44 | theorem tendsto_abs_tan_of_cos_eq_zero {x : ℂ} (hx : cos x = 0) :
Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop := by |
simp only [tan_eq_sin_div_cos, ← norm_eq_abs, norm_div]
have A : sin x ≠ 0 := fun h => by simpa [*, sq] using sin_sq_add_cos_sq x
have B : Tendsto cos (𝓝[≠] x) (𝓝[≠] 0) :=
hx ▸ (hasDerivAt_cos x).tendsto_punctured_nhds (neg_ne_zero.2 A)
exact continuous_sin.continuousWithinAt.norm.mul_atTop (norm_pos_iff... | [
" HasStrictDerivAt tan (1 / x.cos ^ 2) x",
" 1 / x.cos ^ 2 = (x.cos * x.cos - x.sin * -x.sin) / x.cos ^ 2",
" (x.sin ^ 2 + x.cos ^ 2) / x.cos ^ 2 = (x.cos * x.cos - x.sin * -x.sin) / x.cos ^ 2",
" Tendsto (fun x => abs x.tan) (𝓝[≠] x) atTop",
" Tendsto (fun x => ‖x.sin‖ / ‖x.cos‖) (𝓝[≠] x) atTop",
" Fal... | [
" HasStrictDerivAt tan (1 / x.cos ^ 2) x",
" 1 / x.cos ^ 2 = (x.cos * x.cos - x.sin * -x.sin) / x.cos ^ 2",
" (x.sin ^ 2 + x.cos ^ 2) / x.cos ^ 2 = (x.cos * x.cos - x.sin * -x.sin) / x.cos ^ 2"
] |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 249 | 251 | theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by |
ext x
exact exists_congr fun a ↦ and_congr_right fun ha ↦ by rw [h a ha]
| [
" (∀ y ∈ f '' s, p y) ↔ ∀ ⦃x : α⦄, x ∈ s → p (f x)",
" (∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x)",
" f '' s = g '' s",
" x ∈ f '' s ↔ x ∈ g '' s",
" f a = x ↔ g a = x"
] | [
" (∀ y ∈ f '' s, p y) ↔ ∀ ⦃x : α⦄, x ∈ s → p (f x)",
" (∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x)"
] |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 35 | 35 | theorem hasProd_one : HasProd (fun _ ↦ 1 : β → α) 1 := by | simp [HasProd, tendsto_const_nhds]
| [
" HasProd (fun x => 1) 1"
] | [] |
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ι : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 134 | 153 | theorem Lex.acc_single [DecidableEq ι] {i : ι} (hi : Acc (rᶜ ⊓ (· ≠ ·)) i) :
∀ a, Acc (DFinsupp.Lex r s) (single i a) := by |
induction' hi with i _ ih
refine fun a => WellFounded.induction (hs i)
(C := fun x ↦ Acc (DFinsupp.Lex r s) (single i x)) a fun a ha ↦ ?_
refine Acc.intro _ fun x ↦ ?_
rintro ⟨k, hr, hs⟩
rw [single_apply] at hs
split_ifs at hs with hik
swap
· exact (hbot hs).elim
subst hik
classical
refine ... | [
" Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s) fun x =>\n x.2.1.piecewise x.2.2 x.1",
" ∃ a',\n InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => x.2.1.piecewise x.2.2 x.1) a' = x",
" (x₁.piecewise x {j | r j i}) j... | [
" Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s) fun x =>\n x.2.1.piecewise x.2.2 x.1",
" ∃ a',\n InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧\n (fun x => x.2.1.piecewise x.2.2 x.1) a' = x",
" (x₁.piecewise x {j | r j i}) j... |
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 | 75 | 79 | theorem le_suffixLevenshtein_append_minimum (xs : List α) (ys₁ ys₂) :
(suffixLevenshtein C xs ys₂).1.minimum ≤ (suffixLevenshtein C xs (ys₁ ++ ys₂)).1.minimum := by |
induction ys₁ with
| nil => exact le_refl _
| cons y ys₁ ih => exact ih.trans (le_suffixLevenshtein_cons_minimum _ _ _)
| [
" (↑(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.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 329 | 330 | theorem toDual_apply_left (m : M) (i : ι) : b.toDual m (b i) = b.repr m i := by |
rw [← b.toDual_total_left, b.total_repr]
| [
" (b.toDual (b i)) (b j) = if i = j then 1 else 0",
" (if j = i then 1 else 0) = if i = j then 1 else 0",
" (b.toDual ((Finsupp.total ι M R ⇑b) f)) (b i) = f i",
" ∑ d ∈ f.support, (b.toDual (f d • b d)) (b i) = f i",
" (if i ∈ f.support then f i else 0) = f i",
" f i = f i",
" 0 = f i",
" (b.toDual (... | [
" (b.toDual (b i)) (b j) = if i = j then 1 else 0",
" (if j = i then 1 else 0) = if i = j then 1 else 0",
" (b.toDual ((Finsupp.total ι M R ⇑b) f)) (b i) = f i",
" ∑ d ∈ f.support, (b.toDual (f d • b d)) (b i) = f i",
" (if i ∈ f.support then f i else 0) = f i",
" f i = f i",
" 0 = f i",
" (b.toDual (... |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 49 | 52 | theorem norm_smul_eq (h : SameRay ℝ x y) : ‖x‖ • y = ‖y‖ • x := by |
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
simp only [norm_smul_of_nonneg, *, mul_smul]
rw [smul_comm, smul_comm b, smul_comm a b u]
| [
" ‖x + y‖ = ‖x‖ + ‖y‖",
" ‖a • u + b • u‖ = ‖a • u‖ + ‖b • u‖",
" ‖x - y‖ = |‖x‖ - ‖y‖|",
" ‖a • u - b • u‖ = |‖a • u‖ - ‖b • u‖|",
" ‖b • u - a • u‖ = |‖b • u‖ - ‖a • u‖|",
" ‖x‖ • y = ‖y‖ • x",
" ‖a • u‖ • b • u = ‖b • u‖ • a • u",
" a • ‖u‖ • b • u = b • ‖u‖ • a • u"
] | [
" ‖x + y‖ = ‖x‖ + ‖y‖",
" ‖a • u + b • u‖ = ‖a • u‖ + ‖b • u‖",
" ‖x - y‖ = |‖x‖ - ‖y‖|",
" ‖a • u - b • u‖ = |‖a • u‖ - ‖b • u‖|",
" ‖b • u - a • u‖ = |‖b • u‖ - ‖a • u‖|"
] |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 126 | 127 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ioc]
| [
" x ∈ (fun a b => { val := ↑(List.range' a (b + 1 - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x ≤ b",
" a ≤ x ∧ x < a + (b + 1 - a) ↔ a ≤ x ∧ x ≤ b",
" x ∈ (fun a b => { val := ↑(List.range' a (b - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x < b",
" a ≤ x ∧ x < a + (b - a) ↔ a ≤ x ∧ x < b",
" x ∈ (fun a b => { val := ↑(List.... | [
" x ∈ (fun a b => { val := ↑(List.range' a (b + 1 - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x ≤ b",
" a ≤ x ∧ x < a + (b + 1 - a) ↔ a ≤ x ∧ x ≤ b",
" x ∈ (fun a b => { val := ↑(List.range' a (b - a)), nodup := ⋯ }) a b ↔ a ≤ x ∧ x < b",
" a ≤ x ∧ x < a + (b - a) ↔ a ≤ x ∧ x < b",
" x ∈ (fun a b => { val := ↑(List.... |
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 | 112 | 115 | theorem Path.cast_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v) :
HEq (p.cast hu hv) p := by |
rw [Path.cast_eq_cast]
exact _root_.cast_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'",
" 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",
" cast hu hv e = e' ↔ HEq e e'",
" _root_.cast ⋯ e = e' ↔ HEq e e'",
" e' ... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 473 | 483 | theorem A_mem_nhdsWithin_Ioi {L : F} {r ε x : ℝ} (hx : x ∈ A f L r ε) : A f L r ε ∈ 𝓝[>] x := by |
rcases hx with ⟨r', rr', hr'⟩
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between rr'.1
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le rr'.2)⟩
refine ⟨x + r' - s, by simp only [mem_Ioi]; linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
have... | [
" A f L r ε ∈ 𝓝[>] x",
" ∃ u ∈ Ioi x, Ioo x u ⊆ A f L r ε",
" x + r' - s ∈ Ioi x",
" x < x + r' - s",
" ∀ y ∈ Icc x' (x' + s), ∀ z ∈ Icc x' (x' + s), ‖f z - f y - (z - y) • L‖ ≤ ε * r",
" Icc x' (x' + s) ⊆ Icc x (x + r')",
" x' + s ≤ x + r'",
" ‖f z - f y - (z - y) • L‖ ≤ ε * r"
] | [] |
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
import Mathlib.Data.List.Basic
#align_import data.list.pairwise from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Nat Function
namespace List
variable {α β : Type*} {R S T : α → α → Prop} {a : α} {l : List α}
mk_iff_o... | Mathlib/Data/List/Pairwise.lean | 124 | 133 | theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
Pairwise R (l.pmap f h) ↔
Pairwise (fun b₁ b₂ => ∀ (h₁ : p b₁) (h₂ : p b₂), R (f b₁ h₁) (f b₂ h₂)) l := by |
induction' l with a l ihl
· simp
obtain ⟨_, hl⟩ : p a ∧ ∀ b, b ∈ l → p b := by simpa using h
simp only [ihl hl, pairwise_cons, exists₂_imp, pmap, and_congr_left_iff, mem_pmap]
refine fun _ => ⟨fun H b hb _ hpb => H _ _ hb rfl, ?_⟩
rintro H _ b hb rfl
exact H b hb _ _
| [
" Pairwise (flip fun x => R x) l",
" ∀ ⦃a : α⦄, a ∈ l → ∀ ⦃b : α⦄, b ∈ l → a ≠ b → R a b",
" Symmetric fun x y => x ≠ y → R x y",
" ∀ (x : α), x ∈ l → x ≠ x → R x x",
" Pairwise (fun x y => x ≠ y → R x y) l",
" R a b",
" Pairwise R (map f []) ↔ Pairwise (fun a b => R (f a) (f b)) []",
" Pairwise R (ma... | [
" Pairwise (flip fun x => R x) l",
" ∀ ⦃a : α⦄, a ∈ l → ∀ ⦃b : α⦄, b ∈ l → a ≠ b → R a b",
" Symmetric fun x y => x ≠ y → R x y",
" ∀ (x : α), x ∈ l → x ≠ x → R x x",
" Pairwise (fun x y => x ≠ y → R x y) l",
" R a b",
" Pairwise R (map f []) ↔ Pairwise (fun a b => R (f a) (f b)) []",
" Pairwise R (ma... |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 166 | 166 | theorem one_im : im (1 : K) = 0 := by | rw [← ofReal_one, ofReal_im]
| [
" r • x = ↑r • x",
" re 1 = 1",
" im 1 = 0"
] | [
" r • x = ↑r • x",
" re 1 = 1"
] |
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
variable {Ω E : Type*} {m0 : ... | Mathlib/Probability/Martingale/Centering.lean | 93 | 131 | theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
[SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ := by |
refine ⟨adapted_martingalePart hf, fun i j hij => ?_⟩
-- ⊢ μ[martingalePart f ℱ μ j | ℱ i] =ᵐ[μ] martingalePart f ℱ μ i
have h_eq_sum : μ[martingalePart f ℱ μ j|ℱ i] =ᵐ[μ]
f 0 + ∑ k ∈ Finset.range j, (μ[f (k + 1) - f k|ℱ i] - μ[μ[f (k + 1) - f k|ℱ k]|ℱ i]) := by
rw [martingalePart_eq_sum]
refine (c... | [
" predictablePart f ℱ μ 0 = 0",
" martingalePart f ℱ μ = fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])",
" (fun n => f n - ∑ i ∈ Finset.range n, μ[f (i + 1) - f i|↑ℱ i]) = fun n =>\n f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])",
" f n - ∑ i ∈ F... | [
" predictablePart f ℱ μ 0 = 0",
" martingalePart f ℱ μ = fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])",
" (fun n => f n - ∑ i ∈ Finset.range n, μ[f (i + 1) - f i|↑ℱ i]) = fun n =>\n f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])",
" f n - ∑ i ∈ F... |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 33 | 73 | theorem zipWith_eq_zipWith_data (f : α → β → γ) (as : Array α) (bs : Array β) :
(as.zipWith bs f).data = as.data.zipWith f bs.data := by |
let rec loop : ∀ (i : Nat) cs, i ≤ as.size → i ≤ bs.size →
(zipWithAux f as bs i cs).data = cs.data ++ (as.data.drop i).zipWith f (bs.data.drop i) := by
intro i cs hia hib
unfold zipWithAux
by_cases h : i = as.size ∨ i = bs.size
case pos =>
have : ¬(i < as.size) ∨ ¬(i < bs.size) := by
... | [
" forIn as b f = forIn as.data b f",
" forIn.loop as f 0 x✝¹ x✝ = forIn (List.drop as.data.length as.data) x✝ f",
" forIn.loop as f 0 x✝¹ x✝ = forIn [] x✝ f",
" forIn.loop as f (i + 1) x✝¹ x✝ = forIn (List.drop j as.data) x✝ f",
" (do\n let __do_lift ← f as[as.size - 1 - i] x✝\n match __do_lift wi... | [
" forIn as b f = forIn as.data b f",
" forIn.loop as f 0 x✝¹ x✝ = forIn (List.drop as.data.length as.data) x✝ f",
" forIn.loop as f 0 x✝¹ x✝ = forIn [] x✝ f",
" forIn.loop as f (i + 1) x✝¹ x✝ = forIn (List.drop j as.data) x✝ f",
" (do\n let __do_lift ← f as[as.size - 1 - i] x✝\n match __do_lift wi... |
import Mathlib.Data.Vector.Basic
set_option autoImplicit true
namespace Vector
def snoc : Vector α n → α → Vector α (n+1) :=
fun xs x => append xs (x ::ᵥ Vector.nil)
section Simp
variable (xs : Vector α n)
@[simp]
theorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=
rfl
@[simp]
theorem snoc_nil... | Mathlib/Data/Vector/Snoc.lean | 54 | 62 | theorem replicate_succ_to_snoc (val : α) :
replicate (n+1) val = (replicate n val).snoc val := by |
clear xs
induction n with
| zero => rfl
| succ n ih =>
rw [replicate_succ]
conv => rhs; rw [replicate_succ]
rw [snoc_cons, ih]
| [
" (x ::ᵥ xs).reverse = xs.reverse.snoc x",
" (x ::ᵥ ⟨val✝, property✝⟩).reverse = (reverse ⟨val✝, property✝⟩).snoc x",
" ⟨val✝.reverse ++ [x], ⋯⟩ = append ⟨val✝.reverse, ⋯⟩ ⟨[x], ⋯⟩",
" (xs.snoc x).reverse = x ::ᵥ xs.reverse",
" (snoc ⟨val✝, property✝⟩ x).reverse = x ::ᵥ reverse ⟨val✝, property✝⟩",
" ⟨(app... | [
" (x ::ᵥ xs).reverse = xs.reverse.snoc x",
" (x ::ᵥ ⟨val✝, property✝⟩).reverse = (reverse ⟨val✝, property✝⟩).snoc x",
" ⟨val✝.reverse ++ [x], ⋯⟩ = append ⟨val✝.reverse, ⋯⟩ ⟨[x], ⋯⟩",
" (xs.snoc x).reverse = x ::ᵥ xs.reverse",
" (snoc ⟨val✝, property✝⟩ x).reverse = x ::ᵥ reverse ⟨val✝, property✝⟩",
" ⟨(app... |
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprov... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | 185 | 211 | theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by |
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note: Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
... | [
" IsLocalRingHom ((coequalizer.π f.val g.val).c.app { unop := U })",
" IsLocalRingHom\n ((coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).c.app\n ... | [
" IsLocalRingHom ((coequalizer.π f.val g.val).c.app { unop := U })",
" IsLocalRingHom\n ((coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).c.app\n ... |
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {α : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 226 | 228 | theorem _root_.Filter.HasBasis.vitaliFamily {ι : Sort*} {p : ι → Prop} {s : ι → Set α} {x : α}
(h : (𝓝 x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i ↦ {t ∈ v.setsAt x | t ⊆ s i}) := by |
simpa only [← Set.setOf_inter_eq_sep] using h.smallSets.inf_principal _
| [
" MeasurableSet a",
" (interior a).Nonempty",
" ∀ (x : α),\n ∀ ε > 0,\n ∃ s ∈ (fun x => v.setsAt x ∪ {a | MeasurableSet a ∧ (interior a).Nonempty ∧ ¬a ⊆ closedBall x δ}) x,\n s ⊆ closedBall x ε",
" ∃ s ∈ (fun x => v.setsAt x ∪ {a | MeasurableSet a ∧ (interior a).Nonempty ∧ ¬a ⊆ closedBall x δ})... | [
" MeasurableSet a",
" (interior a).Nonempty",
" ∀ (x : α),\n ∀ ε > 0,\n ∃ s ∈ (fun x => v.setsAt x ∪ {a | MeasurableSet a ∧ (interior a).Nonempty ∧ ¬a ⊆ closedBall x δ}) x,\n s ⊆ closedBall x ε",
" ∃ s ∈ (fun x => v.setsAt x ∪ {a | MeasurableSet a ∧ (interior a).Nonempty ∧ ¬a ⊆ closedBall x δ})... |
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import topology.c... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 127 | 128 | theorem sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i := by | simp [sigmaIsoSigma]
| [
" ∀ (s : Cone (Discrete.functor α)) (m : s.pt ⟶ (piFan α).pt),\n (∀ (j : Discrete ι), m ≫ (piFan α).π.app j = s.π.app j) →\n m = (fun S => { toFun := fun s i => (S.π.app { as := i }) s, continuous_toFun := ⋯ }) s",
" m = (fun S => { toFun := fun s i => (S.π.app { as := i }) s, continuous_toFun := ⋯ }) S",... | [
" ∀ (s : Cone (Discrete.functor α)) (m : s.pt ⟶ (piFan α).pt),\n (∀ (j : Discrete ι), m ≫ (piFan α).π.app j = s.π.app j) →\n m = (fun S => { toFun := fun s i => (S.π.app { as := i }) s, continuous_toFun := ⋯ }) s",
" m = (fun S => { toFun := fun s i => (S.π.app { as := i }) s, continuous_toFun := ⋯ }) S",... |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 172 | 173 | theorem vsub_sub_vsub_cancel_right (p₁ p₂ p₃ : P) : p₁ -ᵥ p₃ - (p₂ -ᵥ p₃) = p₁ -ᵥ p₂ := by |
rw [← vsub_vadd_eq_vsub_sub, vsub_vadd]
| [
" g₁ = g₂",
" g +ᵥ p₁ -ᵥ p₂ = g + (p₁ -ᵥ p₂)",
" g +ᵥ p₁ -ᵥ p₂ +ᵥ p₂ = g + (p₁ -ᵥ p₂) +ᵥ p₂",
" p -ᵥ p = 0",
" p₁ = p₂",
" p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) = p₁ -ᵥ p₃",
" p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) +ᵥ p₃ = p₁ -ᵥ p₃ +ᵥ p₃",
" -(p₁ -ᵥ p₂) = p₂ -ᵥ p₁",
" p₁ -ᵥ p₂ + (p₂ -ᵥ p₁) +ᵥ p₁ = 0 +ᵥ p₁",
" g +ᵥ p -ᵥ q = g - (q... | [
" g₁ = g₂",
" g +ᵥ p₁ -ᵥ p₂ = g + (p₁ -ᵥ p₂)",
" g +ᵥ p₁ -ᵥ p₂ +ᵥ p₂ = g + (p₁ -ᵥ p₂) +ᵥ p₂",
" p -ᵥ p = 0",
" p₁ = p₂",
" p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) = p₁ -ᵥ p₃",
" p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) +ᵥ p₃ = p₁ -ᵥ p₃ +ᵥ p₃",
" -(p₁ -ᵥ p₂) = p₂ -ᵥ p₁",
" p₁ -ᵥ p₂ + (p₂ -ᵥ p₁) +ᵥ p₁ = 0 +ᵥ p₁",
" g +ᵥ p -ᵥ q = g - (q... |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 133 | 135 | theorem apply_of_ne (h : ¬(i = i' ∧ j = j')) : stdBasisMatrix i j c i' j' = 0 := by |
simp only [stdBasisMatrix, and_imp, ite_eq_right_iff]
tauto
| [
" r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a)",
" (r • fun i' j' => if i = i' ∧ j = j' then a else 0) = fun i' j' => if i = i' ∧ j = j' then r • a else 0",
" (r • fun i' j' => if i = i' ∧ j = j' then a else 0) i✝ j✝ = if i = i✝ ∧ j = j✝ then r • a else 0",
" stdBasisMatrix i j 0 = 0",
" (fun i' j'... | [
" r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a)",
" (r • fun i' j' => if i = i' ∧ j = j' then a else 0) = fun i' j' => if i = i' ∧ j = j' then r • a else 0",
" (r • fun i' j' => if i = i' ∧ j = j' then a else 0) i✝ j✝ = if i = i✝ ∧ j = j✝ then r • a else 0",
" stdBasisMatrix i j 0 = 0",
" (fun i' j'... |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 112 | 113 | theorem rdropWhile_concat_pos (x : α) (h : p x) : rdropWhile p (l ++ [x]) = rdropWhile p l := by |
rw [rdropWhile_concat, if_pos h]
| [
" [].rdrop n = []",
" l.rdrop 0 = l",
" l.rdrop n = (drop n l.reverse).reverse",
" take (l.length - n) l = (drop n l.reverse).reverse",
" take ([].length - n) [] = (drop n [].reverse).reverse",
" take ((xs ++ [x]).length - n) (xs ++ [x]) = (drop n (xs ++ [x]).reverse).reverse",
" take ((xs ++ [x]).lengt... | [
" [].rdrop n = []",
" l.rdrop 0 = l",
" l.rdrop n = (drop n l.reverse).reverse",
" take (l.length - n) l = (drop n l.reverse).reverse",
" take ([].length - n) [] = (drop n [].reverse).reverse",
" take ((xs ++ [x]).length - n) (xs ++ [x]) = (drop n (xs ++ [x]).reverse).reverse",
" take ((xs ++ [x]).lengt... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 119 | 125 | theorem angle_eq_iff_two_pi_dvd_sub {ψ θ : ℝ} : (θ : Angle) = ψ ↔ ∃ k : ℤ, θ - ψ = 2 * π * k := by |
simp only [QuotientAddGroup.eq, AddSubgroup.zmultiples_eq_closure,
AddSubgroup.mem_closure_singleton, zsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm]
-- Porting note: added `rw`, `simp [Angle.coe, QuotientAddGroup.eq]` doesn't fire otherwise
rw [Angle.coe, Angle.coe, QuotientAddGroup.eq]
simp only [AddS... | [
" 0 < 2 * π",
" ↑(↑n * x) = n • ↑x",
" ↑θ = ↑ψ ↔ ∃ k, θ - ψ = 2 * π * ↑k",
" ↑ψ = ↑θ ↔ ∃ k, θ - ψ = 2 * π * ↑k",
" -ψ + θ ∈ AddSubgroup.zmultiples (2 * π) ↔ ∃ k, θ - ψ = 2 * π * ↑k"
] | [
" 0 < 2 * π",
" ↑(↑n * x) = n • ↑x"
] |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 108 | 111 | theorem to_continuousMap_injective {f g : P →ᴬ[R] Q} (h : (f : C(P, Q)) = (g : C(P, Q))) :
f = g := by |
ext a
exact ContinuousMap.congr_fun h a
| [
" f = g",
" { toAffineMap := toAffineMap✝, cont := cont✝ } = g",
" { toAffineMap := toAffineMap✝¹, cont := cont✝¹ } = { toAffineMap := toAffineMap✝, cont := cont✝ }",
" f a = g a"
] | [
" f = g",
" { toAffineMap := toAffineMap✝, cont := cont✝ } = g",
" { toAffineMap := toAffineMap✝¹, cont := cont✝¹ } = { toAffineMap := toAffineMap✝, cont := cont✝ }"
] |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 79 | 81 | theorem edist_le_of_fixedPoint (hf : ContractingWith K f) {x y} (h : edist x y ≠ ∞)
(hy : IsFixedPt f y) : edist x y ≤ edist x (f x) / (1 - K) := by |
simpa only [hy.eq, edist_self, add_zero] using hf.edist_inequality h
| [
" 0 < 1 - ↑K",
" 1 - ↑K ≠ ⊤",
" ¬1 - ↑K = ⊤",
" edist x (f x) + edist (f x) (f y) + edist (f y) y = edist x (f x) + edist y (f y) + edist (f x) (f y)",
" edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - ↑K)",
" edist x y ≤ edist x (f x) / (1 - ↑K)"
] | [
" 0 < 1 - ↑K",
" 1 - ↑K ≠ ⊤",
" ¬1 - ↑K = ⊤",
" edist x (f x) + edist (f x) (f y) + edist (f y) y = edist x (f x) + edist y (f y) + edist (f x) (f y)",
" edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - ↑K)"
] |
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