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.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 111 | 194 | theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
(hf2 : HasCompactSupport f) :
Tendsto (fun w : V => ∫ v : V, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) := by |
refine NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => ?_
suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, 𝐞 (-⟪v, w⟫) • f v‖ < ε by
simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
dist_eq_norm', sub_zero]
exact
let ⟨T, hT⟩ := this
⟨T, fun b ... | [
" ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v",
" ⟪i w, w⟫_ℝ = 1 / 2",
" ‖w‖ ^ 2 ≠ 0",
" (fun v => 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w)) = fun v => (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" cexp (... | [
" ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v",
" ⟪i w, w⟫_ℝ = 1 / 2",
" ‖w‖ ^ 2 ≠ 0",
" (fun v => 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w)) = fun v => (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" cexp (... |
import Batteries.Data.Nat.Gcd
import Batteries.Data.Int.DivMod
import Batteries.Lean.Float
-- `Rat` is not tagged with the `ext` attribute, since this is more often than not undesirable
structure Rat where
mk' ::
num : Int
den : Nat := 1
den_nz : den ≠ 0 := by decide
reduced : num.natAbs.C... | .lake/packages/batteries/Batteries/Data/Rat/Basic.lean | 60 | 66 | theorem Rat.normalize.reduced {num : Int} {den g : Nat} (den_nz : den ≠ 0)
(e : g = num.natAbs.gcd den) : (num.div g).natAbs.Coprime (den / g) :=
have : Int.natAbs (num.div ↑g) = num.natAbs / g := by |
match num, num.eq_nat_or_neg with
| _, ⟨_, .inl rfl⟩ => rfl
| _, ⟨_, .inr rfl⟩ => rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg]; rfl
this ▸ e ▸ Nat.coprime_div_gcd_div_gcd (Nat.gcd_pos_of_pos_right _ (Nat.pos_of_ne_zero den_nz))
| [
" den ≠ 0",
" num.natAbs.Coprime den",
" (num.div ↑g).natAbs = num.natAbs / g",
" ((↑w✝).div ↑g).natAbs = (↑w✝).natAbs / g",
" ((-↑w✝).div ↑g).natAbs = (-↑w✝).natAbs / g"
] | [
" den ≠ 0",
" num.natAbs.Coprime den"
] |
import Mathlib.Geometry.Manifold.PartitionOfUnity
import Mathlib.Geometry.Manifold.Metrizable
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
open MeasureTheory Filter Metric Function Set TopologicalSpace
open scoped Topology Manifold
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimen... | Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean | 41 | 112 | theorem ae_eq_zero_of_integral_smooth_smul_eq_zero (hf : LocallyIntegrable f μ)
(h : ∀ g : M → ℝ, Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
∀ᵐ x ∂μ, f x = 0 := by |
-- record topological properties of `M`
have := I.locallyCompactSpace
have := ChartedSpace.locallyCompactSpace H M
have := I.secondCountableTopology
have := ChartedSpace.secondCountable_of_sigma_compact H M
have := ManifoldWithCorners.metrizableSpace I M
let _ : MetricSpace M := TopologicalSpace.metrizab... | [
" ∀ᵐ (x : M) ∂μ, f x = 0",
" 0 = ∫ (x : M) in s, f x ∂μ",
" ∀ (n : ℕ), ∃ g, support g = v n ∧ Smooth I 𝓘(ℝ, ℝ) g ∧ range g ⊆ Icc 0 1 ∧ ∀ x ∈ s, g x = 1",
" ∃ g, support g = v n ∧ Smooth I 𝓘(ℝ, ℝ) g ∧ range g ⊆ Icc 0 1 ∧ ∀ x ∈ s, g x = 1",
" Tendsto (fun n => ∫ (x : M), g n x • f x ∂μ) atTop (𝓝 (∫ (x : M)... | [] |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 56 | 57 | theorem log_pos_iff {b n : ℕ} : 0 < log b n ↔ b ≤ n ∧ 1 < b := by |
rw [Nat.pos_iff_ne_zero, Ne, log_eq_zero_iff, not_or, not_lt, not_le]
| [
" (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 (n / b) a✝",
" b.log n = 0 ↔ n < b ∨ b ≤ 1",
" (∀ (h : b ≤ n ∧ 1 < b), b.log (n / b) + 1 = 0) ↔ n < b ∨ b ≤ 1",
" 0 < b.log n ↔ b ≤ n ∧ 1 < b"
] | [
" (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 (n / b) a✝",
" b.log n = 0 ↔ n < b ∨ b ≤ 1",
" (∀ (h : b ≤ n ∧ 1 < b), b.log (n / b) + 1 = 0) ↔ n < b ∨ b ≤ 1"
] |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 41 | 42 | theorem mem_nhdsSet_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ x : X, x ∈ t → s ∈ 𝓝 x := by |
simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image]
| [
" 𝓝ˢ (diagonal X) = ⨆ x, 𝓝 (x, x)",
" sSup (range (𝓝 ∘ fun x => (x, x))) = ⨆ x, 𝓝 (x, x)",
" s ∈ 𝓝ˢ t ↔ ∀ x ∈ t, s ∈ 𝓝 x"
] | [
" 𝓝ˢ (diagonal X) = ⨆ x, 𝓝 (x, x)",
" sSup (range (𝓝 ∘ fun x => (x, x))) = ⨆ x, 𝓝 (x, x)"
] |
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 62 | 64 | theorem mem_iff_one_sub_mem {t : ℝ} : t ∈ I ↔ 1 - t ∈ I := by |
rw [mem_Icc, mem_Icc]
constructor <;> intro <;> constructor <;> linarith
| [
" t ∈ I ↔ 1 - t ∈ I",
" 0 ≤ t ∧ t ≤ 1 ↔ 0 ≤ 1 - t ∧ 1 - t ≤ 1",
" 0 ≤ t ∧ t ≤ 1 → 0 ≤ 1 - t ∧ 1 - t ≤ 1",
" 0 ≤ 1 - t ∧ 1 - t ≤ 1 → 0 ≤ t ∧ t ≤ 1",
" 0 ≤ 1 - t ∧ 1 - t ≤ 1",
" 0 ≤ t ∧ t ≤ 1",
" 0 ≤ 1 - t",
" 1 - t ≤ 1",
" 0 ≤ t",
" t ≤ 1"
] | [] |
import Mathlib.Mathport.Rename
#align_import init.data.list.instances from "leanprover-community/lean"@"9af482290ef68e8aaa5ead01aa7b09b7be7019fd"
universe u v w
namespace List
variable {α : Type u} {β : Type v} {γ : Type w}
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem bind_singleton (f : α →... | Mathlib/Init/Data/List/Instances.lean | 35 | 36 | theorem bind_assoc {α β} (l : List α) (f : α → List β) (g : β → List γ) :
(l.bind f).bind g = l.bind fun x => (f x).bind g := by | induction l <;> simp [*]
| [
" (l.bind fun x => [x]) = l",
" ([].bind fun x => [x]) = []",
" ((head✝ :: tail✝).bind fun x => [x]) = head✝ :: tail✝",
" map f l = l.bind fun x => [f x]",
" map f l = l.bind fun x => map f [x]",
" (l.bind f).bind g = l.bind fun x => (f x).bind g",
" ([].bind f).bind g = [].bind fun x => (f x).bind g",
... | [
" (l.bind fun x => [x]) = l",
" ([].bind fun x => [x]) = []",
" ((head✝ :: tail✝).bind fun x => [x]) = head✝ :: tail✝",
" map f l = l.bind fun x => [f x]",
" map f l = l.bind fun x => map f [x]"
] |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 140 | 143 | theorem EuclideanSpace.closedBall_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) :
Metric.closedBall (0 : EuclideanSpace ℝ n) r = {x | ∑ i, x i ^ 2 ≤ r ^ 2} := by |
ext
simp_rw [mem_setOf, mem_closedBall_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_le_left hr]
| [
" ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜",
" ∀ (x y : PiLp 2 f), (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ({ inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n { inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i)... | [
" ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜",
" ∀ (x y : PiLp 2 f), (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜",
" (starRingEnd 𝕜) ({ inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n { inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i)... |
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 | 67 | 67 | theorem Ico_zero_eq_range : Ico 0 = range := by | rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
| [
" 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.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 | 112 | 113 | theorem intCast_mul_eq_zsmul (x : ℝ) (n : ℤ) : ↑((n : ℝ) * x : ℝ) = n • (↑x : Angle) := by |
simpa only [zsmul_eq_mul] using coeHom.map_zsmul x n
| [
" 0 < 2 * π",
" ↑(↑n * x) = n • ↑x"
] | [
" 0 < 2 * π",
" ↑(↑n * x) = n • ↑x"
] |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 63 | 65 | theorem tsub_add_tsub_cancel (hab : b ≤ a) (hcb : c ≤ b) : a - b + (b - c) = a - c := by |
convert tsub_add_cancel_of_le (tsub_le_tsub_right hab c) using 2
rw [tsub_tsub, add_tsub_cancel_of_le hcb]
| [
" a + (b - a) = b",
" a + (b - a) ≤ b",
" a + (a + c - a) ≤ a + c",
" b - a + a = b",
" a - c ≤ b - c ↔ a ≤ b",
" a - c = b - c ↔ a = b",
" a < b",
" a ≠ b",
" False",
" a - b + (b - c) = a - c",
" a - b = a - c - (b - c)"
] | [
" a + (b - a) = b",
" a + (b - a) ≤ b",
" a + (a + c - a) ≤ a + c",
" b - a + a = b",
" a - c ≤ b - c ↔ a ≤ b",
" a - c = b - c ↔ a = b",
" a < b",
" a ≠ b",
" False"
] |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 78 | 79 | theorem sinh_arsinh (x : ℝ) : sinh (arsinh x) = x := by |
rw [sinh_eq, ← arsinh_neg, exp_arsinh, exp_arsinh, neg_sq]; field_simp
| [
" rexp x.arsinh = x + √(1 + x ^ 2)",
" 0 < x + √(1 + x ^ 2)",
" -x < √(1 + x ^ 2)",
" (-x) ^ 2 < 1 + x ^ 2",
" arsinh 0 = 0",
" (-x).arsinh = -x.arsinh",
" -x + √(1 + (-x) ^ 2) = (x + √(1 + x ^ 2))⁻¹",
" (-x + √(1 + (-x) ^ 2)) * (x + √(1 + x ^ 2)) = 1",
" 0 ≤ 1 + x ^ 2",
" x.arsinh.sinh = x",
" ... | [
" rexp x.arsinh = x + √(1 + x ^ 2)",
" 0 < x + √(1 + x ^ 2)",
" -x < √(1 + x ^ 2)",
" (-x) ^ 2 < 1 + x ^ 2",
" arsinh 0 = 0",
" (-x).arsinh = -x.arsinh",
" -x + √(1 + (-x) ^ 2) = (x + √(1 + x ^ 2))⁻¹",
" (-x + √(1 + (-x) ^ 2)) * (x + √(1 + x ^ 2)) = 1",
" 0 ≤ 1 + x ^ 2"
] |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 60 | 60 | theorem exp_mul (x y : ℝ) : exp (x * y) = exp x ^ y := by | rw [rpow_def_of_pos (exp_pos _), log_exp]
| [
" x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" (if ↑x = 0 then if ↑y = 0 then 1 else 0 else ((↑x).log * ↑y).exp).re =\n if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" Complex.re 1 = 1",
" Complex.re 1 = 0",
" Complex.re 1 = rexp (x.log * y)",
" Complex.re 0 = 1... | [
" x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" (if ↑x = 0 then if ↑y = 0 then 1 else 0 else ((↑x).log * ↑y).exp).re =\n if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" Complex.re 1 = 1",
" Complex.re 1 = 0",
" Complex.re 1 = rexp (x.log * y)",
" Complex.re 0 = 1... |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 106 | 111 | theorem hasBasis_nhds_adic (I : Ideal R) (x : R) :
HasBasis (@nhds R I.adicTopology x) (fun _n : ℕ => True) fun n =>
(fun y => x + y) '' (I ^ n : Ideal R) := by |
letI := I.adicTopology
have := I.hasBasis_nhds_zero_adic.map fun y => x + y
rwa [map_add_left_nhds_zero x] at this
| [
" ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
" ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
" ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
" ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j • ⊤ ≤ I ^ i • ⊤",
" ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j ≤ I ^ i",
" ∃ j, r • I ^ j ≤ I ^ n",
" r • I ^ n ≤ I ^ n",
" (Dis... | [
" ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
" ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
" ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j",
" ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j • ⊤ ≤ I ^ i • ⊤",
" ∀ (a : R) (i : ℕ), ∃ j, a • I ^ j ≤ I ^ i",
" ∃ j, r • I ^ j ≤ I ^ n",
" r • I ^ n ≤ I ^ n",
" (Dis... |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 163 | 205 | theorem affineLocally_iff_affineOpens_le
(hP : RingHom.RespectsIso @P) {X Y : Scheme.{u}} (f : X ⟶ Y) :
affineLocally.{u} (@P) f ↔
∀ (U : Y.affineOpens) (V : X.affineOpens) (e : V.1 ≤ (Opens.map f.1.base).obj U.1),
P (Scheme.Hom.appLe f e) := by |
apply forall_congr'
intro U
delta sourceAffineLocally
simp_rw [op_comp, Scheme.Γ.map_comp, Γ_map_morphismRestrict, Category.assoc, Scheme.Γ_map_op,
hP.cancel_left_isIso (Y.presheaf.map (eqToHom _).op)]
constructor
· intro H V e
let U' := (Opens.map f.val.base).obj U.1
have e'' : (Scheme.Hom.ope... | [
" (sourceAffineLocally P).toProperty.RespectsIso",
" ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z],\n sourceAffineLocally P f → sourceAffineLocally P (e.hom ≫ f)",
" P (Scheme.Γ.map (X.ofRestrict ⋯ ≫ e.hom ≫ f).op)",
" P (Scheme.Γ.map ((Scheme.restrictMapIso e.inv ↑U).hom ≫ X.ofRestrict ⋯ ... | [
" (sourceAffineLocally P).toProperty.RespectsIso",
" ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z],\n sourceAffineLocally P f → sourceAffineLocally P (e.hom ≫ f)",
" P (Scheme.Γ.map (X.ofRestrict ⋯ ≫ e.hom ≫ f).op)",
" P (Scheme.Γ.map ((Scheme.restrictMapIso e.inv ↑U).hom ≫ X.ofRestrict ⋯ ... |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 194 | 198 | theorem ofComplex_comp_toComplex : ofComplex.comp toComplex = AlgHom.id ℝ (CliffordAlgebra Q) := by |
ext
dsimp only [LinearMap.comp_apply, Subtype.coe_mk, AlgHom.id_apply, AlgHom.toLinearMap_apply,
AlgHom.comp_apply]
rw [toComplex_ι, one_smul, ofComplex_I]
| [
" (LinearMap.toSpanSingleton ℝ ℂ Complex.I) r * (LinearMap.toSpanSingleton ℝ ℂ Complex.I) r = (algebraMap ℝ ℂ) (Q r)",
" ↑r * Complex.I * (↑r * Complex.I) = ↑(-(r * r))",
" ↑r * ↑r * (Complex.I * Complex.I) = ↑(-(r * r))",
" toComplex (involute c) = (starRingEnd ℂ) (toComplex c)",
" toComplex (involute ((ι ... | [
" (LinearMap.toSpanSingleton ℝ ℂ Complex.I) r * (LinearMap.toSpanSingleton ℝ ℂ Complex.I) r = (algebraMap ℝ ℂ) (Q r)",
" ↑r * Complex.I * (↑r * Complex.I) = ↑(-(r * r))",
" ↑r * ↑r * (Complex.I * Complex.I) = ↑(-(r * r))",
" toComplex (involute c) = (starRingEnd ℂ) (toComplex c)",
" toComplex (involute ((ι ... |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
namespace NonUnitalSubalgebra
| Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 145 | 157 | theorem _root_.AlgHomClass.unitization_injective' {F R S A : Type*} [CommRing R] [Ring A]
[Algebra R A] [SetLike S A] [hSA : NonUnitalSubringClass S A] [hSRA : SMulMemClass S R A]
(s : S) (h : ∀ r, r ≠ 0 → algebraMap R A r ∉ s)
[FunLike F (Unitization R s) A] [AlgHomClass F R (Unitization R s) A]
(f : F... |
refine (injective_iff_map_eq_zero f).mpr fun x hx => ?_
induction' x with r a
simp_rw [map_add, hf, ← Unitization.algebraMap_eq_inl, AlgHomClass.commutes] at hx
rw [add_eq_zero_iff_eq_neg] at hx ⊢
by_cases hr : r = 0
· ext <;> simp [hr] at hx ⊢
exact hx
· exact (h r hr <| hx ▸ (neg_mem a.property)).e... | [
" Function.Injective ⇑f",
" x = 0",
" Unitization.inl r + ↑a = 0",
" Unitization.inl r = -↑a",
" (Unitization.inl r).fst = (-↑a).fst",
" ↑(Unitization.inl r).snd = ↑(-↑a).snd",
" a = 0"
] | [] |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 358 | 360 | theorem toQuaternion_comp_ofQuaternion :
toQuaternion.comp ofQuaternion = AlgHom.id R ℍ[R,c₁,c₂] := by |
ext : 1 <;> simp
| [
" (ι (Q c₁ c₂)) (1, 0) * (ι (Q c₁ c₂)) (1, 0) = c₁ • 1",
" (algebraMap R (CliffordAlgebra (Q c₁ c₂))) (c₁ * ((1, 0).1 * (1, 0).1) + c₂ * ((1, 0).2 * (1, 0).2)) =\n (algebraMap R (CliffordAlgebra (Q c₁ c₂))) c₁",
" (ι (Q c₁ c₂)) (0, 1) * (ι (Q c₁ c₂)) (0, 1) = c₂ • 1",
" (algebraMap R (CliffordAlgebra (Q c₁... | [
" (ι (Q c₁ c₂)) (1, 0) * (ι (Q c₁ c₂)) (1, 0) = c₁ • 1",
" (algebraMap R (CliffordAlgebra (Q c₁ c₂))) (c₁ * ((1, 0).1 * (1, 0).1) + c₂ * ((1, 0).2 * (1, 0).2)) =\n (algebraMap R (CliffordAlgebra (Q c₁ c₂))) c₁",
" (ι (Q c₁ c₂)) (0, 1) * (ι (Q c₁ c₂)) (0, 1) = c₂ • 1",
" (algebraMap R (CliffordAlgebra (Q c₁... |
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Convex.Gauge
#align_import analysis.locally_convex.abs_convex from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open NormedField Set
open NNReal Pointwis... | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 65 | 74 | theorem nhds_basis_abs_convex_open :
(𝓝 (0 : E)).HasBasis (fun s => (0 : E) ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s) id := by |
refine (nhds_basis_abs_convex 𝕜 E).to_hasBasis ?_ ?_
· rintro s ⟨hs_nhds, hs_balanced, hs_convex⟩
refine ⟨interior s, ?_, interior_subset⟩
exact
⟨mem_interior_iff_mem_nhds.mpr hs_nhds, isOpen_interior,
hs_balanced.interior (mem_interior_iff_mem_nhds.mpr hs_nhds), hs_convex.interior⟩
rintro... | [
" (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Balanced 𝕜 s ∧ Convex ℝ s) id",
" ∃ i', (i' ∈ 𝓝 0 ∧ Balanced 𝕜 i' ∧ Convex ℝ i') ∧ id i' ⊆ id s",
" (convexHull ℝ) (balancedCore 𝕜 s) ∈ 𝓝 0 ∧\n Balanced 𝕜 ((convexHull ℝ) (balancedCore 𝕜 s)) ∧ Convex ℝ ((convexHull ℝ) (balancedCore 𝕜 s))",
" Balanced 𝕜 ((conv... | [
" (𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ Balanced 𝕜 s ∧ Convex ℝ s) id",
" ∃ i', (i' ∈ 𝓝 0 ∧ Balanced 𝕜 i' ∧ Convex ℝ i') ∧ id i' ⊆ id s",
" (convexHull ℝ) (balancedCore 𝕜 s) ∈ 𝓝 0 ∧\n Balanced 𝕜 ((convexHull ℝ) (balancedCore 𝕜 s)) ∧ Convex ℝ ((convexHull ℝ) (balancedCore 𝕜 s))",
" Balanced 𝕜 ((conv... |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 124 | 134 | theorem int_valuation_lt_one_iff_dvd (r : R) :
v.intValuationDef r < 1 ↔ v.asIdeal ∣ Ideal.span {r} := by |
rw [intValuationDef]
split_ifs with hr
· simp [hr]
· rw [← WithZero.coe_one, ← ofAdd_zero, WithZero.coe_lt_coe, ofAdd_lt, neg_lt_zero, ←
Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
exact h... | [
" v.intValuationDef x ≠ 0",
" ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) ≠ 0",
" 0 < v.intValuationDef ↑x",
" 0 < ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑x})).factors)))",
" v.intValuationDef x ≤ 1",
" (if x = 0 then 0 else ↑(of... | [
" v.intValuationDef x ≠ 0",
" ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) ≠ 0",
" 0 < v.intValuationDef ↑x",
" 0 < ↑(ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑x})).factors)))",
" v.intValuationDef x ≤ 1",
" (if x = 0 then 0 else ↑(of... |
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection
#align_import order.modular_lattice from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set
variable {α : Type*}
class IsWeakUpperModularLattice (α : Type*) [Lattice α] : Prop ... | Mathlib/Order/ModularLattice.lean | 216 | 217 | theorem inf_sup_assoc_of_le {x : α} (y : α) {z : α} (h : z ≤ x) : x ⊓ y ⊔ z = x ⊓ (y ⊔ z) := by |
rw [inf_comm, sup_comm, ← sup_inf_assoc_of_le y h, inf_comm, sup_comm]
| [
" x ⊓ y ⊔ z = x ⊓ (y ⊔ z)"
] | [] |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 90 | 98 | theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) := by |
constructor
· exact fun H i => H.inter (U i).2
· intro H
have : ⋃ i, (U i : Set β) = Set.univ := by
convert congr_arg (SetLike.coe) hU
simp
rw [← s.inter_univ, ← this, Set.inter_iUnion]
exact isOpen_iUnion H
| [
" Inducing (s.restrictPreimage f)",
" ∀ (x : ↑(f ⁻¹' s)), 𝓝 x = comap Subtype.val (comap f (𝓝 (f ↑x)))",
" 𝓝 a = comap Subtype.val (comap f (𝓝 (f ↑a)))",
" IsClosedMap (s.restrictPreimage f)",
" IsClosed t → IsClosed (s.restrictPreimage f '' t)",
" ∀ (u : Set α), IsClosed u → Subtype.val ⁻¹' u = t → ∃... | [
" Inducing (s.restrictPreimage f)",
" ∀ (x : ↑(f ⁻¹' s)), 𝓝 x = comap Subtype.val (comap f (𝓝 (f ↑x)))",
" 𝓝 a = comap Subtype.val (comap f (𝓝 (f ↑a)))",
" IsClosedMap (s.restrictPreimage f)",
" IsClosed t → IsClosed (s.restrictPreimage f '' t)",
" ∀ (u : Set α), IsClosed u → Subtype.val ⁻¹' u = t → ∃... |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.infinite_sum.ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filter Finset Function
open... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 208 | 213 | theorem summable_sum_mul_antidiagonal_of_summable_mul
(h : Summable fun x : A × A ↦ f x.1 * g x.2) :
Summable fun n ↦ ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 := by |
rw [summable_mul_prod_iff_summable_mul_sigma_antidiagonal] at h
conv => congr; ext; rw [← Finset.sum_finset_coe, ← tsum_fintype]
exact h.sigma' fun n ↦ (hasSum_fintype _).summable
| [
" HasSum (fun i => a₂ * f i) (a₂ * a₁)",
" HasSum (fun i => f i * a₂) (a₁ * a₂)",
" Summable fun n => ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2",
"ι : Type u_1\nκ : Type u_2\nR : Type u_3\nα : Type u_4\ninst✝⁸ : NonUnitalNonAssocSemiring α\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSemiring α\nf✝ g✝ : ι →... | [
" HasSum (fun i => a₂ * f i) (a₂ * a₁)",
" HasSum (fun i => f i * a₂) (a₁ * a₂)"
] |
import Mathlib.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adhesive
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.subsheaf from "leanprover-community/mathl... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 146 | 149 | theorem Subpresheaf.lift_ι (f : F' ⟶ F) (hf : ∀ U x, f.app U x ∈ G.obj U) :
G.lift f hf ≫ G.ι = f := by |
ext
rfl
| [
" x ∈ F.map x✝¹ ⁻¹' (fun U => ⊤) V",
" { obj := fun U => ↑(G.obj U), map := fun U V i x => ⟨F.map i ↑x, ⋯⟩ }.map (𝟙 X) =\n 𝟙 ({ obj := fun U => ↑(G.obj U), map := fun U V i x => ⟨F.map i ↑x, ⋯⟩ }.obj X)",
" ↑({ obj := fun U => ↑(G.obj U), map := fun U V i x => ⟨F.map i ↑x, ⋯⟩ }.map (𝟙 X) ⟨x, property✝⟩) =... | [
" x ∈ F.map x✝¹ ⁻¹' (fun U => ⊤) V",
" { obj := fun U => ↑(G.obj U), map := fun U V i x => ⟨F.map i ↑x, ⋯⟩ }.map (𝟙 X) =\n 𝟙 ({ obj := fun U => ↑(G.obj U), map := fun U V i x => ⟨F.map i ↑x, ⋯⟩ }.obj X)",
" ↑({ obj := fun U => ↑(G.obj U), map := fun U V i x => ⟨F.map i ↑x, ⋯⟩ }.map (𝟙 X) ⟨x, property✝⟩) =... |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 55 | 56 | theorem exp_log_eq_abs (hx : x ≠ 0) : exp (log x) = |x| := by |
rw [log_of_ne_zero hx, ← coe_expOrderIso_apply, OrderIso.apply_symm_apply, Subtype.coe_mk]
| [
" x.log = expOrderIso.symm ⟨x, hx⟩",
" expOrderIso.symm ⟨|x|, ⋯⟩ = expOrderIso.symm ⟨x, hx⟩",
" |x| = x",
" rexp x.log = |x|"
] | [
" x.log = expOrderIso.symm ⟨x, hx⟩",
" expOrderIso.symm ⟨|x|, ⋯⟩ = expOrderIso.symm ⟨x, hx⟩",
" |x| = x"
] |
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 163 | 165 | theorem trace_comp_trace [Algebra K T] [Algebra L T] [IsScalarTower K L T] [FiniteDimensional K L]
[FiniteDimensional L T] : (trace K L).comp ((trace L T).restrictScalars K) = trace K T := by |
ext; rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace]
| [
" trace R S = 0",
" (trace R S) s = 0 s",
" (trace R S) s = ((leftMulMatrix b) s).trace",
" ((toMatrix b b) ((lmul R S) s)).trace = ((toMatrix b b) (mulLeft R s)).trace",
" (trace R S) ((algebraMap R S) x) = Fintype.card ι • x",
" ∑ i : ι, ((toMatrix b b) ((lmul R S) ((algebraMap R S) x))).diag i = Fintyp... | [
" trace R S = 0",
" (trace R S) s = 0 s",
" (trace R S) s = ((leftMulMatrix b) s).trace",
" ((toMatrix b b) ((lmul R S) s)).trace = ((toMatrix b b) (mulLeft R s)).trace",
" (trace R S) ((algebraMap R S) x) = Fintype.card ι • x",
" ∑ i : ι, ((toMatrix b b) ((lmul R S) ((algebraMap R S) x))).diag i = Fintyp... |
import Mathlib.Topology.Separation
#align_import topology.extend_from from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
noncomputable section
open Topology
open Filter Set
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
def extendFrom (A : Set X) (f : X → Y) : X ... | Mathlib/Topology/ExtendFrom.lean | 63 | 81 | theorem continuousOn_extendFrom [RegularSpace Y] {f : X → Y} {A B : Set X} (hB : B ⊆ closure A)
(hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : ContinuousOn (extendFrom A f) B := by |
set φ := extendFrom A f
intro x x_in
suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝[B] x by
simpa [ContinuousWithinAt, (closed_nhds_basis (φ x)).tendsto_right_iff]
intro V' V'_in V'_closed
obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ V ∩ A ⊆ f ⁻¹' V' := by
have := tendsto_extendFrom (hf... | [
" ContinuousOn (extendFrom A f) B",
" ContinuousOn φ B",
" ContinuousWithinAt φ B x",
" ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝[B] x",
" φ ⁻¹' V' ∈ 𝓝[B] x",
" ∃ V ∈ 𝓝 x, IsOpen V ∧ V ∩ A ⊆ f ⁻¹' V'",
" ∀ y ∈ V ∩ B, φ y ∈ V'",
" φ y ∈ V'",
" V ∩ A ∈ 𝓝[A] y"
] | [] |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : ℕ → Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 113 | 119 | theorem image_nth_Iio_card (hf : (setOf p).Finite) : nth p '' Set.Iio hf.toFinset.card = setOf p :=
calc
nth p '' Set.Iio hf.toFinset.card = Set.range (hf.toFinset.orderEmbOfFin rfl) := by |
ext x
simp only [Set.mem_image, Set.mem_range, Fin.exists_iff, ← nth_eq_orderEmbOfFin hf,
Set.mem_Iio, exists_prop]
_ = setOf p := by rw [range_orderEmbOfFin, Set.Finite.coe_toFinset]
| [
" ℕ",
" nth p n = 0",
" (sort (fun x x_1 => x ≤ x_1) hf.toFinset).length ≤ n",
" nth p n = (hf.toFinset.orderEmbOfFin ⋯) ⟨n, hn⟩",
" StrictMonoOn (nth p) (Set.Iio hf.toFinset.card)",
" nth p m < nth p n",
" (⋯.toFinset.orderEmbOfFin ⋯) ⟨m, ⋯⟩ < (⋯.toFinset.orderEmbOfFin ⋯) ⟨n, ⋯⟩",
" Set.range (nth p)... | [
" ℕ",
" nth p n = 0",
" (sort (fun x x_1 => x ≤ x_1) hf.toFinset).length ≤ n",
" nth p n = (hf.toFinset.orderEmbOfFin ⋯) ⟨n, hn⟩",
" StrictMonoOn (nth p) (Set.Iio hf.toFinset.card)",
" nth p m < nth p n",
" (⋯.toFinset.orderEmbOfFin ⋯) ⟨m, ⋯⟩ < (⋯.toFinset.orderEmbOfFin ⋯) ⟨n, ⋯⟩",
" Set.range (nth p)... |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 102 | 124 | theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0) := by |
simp_rw [ρ.IicSnd_apply _ hs]
have h_empty : ρ (s ×ˢ ∅) = 0 := by simp only [prod_empty, measure_empty]
rw [← h_empty, ← Real.iInter_Iic_rat, prod_iInter]
suffices h_neg :
Tendsto (fun r : ℚ ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r : ℚ, s ×ˢ Iic ↑(-r)))) by
have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ... | [
" (ρ.IicSnd r) s = ρ (s ×ˢ Iic r)",
" ρ.IicSnd r ≤ ρ.IicSnd r'",
" (ρ.IicSnd r) s ≤ (ρ.IicSnd r') s",
" ρ (s ×ˢ Iic r) ≤ ρ (s ×ˢ Iic r')",
" r ≤ r'",
" ρ.IicSnd r ≤ ρ.fst",
" (ρ.IicSnd r) s ≤ ρ.fst s",
" ρ (s ×ˢ Iic r) ≤ ρ (Prod.fst ⁻¹' s)",
" ⨅ r, (ρ.IicSnd ↑↑r) s = (ρ.IicSnd ↑t) s",
" Tendsto (f... | [
" (ρ.IicSnd r) s = ρ (s ×ˢ Iic r)",
" ρ.IicSnd r ≤ ρ.IicSnd r'",
" (ρ.IicSnd r) s ≤ (ρ.IicSnd r') s",
" ρ (s ×ˢ Iic r) ≤ ρ (s ×ˢ Iic r')",
" r ≤ r'",
" ρ.IicSnd r ≤ ρ.fst",
" (ρ.IicSnd r) s ≤ ρ.fst s",
" ρ (s ×ˢ Iic r) ≤ ρ (Prod.fst ⁻¹' s)",
" ⨅ r, (ρ.IicSnd ↑↑r) s = (ρ.IicSnd ↑t) s",
" Tendsto (f... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 145 | 146 | theorem mul_star_self_eq_zero_iff (x : E) : x * x⋆ = 0 ↔ x = 0 := by |
simpa only [star_eq_zero, star_star] using @star_mul_self_eq_zero_iff _ _ _ _ (star x)
| [
" ‖x⋆ * x‖ = ‖x‖ * ‖x‖",
" ∀ (x : E), ‖x⋆‖ = ‖x‖",
" ‖x⋆‖ = ‖x‖",
" ‖x⋆‖ * ‖x⋆‖ = ‖x * x⋆‖",
" ‖x * x⋆‖ = ‖x‖ * ‖x‖",
" ‖x⋆⋆ * x⋆‖ = ‖x‖ * ‖x‖",
" ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖",
" x⋆ * x = 0 ↔ x = 0",
" ‖x‖ * ‖x‖ = 0 ↔ x = 0",
" x⋆ * x ≠ 0 ↔ x ≠ 0",
" x * x⋆ = 0 ↔ x = 0"
] | [
" ‖x⋆ * x‖ = ‖x‖ * ‖x‖",
" ∀ (x : E), ‖x⋆‖ = ‖x‖",
" ‖x⋆‖ = ‖x‖",
" ‖x⋆‖ * ‖x⋆‖ = ‖x * x⋆‖",
" ‖x * x⋆‖ = ‖x‖ * ‖x‖",
" ‖x⋆⋆ * x⋆‖ = ‖x‖ * ‖x‖",
" ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖",
" x⋆ * x = 0 ↔ x = 0",
" ‖x‖ * ‖x‖ = 0 ↔ x = 0",
" x⋆ * x ≠ 0 ↔ x ≠ 0"
] |
import Mathlib.Data.Matrix.PEquiv
import Mathlib.Data.Set.Card
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
open BigOperators Matrix Equiv
variable {n R : Type*} [DecidableEq n] [Fintype n] (σ : Perm n)
variable (R) in
abbrev Equiv.Perm.permMatrix [Zero R] [One... | Mathlib/LinearAlgebra/Matrix/Permutation.lean | 41 | 43 | theorem det_permutation [CommRing R] : det (σ.permMatrix R) = Perm.sign σ := by |
rw [← Matrix.mul_one (σ.permMatrix R), PEquiv.toPEquiv_mul_matrix,
det_permute, det_one, mul_one]
| [
" (Perm.permMatrix R σ).det = ↑↑(Perm.sign σ)"
] | [] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 97 | 108 | theorem derivative_bernoulli_add_one (k : ℕ) :
Polynomial.derivative (bernoulli (k + 1)) = (k + 1) * bernoulli k := by |
simp_rw [bernoulli, derivative_sum, derivative_monomial, Nat.sub_sub, Nat.add_sub_add_right]
-- LHS sum has an extra term, but the coefficient is zero:
rw [range_add_one, sum_insert not_mem_range_self, tsub_self, cast_zero, mul_zero,
map_zero, zero_add, mul_sum]
-- the rest of the sum is termwise equal:
... | [
" bernoulli n = ∑ i ∈ range (n + 1), (monomial i) (_root_.bernoulli (n - i) * ↑(n.choose i))",
" ∑ i ∈ range (n + 1), (monomial (n - i)) (_root_.bernoulli i * ↑(n.choose i)) =\n ∑ j ∈ range (n + 1), (monomial (n - j)) (_root_.bernoulli (n - (n - j)) * ↑(n.choose (n - j)))",
" ∀ x ∈ range (n + 1),\n (monom... | [
" bernoulli n = ∑ i ∈ range (n + 1), (monomial i) (_root_.bernoulli (n - i) * ↑(n.choose i))",
" ∑ i ∈ range (n + 1), (monomial (n - i)) (_root_.bernoulli i * ↑(n.choose i)) =\n ∑ j ∈ range (n + 1), (monomial (n - j)) (_root_.bernoulli (n - (n - j)) * ↑(n.choose (n - j)))",
" ∀ x ∈ range (n + 1),\n (monom... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 89 | 93 | theorem boundary_sup_le : ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b := by |
rw [boundary, inf_sup_right]
exact
sup_le_sup (inf_le_inf_left _ <| hnot_anti le_sup_left)
(inf_le_inf_left _ <| hnot_anti le_sup_right)
| [
" ∂ ⊤ = ⊥",
" ∂ (￢￢a) = ∂ (￢a)",
" ￢∂ a = ⊤",
" ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b",
" a ⊓ b ⊓ ￢(a ⊓ b) = a ⊓ ￢a ⊓ b ⊔ a ⊓ (b ⊓ ￢b)",
" ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b",
" a ⊓ ￢(a ⊔ b) ⊔ b ⊓ ￢(a ⊔ b) ≤ ∂ a ⊔ ∂ b"
] | [
" ∂ ⊤ = ⊥",
" ∂ (￢￢a) = ∂ (￢a)",
" ￢∂ a = ⊤",
" ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b",
" a ⊓ b ⊓ ￢(a ⊓ b) = a ⊓ ￢a ⊓ b ⊔ a ⊓ (b ⊓ ￢b)"
] |
import Mathlib.Algebra.Periodic
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.periodic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Nat
open Nat Function
| Mathlib/Data/Nat/Periodic.lean | 25 | 26 | theorem periodic_gcd (a : ℕ) : Periodic (gcd a) a := by |
simp only [forall_const, gcd_add_self_right, eq_self_iff_true, Periodic]
| [
" Periodic a.gcd a"
] | [] |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 113 | 129 | theorem induction_on {motive : HNNExtension G A B φ → Prop}
(x : HNNExtension G A B φ) (of : ∀ g, motive (of g))
(t : motive t) (mul : ∀ x y, motive x → motive y → motive (x * y))
(inv : ∀ x, motive x → motive x⁻¹) : motive x := by |
let S : Subgroup (HNNExtension G A B φ) :=
{ carrier := setOf motive
one_mem' := by simpa using of 1
mul_mem' := mul _ _
inv_mem' := inv _ }
let f : HNNExtension G A B φ →* S :=
lift (HNNExtension.of.codRestrict S of)
⟨HNNExtension.t, t⟩ (by intro a; ext; simp [equiv_eq_conj, mul_as... | [
" Group (HNNExtension G A B φ)",
" Group (HNNExtension.con G A B φ).Quotient",
" (fun x x_1 => x * x_1) (inr (ofAdd 1)) (inl ↑a) = inr (ofAdd 1) * inl ↑a ∧\n (fun x x_1 => x * x_1) (inl ↑(φ a)) (inr (ofAdd 1)) = inl ↑(φ a) * inr (ofAdd 1)",
" of ↑b * t = t * of ↑(φ.symm b)",
" of ↑b * t = of ↑(φ (φ.symm ... | [
" Group (HNNExtension G A B φ)",
" Group (HNNExtension.con G A B φ).Quotient",
" (fun x x_1 => x * x_1) (inr (ofAdd 1)) (inl ↑a) = inr (ofAdd 1) * inl ↑a ∧\n (fun x x_1 => x * x_1) (inl ↑(φ a)) (inr (ofAdd 1)) = inl ↑(φ a) * inr (ofAdd 1)",
" of ↑b * t = t * of ↑(φ.symm b)",
" of ↑b * t = of ↑(φ (φ.symm ... |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 158 | 164 | theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
(hac : map X ℙ ≪ μ) (hpdf : ¬pdf X ℙ μ =ᵐ[μ] 0) : HasPDF X ℙ μ := by |
refine ⟨?_, ?_, hac⟩
· exact aemeasurable_of_pdf_ne_zero X hpdf
· contrapose! hpdf
have := pdf_of_not_haveLebesgueDecomposition hpdf
filter_upwards using congrFun this
| [
" pdf X ℙ μ =ᶠ[ae μ] 0",
" rnDeriv 0 μ =ᶠ[ae μ] 0",
" AEMeasurable X ℙ",
" HasPDF X ℙ μ",
" (map X ℙ).HaveLebesgueDecomposition μ"
] | [
" pdf X ℙ μ =ᶠ[ae μ] 0",
" rnDeriv 0 μ =ᶠ[ae μ] 0",
" AEMeasurable X ℙ"
] |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.instances from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
section OrderedSemiring
variable [OrderedSe... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 89 | 91 | theorem coe_eq_one {x : Icc (0 : α) 1} : (x : α) = 1 ↔ x = 1 := by |
symm
exact Subtype.ext_iff
| [
" ↑x = 0 ↔ x = 0",
" x = 0 ↔ ↑x = 0",
" ↑x = 1 ↔ x = 1",
" x = 1 ↔ ↑x = 1"
] | [
" ↑x = 0 ↔ x = 0",
" x = 0 ↔ ↑x = 0"
] |
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 51 | 55 | theorem Monic.as_sum (hp : p.Monic) :
p = X ^ p.natDegree + ∑ i ∈ range p.natDegree, C (p.coeff i) * X ^ i := by |
conv_lhs => rw [p.as_sum_range_C_mul_X_pow, sum_range_succ_comm]
suffices C (p.coeff p.natDegree) = 1 by rw [this, one_mul]
exact congr_arg C hp
| [
" Subsingleton R → (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
" (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
" p = X ^ p.natDegree + ∑ i ∈ range p.natDegree, C (p.coeff i) * X ^ i",
"R : Type u S : Type v a b : R m n : ℕ ι : Type y inst✝ : Semiring R p q r : R[X] hp : p.Monic | p",
" C (p.coeff p.natD... | [
" Subsingleton R → (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
" (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b"
] |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 99 | 103 | theorem abs_truncation_le_abs_self (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |f x| := by |
simp only [truncation, indicator, Set.mem_Icc, id, Function.comp_apply]
split_ifs
· exact le_rfl
· simp [abs_nonneg]
| [
" AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ",
" AEStronglyMeasurable ((Set.Ioc (-A) A).indicator id) (Measure.map f μ)",
" |truncation f A x| ≤ |A|",
" |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |A|",
" |f x| ≤ |A|",
" |0| ≤ |A|",
" truncation f 0 = 0",
" (fun x => 0) ∘ f = 0",
" |t... | [
" AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ",
" AEStronglyMeasurable ((Set.Ioc (-A) A).indicator id) (Measure.map f μ)",
" |truncation f A x| ≤ |A|",
" |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |A|",
" |f x| ≤ |A|",
" |0| ≤ |A|",
" truncation f 0 = 0",
" (fun x => 0) ∘ f = 0"
] |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 115 | 117 | theorem inr_zeroCoprodIso_hom (X : C) : coprod.inr ≫ (zeroCoprodIso X).hom = 𝟙 X := by |
dsimp [zeroCoprodIso, binaryCofanZeroLeft]
simp
| [
" ∀ (s : BinaryFan 0 X), (fun s => s.snd) s ≫ 0 = s.fst",
" ∀ (s : BinaryFan 0 X), (fun s => s.snd) s ≫ 𝟙 X = s.snd",
" m = (fun s => s.snd) s",
" (zeroProdIso X).inv ≫ prod.snd = 𝟙 X",
" (limit.isoLimitCone { cone := BinaryFan.mk 0 (𝟙 X), isLimit := binaryFanZeroLeftIsLimit X }).inv ≫ prod.snd = 𝟙 X",
... | [
" ∀ (s : BinaryFan 0 X), (fun s => s.snd) s ≫ 0 = s.fst",
" ∀ (s : BinaryFan 0 X), (fun s => s.snd) s ≫ 𝟙 X = s.snd",
" m = (fun s => s.snd) s",
" (zeroProdIso X).inv ≫ prod.snd = 𝟙 X",
" (limit.isoLimitCone { cone := BinaryFan.mk 0 (𝟙 X), isLimit := binaryFanZeroLeftIsLimit X }).inv ≫ prod.snd = 𝟙 X",
... |
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.BilinearForm.DualLattice
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Trace
#align_import ring_theory.dedekind_domain.... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 145 | 167 | theorem FiniteDimensional.exists_is_basis_integral :
∃ (s : Finset L) (b : Basis s K L), ∀ x, IsIntegral A (b x) := by |
letI := Classical.decEq L
letI : IsNoetherian K L := IsNoetherian.iff_fg.2 inferInstance
let s' := IsNoetherian.finsetBasisIndex K L
let bs' := IsNoetherian.finsetBasis K L
obtain ⟨y, hy, his'⟩ := exists_integral_multiples A K (Finset.univ.image bs')
have hy' : algebraMap A L y ≠ 0 := by
refine mt ((in... | [
" IsLocalization (algebraMapSubmonoid C A⁰) L",
" ∀ (y : ↥(algebraMapSubmonoid C A⁰)), IsUnit ((algebraMap C L) ↑y)",
" IsUnit ((algebraMap C L) ↑⟨(algebraMap A C) x, ⋯⟩)",
" x ≠ 0",
" ∃ x, z * (algebraMap C L) ↑x.2 = (algebraMap C L) x.1",
" z * (algebraMap C L) ↑(x, ⟨(algebraMap A C) ↑m, ⋯⟩).2 = (algebr... | [
" IsLocalization (algebraMapSubmonoid C A⁰) L",
" ∀ (y : ↥(algebraMapSubmonoid C A⁰)), IsUnit ((algebraMap C L) ↑y)",
" IsUnit ((algebraMap C L) ↑⟨(algebraMap A C) x, ⋯⟩)",
" x ≠ 0",
" ∃ x, z * (algebraMap C L) ↑x.2 = (algebraMap C L) x.1",
" z * (algebraMap C L) ↑(x, ⟨(algebraMap A C) ↑m, ⋯⟩).2 = (algebr... |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 72 | 77 | theorem coeff_select (x : 𝕎 R) (n : ℕ) :
(select P x).coeff n = aeval x.coeff (selectPoly P n) := by |
dsimp [select, selectPoly]
split_ifs with hi
· rw [aeval_X, mk]; simp only [hi]; rfl
· rw [AlgHom.map_zero, mk]; simp only [hi]; rfl
| [
" (select P x).coeff n = (aeval x.coeff) (selectPoly P n)",
" (mk p fun n => if P n then x.coeff n else 0).coeff n = (aeval x.coeff) (if P n then X n else 0)",
" (mk p fun n => if P n then x.coeff n else 0).coeff n = (aeval x.coeff) (X n)",
" { coeff := fun n => if P n then x.coeff n else 0 }.coeff n = x.coef... | [] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 61 | 62 | theorem mem_lifts (p : S[X]) : p ∈ lifts f ↔ ∃ q : R[X], map f q = p := by |
simp only [coe_mapRingHom, lifts, RingHom.mem_rangeS]
| [
" p ∈ lifts f ↔ ∃ q, map f q = p"
] | [] |
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.M... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 96 | 104 | theorem fourierIntegral_eq_half_sub_half_period_translate {w : V} (hw : w ≠ 0)
(hf : Integrable f) :
∫ v : V, 𝐞 (-⟪v, w⟫) • f v = (1 / (2 : ℂ)) • ∫ v : V, 𝐞 (-⟪v, w⟫) • (f v - f (v + i w)) := by |
simp_rw [smul_sub]
rw [integral_sub, fourierIntegral_half_period_translate hw, sub_eq_add_neg, neg_neg, ←
two_smul ℂ _, ← @smul_assoc _ _ _ _ _ _ (IsScalarTower.left ℂ), smul_eq_mul]
· norm_num
exacts [(Real.fourierIntegral_convergent_iff w).2 hf,
(Real.fourierIntegral_convergent_iff w).2 (hf.comp_add_... | [
" ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v",
" ⟪i w, w⟫_ℝ = 1 / 2",
" ‖w‖ ^ 2 ≠ 0",
" (fun v => 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w)) = fun v => (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" cexp (... | [
" ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v",
" ⟪i w, w⟫_ℝ = 1 / 2",
" ‖w‖ ^ 2 ≠ 0",
" (fun v => 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w)) = fun v => (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w) = (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)",
" cexp (... |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 132 | 134 | theorem update_inl_apply_inl [DecidableEq α] [DecidableEq (Sum α β)] {f : Sum α β → γ} {i j : α}
{x : γ} : update f (inl i) x (inl j) = update (f ∘ inl) i x j := by |
rw [← update_inl_comp_inl, Function.comp_apply]
| [
" (∃ fab, p fab) ↔ ∃ fa fb, p fun t => rec fa fb t",
" (¬∀ (fa : (val : α) → γ (inl val)) (fb : (val : β) → γ (inr val)), ¬p fun t => rec fa fb t) ↔\n ∃ fa fb, p fun t => rec fa fb t",
" rec f g x = cast ⋯ (rec f g y)",
" rec f g x = cast ⋯ (rec f g x)",
" x = Sum.elim (update f i x) g (inl i)",
" ∀ (x... | [
" (∃ fab, p fab) ↔ ∃ fa fb, p fun t => rec fa fb t",
" (¬∀ (fa : (val : α) → γ (inl val)) (fb : (val : β) → γ (inr val)), ¬p fun t => rec fa fb t) ↔\n ∃ fa fb, p fun t => rec fa fb t",
" rec f g x = cast ⋯ (rec f g y)",
" rec f g x = cast ⋯ (rec f g x)",
" x = Sum.elim (update f i x) g (inl i)",
" ∀ (x... |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
open s... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 38 | 59 | theorem condexp_ae_eq_restrict_zero (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict s] 0) :
μ[f|m] =ᵐ[μ.restrict s] 0 := by |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
have : SigmaFinite ((μ.restrict s).trim hm) := by
rw [← restrict_trim hm _ hs]
exact Restrict.sigma... | [
" μ[f|m] =ᶠ[ae (μ.restrict s)] 0",
" 0 =ᶠ[ae (μ.restrict s)] 0",
" SigmaFinite ((μ.restrict s).trim hm)",
" SigmaFinite ((μ.trim hm).restrict s)",
" ∀ (s_1 : Set α), MeasurableSet s_1 → (μ.restrict s) s_1 < ⊤ → IntegrableOn (μ[f|m]) s_1 (μ.restrict s)",
" ∀ (s_1 : Set α), MeasurableSet s_1 → (μ.restrict s... | [] |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 494 | 503 | theorem schur_complement_eq₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)
(B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (x : m → 𝕜) (y : n → 𝕜) [Invertible D]
(hD : D.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star ((D⁻¹ * Bᴴ) *ᵥ x + y)) ᵥ* D ⬝ᵥ ((D⁻¹ * Bᴴ) *ᵥ x... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| [
" A.fromBlocks B C D = fromBlocks 1 0 (C * ⅟A) 1 * A.fromBlocks 0 0 (D - C * ⅟A * B) * fromBlocks 1 (⅟A * B) 0 1",
" (reindex (Equiv.sumComm l n) (Equiv.sumComm m n)) (A.fromBlocks B C D) =\n (reindex (Equiv.sumComm l n) (Equiv.sumComm m n))\n (fromBlocks 1 (B * ⅟D) 0 1 * (A - B * ⅟D * C).fromBlocks 0 0 D... | [
" A.fromBlocks B C D = fromBlocks 1 0 (C * ⅟A) 1 * A.fromBlocks 0 0 (D - C * ⅟A * B) * fromBlocks 1 (⅟A * B) 0 1",
" (reindex (Equiv.sumComm l n) (Equiv.sumComm m n)) (A.fromBlocks B C D) =\n (reindex (Equiv.sumComm l n) (Equiv.sumComm m n))\n (fromBlocks 1 (B * ⅟D) 0 1 * (A - B * ⅟D * C).fromBlocks 0 0 D... |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Int
#align_import data.int.associated from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
| Mathlib/Data/Int/Associated.lean | 21 | 30 | theorem Int.natAbs_eq_iff_associated {a b : ℤ} : a.natAbs = b.natAbs ↔ Associated a b := by |
refine Int.natAbs_eq_natAbs_iff.trans ?_
constructor
· rintro (rfl | rfl)
· rfl
· exact ⟨-1, by simp⟩
· rintro ⟨u, rfl⟩
obtain rfl | rfl := Int.units_eq_one_or u
· exact Or.inl (by simp)
· exact Or.inr (by simp)
| [
" a.natAbs = b.natAbs ↔ Associated a b",
" a = b ∨ a = -b ↔ Associated a b",
" a = b ∨ a = -b → Associated a b",
" Associated a a",
" Associated (-b) b",
" -b * ↑(-1) = b",
" Associated a b → a = b ∨ a = -b",
" a = a * ↑u ∨ a = -(a * ↑u)",
" a = a * ↑1 ∨ a = -(a * ↑1)",
" a = a * ↑1",
" a = a * ... | [] |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 438 | 440 | theorem det_one_sub_mul_comm (A : Matrix m n α) (B : Matrix n m α) :
det (1 - A * B) = det (1 - B * A) := by |
rw [sub_eq_add_neg, ← Matrix.neg_mul, det_one_add_mul_comm, Matrix.mul_neg, ← sub_eq_add_neg]
| [
" A.fromBlocks B C D = fromBlocks 1 0 (C * ⅟A) 1 * A.fromBlocks 0 0 (D - C * ⅟A * B) * fromBlocks 1 (⅟A * B) 0 1",
" (reindex (Equiv.sumComm l n) (Equiv.sumComm m n)) (A.fromBlocks B C D) =\n (reindex (Equiv.sumComm l n) (Equiv.sumComm m n))\n (fromBlocks 1 (B * ⅟D) 0 1 * (A - B * ⅟D * C).fromBlocks 0 0 D... | [
" A.fromBlocks B C D = fromBlocks 1 0 (C * ⅟A) 1 * A.fromBlocks 0 0 (D - C * ⅟A * B) * fromBlocks 1 (⅟A * B) 0 1",
" (reindex (Equiv.sumComm l n) (Equiv.sumComm m n)) (A.fromBlocks B C D) =\n (reindex (Equiv.sumComm l n) (Equiv.sumComm m n))\n (fromBlocks 1 (B * ⅟D) 0 1 * (A - B * ⅟D * C).fromBlocks 0 0 D... |
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 125 | 131 | theorem Module.finite_of_rank_eq_nat [Module.Free R M] {n : ℕ} (h : Module.rank R M = n) :
Module.Finite R M := by |
nontriviality R
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
have := mk_lt_aleph0_iff.mp <|
b.linearIndependent.cardinal_le_rank |>.trans_eq h |>.trans_lt <| nat_lt_aleph0 n
exact Module.Finite.of_basis b
| [
" Module.rank R M ≤ ↑n",
" ⨆ ι, #↑↑ι ≤ ↑n",
" ∀ (i : { s // LinearIndependent (ι := { x // x ∈ s }) R Subtype.val }), #↑↑i ≤ ↑n",
" #↑↑⟨s, li⟩ ≤ ↑n",
" Finite R M"
] | [
" Module.rank R M ≤ ↑n",
" ⨆ ι, #↑↑ι ≤ ↑n",
" ∀ (i : { s // LinearIndependent (ι := { x // x ∈ s }) R Subtype.val }), #↑↑i ≤ ↑n",
" #↑↑⟨s, li⟩ ≤ ↑n"
] |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 180 | 189 | theorem vars_sum_subset [DecidableEq σ] :
(∑ i ∈ t, φ i).vars ⊆ Finset.biUnion t fun i => (φ i).vars := by |
classical
induction t using Finset.induction_on with
| empty => simp
| insert has hsum =>
rw [Finset.biUnion_insert, Finset.sum_insert has]
refine Finset.Subset.trans
(vars_add_subset _ _) (Finset.union_subset_union (Finset.Subset.refl _) ?_)
assumption
| [
" p.vars = p.degrees.toFinset",
" p.degrees.toFinset = p.degrees.toFinset",
" vars 0 = ∅",
" ((monomial s) r).vars = s.support",
" (C r).vars = ∅",
" (X n).vars = {n}",
" i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support",
" x v = 0",
" v ∈ f.vars",
" (p + q).vars ⊆ p.vars ∪ q.vars",
" x ∈ p.vars ∪ q.... | [
" p.vars = p.degrees.toFinset",
" p.degrees.toFinset = p.degrees.toFinset",
" vars 0 = ∅",
" ((monomial s) r).vars = s.support",
" (C r).vars = ∅",
" (X n).vars = {n}",
" i ∈ p.vars ↔ ∃ d ∈ p.support, i ∈ d.support",
" x v = 0",
" v ∈ f.vars",
" (p + q).vars ⊆ p.vars ∪ q.vars",
" x ∈ p.vars ∪ q.... |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 81 | 88 | theorem length_eq_zero_iff {w : W} : ℓ w = 0 ↔ w = 1 := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
have : ω = [] := eq_nil_of_length_eq_zero (hω.trans h)
rw [this, wordProd_nil]
· rintro rfl
exact cs.length_one
| [
" ∃ n ω, ω.length = n ∧ cs.wordProd ω = w",
" ∃ n ω_1, ω_1.length = n ∧ cs.wordProd ω_1 = cs.wordProd ω",
" ∃ ω, ω.length = cs.length w ∧ w = cs.wordProd ω",
" ω.length = ω.length ∧ cs.wordProd ω = cs.wordProd ω",
" cs.length w = 0 ↔ w = 1",
" cs.length w = 0 → w = 1",
" w = 1",
" cs.wordProd ω = 1",
... | [
" ∃ n ω, ω.length = n ∧ cs.wordProd ω = w",
" ∃ n ω_1, ω_1.length = n ∧ cs.wordProd ω_1 = cs.wordProd ω",
" ∃ ω, ω.length = cs.length w ∧ w = cs.wordProd ω",
" ω.length = ω.length ∧ cs.wordProd ω = cs.wordProd ω"
] |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {𝕜 : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 106 | 114 | theorem applyComposition_ones (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
p.applyComposition (Composition.ones n) = fun v i =>
p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) := by |
funext v i
apply p.congr (Composition.ones_blocksFun _ _)
intro j hjn hj1
obtain rfl : j = 0 := by omega
refine congr_arg v ?_
rw [Fin.ext_iff, Fin.coe_castLE, Composition.ones_embedding, Fin.val_mk]
| [
" p.applyComposition (Composition.ones n) = fun v i => (p 1) fun x => v (Fin.castLE ⋯ i)",
" p.applyComposition (Composition.ones n) v i = (p 1) fun x => v (Fin.castLE ⋯ i)",
" ∀ (i_1 : ℕ) (him : i_1 < (Composition.ones n).blocksFun i),\n i_1 < 1 → (v ∘ ⇑((Composition.ones n).embedding i)) ⟨i_1, him⟩ = v (Fi... | [] |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 61 | 63 | theorem of_subsingleton [Subsingleton M] (x y : M) : SameRay R x y := by |
rw [Subsingleton.elim x 0]
exact zero_left _
| [
" SameRay R x y",
" SameRay R 0 y"
] | [] |
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Preadditive.LeftExact
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.Algebra.Homology.Exact
import Mathli... | Mathlib/CategoryTheory/Abelian/Exact.lean | 97 | 102 | theorem exact_tfae :
TFAE [Exact f g, f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0,
imageSubobject f = kernelSubobject g] := by |
tfae_have 1 ↔ 2; · apply exact_iff
tfae_have 1 ↔ 3; · apply exact_iff_image_eq_kernel
tfae_finish
| [
" Exact f g ↔ imageSubobject f = kernelSubobject g",
" Exact f g → imageSubobject f = kernelSubobject g",
" imageSubobject f = kernelSubobject g",
" (asIso (imageToKernel f g ⋯)).hom ≫ (kernelSubobject g).arrow = (imageSubobject f).arrow",
" imageSubobject f = kernelSubobject g → Exact f g",
" Exact f g ↔... | [
" Exact f g ↔ imageSubobject f = kernelSubobject g",
" Exact f g → imageSubobject f = kernelSubobject g",
" imageSubobject f = kernelSubobject g",
" (asIso (imageToKernel f g ⋯)).hom ≫ (kernelSubobject g).arrow = (imageSubobject f).arrow",
" imageSubobject f = kernelSubobject g → Exact f g",
" Exact f g ↔... |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 132 | 135 | theorem degree_eq_natDegree (hp : p ≠ 0) : degree p = (natDegree p : WithBot ℕ) := by |
let ⟨n, hn⟩ := not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt degree_eq_bot.1 hp))
have hn : degree p = some n := Classical.not_not.1 hn
rw [natDegree, hn]; rfl
| [
" Decidable p.Monic",
" Decidable (p.leadingCoeff = 1)",
" p.degree = ⊥",
" p.natDegree = 0",
" p.degree = ↑p.natDegree",
" Option.some n = ↑(WithBot.unbot' 0 (Option.some n))"
] | [
" Decidable p.Monic",
" Decidable (p.leadingCoeff = 1)",
" p.degree = ⊥",
" p.natDegree = 0"
] |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 119 | 125 | theorem QuotientAddGroup.norm_mk {S : AddSubgroup M} (x : M) :
‖(x : M ⧸ S)‖ = infDist x S := by |
rw [norm_eq_infDist, ← infDist_image (IsometryEquiv.subLeft x).isometry,
IsometryEquiv.subLeft_apply, sub_zero, ← IsometryEquiv.preimage_symm]
congr 1 with y
simp only [mem_preimage, IsometryEquiv.subLeft_symm_apply, mem_setOf_eq, QuotientAddGroup.eq,
neg_add, neg_neg, neg_add_cancel_right, SetLike.mem_c... | [
" ‖x‖ = infDist 0 {m | ↑m = x}",
" ‖↑x‖ = infDist x ↑S",
" infDist x (⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m | ↑m = ↑x}) = infDist x ↑S",
" y ∈ ⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m | ↑m = ↑x} ↔ y ∈ ↑S"
] | [
" ‖x‖ = infDist 0 {m | ↑m = x}"
] |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 245 | 245 | theorem bit1_zero [One M] : bit1 (0 : M) = 1 := by | rw [bit1, bit0_zero, zero_add]
| [
" bit1 0 = 1"
] | [] |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
variable {α : Type*} [LinearOrderedSemiring α] {a : α}
@[simp]
theorem invOf_pos [I... | Mathlib/Algebra/Order/Invertible.lean | 25 | 25 | theorem invOf_nonpos [Invertible a] : ⅟ a ≤ 0 ↔ a ≤ 0 := by | simp only [← not_lt, invOf_pos]
| [
" 0 < a * ⅟a",
" ⅟a ≤ 0 ↔ a ≤ 0"
] | [
" 0 < a * ⅟a"
] |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 75 | 76 | theorem eq_zero_cpow_iff {x : ℂ} {a : ℂ} : a = (0 : ℂ) ^ x ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by |
rw [← zero_cpow_eq_iff, eq_comm]
| [
" x ^ 0 = 1",
" x ^ y = 0 ↔ x = 0 ∧ y ≠ 0",
" (if x = 0 then if y = 0 then 1 else 0 else cexp (x.log * y)) = 0 ↔ x = 0 ∧ y ≠ 0",
" 1 = 0 ↔ x = 0 ∧ y ≠ 0",
" 0 = 0 ↔ x = 0 ∧ y ≠ 0",
" cexp (x.log * y) = 0 ↔ x = 0 ∧ y ≠ 0",
" 0 ^ x = 0",
" 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
" 0 ^ x = a → x ≠ ... | [
" x ^ 0 = 1",
" x ^ y = 0 ↔ x = 0 ∧ y ≠ 0",
" (if x = 0 then if y = 0 then 1 else 0 else cexp (x.log * y)) = 0 ↔ x = 0 ∧ y ≠ 0",
" 1 = 0 ↔ x = 0 ∧ y ≠ 0",
" 0 = 0 ↔ x = 0 ∧ y ≠ 0",
" cexp (x.log * y) = 0 ↔ x = 0 ∧ y ≠ 0",
" 0 ^ x = 0",
" 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
" 0 ^ x = a → x ≠ ... |
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import number_theory.sum_four_squares from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
open Finset Polynomial FiniteField Equiv
the... | Mathlib/NumberTheory/SumFourSquares.lean | 63 | 75 | theorem lt_of_sum_four_squares_eq_mul {a b c d k m : ℕ}
(h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = k * m)
(ha : 2 * a < m) (hb : 2 * b < m) (hc : 2 * c < m) (hd : 2 * d < m) :
k < m := by |
refine _root_.lt_of_mul_lt_mul_right
(_root_.lt_of_mul_lt_mul_left ?_ (zero_le (2 ^ 2))) (zero_le m)
calc
2 ^ 2 * (k * ↑m) = ∑ i : Fin 4, (2 * ![a, b, c, d] i) ^ 2 := by
simp [← h, Fin.sum_univ_succ, mul_add, mul_pow, add_assoc]
_ < ∑ _i : Fin 4, m ^ 2 := Finset.sum_lt_sum_of_nonempty Finset.univ... | [
" (a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 + (a * z - b * w + c * x + d * y) ^ 2 +\n (a * w + b * z - c * y + d * x) ^ 2 =\n (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)",
" (↑a * ↑x - ↑b * ↑y - ↑c * ↑z - ↑d * ↑w).natAbs ^ 2 + (↑a * ↑y + ↑b * ↑x + ↑c *... | [
" (a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 + (a * z - b * w + c * x + d * y) ^ 2 +\n (a * w + b * z - c * y + d * x) ^ 2 =\n (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2)",
" (↑a * ↑x - ↑b * ↑y - ↑c * ↑z - ↑d * ↑w).natAbs ^ 2 + (↑a * ↑y + ↑b * ↑x + ↑c *... |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 54 | 62 | theorem aeval_apply_of_hasEigenvector {f : End K V} {p : K[X]} {μ : K} {x : V}
(h : f.HasEigenvector μ x) : aeval f p x = p.eval μ • x := by |
refine p.induction_on ?_ ?_ ?_
· intro a; simp [Module.algebraMap_end_apply]
· intro p q hp hq; simp [hp, hq, add_smul]
· intro n a hna
rw [mul_comm, pow_succ', mul_assoc, AlgHom.map_mul, LinearMap.mul_apply, mul_comm, hna]
simp only [mem_eigenspace_iff.1 h.1, smul_smul, aeval_X, eval_mul, eval_C, eval... | [
" f.eigenspace (-q.coeff 0 / q.leadingCoeff) =\n LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0))",
" q.leadingCoeff ≠ 0",
" False",
" LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0)) =\n LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0)))... | [
" f.eigenspace (-q.coeff 0 / q.leadingCoeff) =\n LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0))",
" q.leadingCoeff ≠ 0",
" False",
" LinearMap.ker (q.leadingCoeff • f - (algebraMap K (End K V)) (-q.coeff 0)) =\n LinearMap.ker ((aeval f) (C q.leadingCoeff * X + C (q.coeff 0)))... |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 64 | 66 | theorem rpow_intCast (x : ℝ) (n : ℤ) : x ^ (n : ℝ) = x ^ n := by |
simp only [rpow_def, ← Complex.ofReal_zpow, Complex.cpow_intCast, Complex.ofReal_intCast,
Complex.ofReal_re]
| [
" x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" (if ↑x = 0 then if ↑y = 0 then 1 else 0 else ((↑x).log * ↑y).exp).re =\n if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" Complex.re 1 = 1",
" Complex.re 1 = 0",
" Complex.re 1 = rexp (x.log * y)",
" Complex.re 0 = 1... | [
" x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" (if ↑x = 0 then if ↑y = 0 then 1 else 0 else ((↑x).log * ↑y).exp).re =\n if x = 0 then if y = 0 then 1 else 0 else rexp (x.log * y)",
" Complex.re 1 = 1",
" Complex.re 1 = 0",
" Complex.re 1 = rexp (x.log * y)",
" Complex.re 0 = 1... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 254 | 255 | theorem hasDerivAt_mul_const (c : 𝕜) : HasDerivAt (fun x => x * c) c x := by |
simpa only [one_mul] using (hasDerivAt_id' x).mul_const c
| [
" HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x",
" HasDerivAt (fun y => c y * d y) (c' * d x + c x * d') x",
" HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') univ x",
" HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x",
" HasDerivWithinAt (fun y => c y * d) (c' *... | [
" HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x",
" HasDerivAt (fun y => c y * d y) (c' * d x + c x * d') x",
" HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') univ x",
" HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x",
" HasDerivWithinAt (fun y => c y * d) (c' *... |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Logic.Lemmas
#align_import combinatorics.quiver.path from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
open Function
universe v v₁ v₂ u u₁ u₂
namespace Quiver
inductive Path {V : Type u} [Quiver.{v} V] (a : V) : V → Sort max ... | Mathlib/Combinatorics/Quiver/Path.lean | 123 | 134 | theorem comp_inj {p₁ p₂ : Path a b} {q₁ q₂ : Path b c} (hq : q₁.length = q₂.length) :
p₁.comp q₁ = p₂.comp q₂ ↔ p₁ = p₂ ∧ q₁ = q₂ := by |
refine ⟨fun h => ?_, by rintro ⟨rfl, rfl⟩; rfl⟩
induction' q₁ with d₁ e₁ q₁ f₁ ih <;> obtain _ | ⟨q₂, f₂⟩ := q₂
· exact ⟨h, rfl⟩
· cases hq
· cases hq
· simp only [comp_cons, cons.injEq] at h
obtain rfl := h.1
obtain ⟨rfl, rfl⟩ := ih (Nat.succ.inj hq) h.2.1.eq
rw [h.2.2.eq]
exact ⟨rfl, rfl⟩... | [
" False",
" b = c",
" HEq p p'",
" HEq e e'",
" a = b",
" a = a",
" nil.comp (p.cons a✝) = p.cons a✝",
" (p.comp q).comp (r.cons a✝) = p.comp (q.comp (r.cons a✝))",
" p₁.comp q₁ = p₂.comp q₂ ↔ p₁ = p₂ ∧ q₁ = q₂",
" p₁ = p₂ ∧ q₁ = q₂ → p₁.comp q₁ = p₂.comp q₂",
" p₁.comp q₁ = p₁.comp q₁",
" p₁ ... | [
" False",
" b = c",
" HEq p p'",
" HEq e e'",
" a = b",
" a = a",
" nil.comp (p.cons a✝) = p.cons a✝",
" (p.comp q).comp (r.cons a✝) = p.comp (q.comp (r.cons a✝))"
] |
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α R : Type*}
open Filter Set Function
@[simp]
theorem Nat.comap_cast_atTop [S... | Mathlib/Order/Filter/Archimedean.lean | 93 | 95 | theorem Filter.Eventually.intCast_atTop [StrictOrderedRing R] [Archimedean R] {p : R → Prop}
(h : ∀ᶠ (x:R) in atTop, p x) : ∀ᶠ (n:ℤ) in atTop, p n := by |
rw [← Int.comap_cast_atTop (R := R)]; exact h.comap _
| [
" ↑(-↑n) ≤ r",
" Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto f l atTop",
" Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto (Int.cast ∘ f) l atTop",
" Tendsto (fun n => ↑(f n)) l atBot ↔ Tendsto f l atBot",
" Tendsto (fun n => ↑(f n)) l atBot ↔ Tendsto (Int.cast ∘ f) l atBot",
" ∀ᶠ (n : ℤ) in atTop, p ↑n",
"... | [
" ↑(-↑n) ≤ r",
" Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto f l atTop",
" Tendsto (fun n => ↑(f n)) l atTop ↔ Tendsto (Int.cast ∘ f) l atTop",
" Tendsto (fun n => ↑(f n)) l atBot ↔ Tendsto f l atBot",
" Tendsto (fun n => ↑(f n)) l atBot ↔ Tendsto (Int.cast ∘ f) l atBot"
] |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int
#align_import algebra.field.power from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
variable {α : Type*}
section DivisionRing
variable [DivisionRing α] {n : ℤ}
theorem Odd.neg_zpow (h : Odd n) (a : α) : (-a) ^ n = -a... | Mathlib/Algebra/Field/Power.lean | 33 | 33 | theorem Odd.neg_one_zpow (h : Odd n) : (-1 : α) ^ n = -1 := by | rw [h.neg_zpow, one_zpow]
| [
" (-a) ^ n = -a ^ n",
" n ≠ 0",
" False",
" (-a) ^ (2 * k + 1) = -a ^ (2 * k + 1)",
" (-1) ^ n = -1"
] | [
" (-a) ^ n = -a ^ n",
" n ≠ 0",
" False",
" (-a) ^ (2 * k + 1) = -a ^ (2 * k + 1)"
] |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section UnusedInput
variable {xs : Vector α n} {ys : Vector β n}
@[simp]
| Mathlib/Data/Vector/MapLemmas.lean | 342 | 347 | theorem mapAccumr₂_unused_input_left [Inhabited α] (f : α → β → σ → σ × γ)
(h : ∀ a b s, f default b s = f a b s) :
mapAccumr₂ f xs ys s = mapAccumr (fun b s => f default b s) ys s := by |
induction xs, ys using Vector.revInductionOn₂ generalizing s with
| nil => rfl
| snoc xs ys x y ih => simp [h x y s, ih]
| [
" mapAccumr₂ f xs ys s = mapAccumr (fun b s => f default b s) ys s",
" mapAccumr₂ f nil nil s = mapAccumr (fun b s => f default b s) nil s",
" mapAccumr₂ f (xs.snoc x) (ys.snoc y) s = mapAccumr (fun b s => f default b s) (ys.snoc y) s"
] | [] |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 81 | 82 | theorem PartialHomeomorph.univUnitBall_symm_apply_zero : univUnitBall.symm (0 : E) = 0 := by |
simp [PartialHomeomorph.univUnitBall_symm_apply]
| [
" (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x ∈ ball 0 1",
" 0 < 1 + ‖x‖ ^ 2",
" ‖x‖ ^ 2 < 1 + ‖x‖ ^ 2",
" (fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) ((fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x) = x",
" (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) ((fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) y) = y",
" 0 < 1 - ‖y‖ ^ 2",
" ContinuousOn\n ↑{ toFu... | [
" (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x ∈ ball 0 1",
" 0 < 1 + ‖x‖ ^ 2",
" ‖x‖ ^ 2 < 1 + ‖x‖ ^ 2",
" (fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) ((fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x) = x",
" (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) ((fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) y) = y",
" 0 < 1 - ‖y‖ ^ 2",
" ContinuousOn\n ↑{ toFu... |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 66 | 68 | theorem exp_half_dist (z w : ℍ) :
exp (dist z w / 2) = (dist (z : ℂ) w + dist (z : ℂ) (conj ↑w)) / (2 * √(z.im * w.im)) := by |
rw [← sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div]
| [
" (dist z w / 2).sinh = dist ↑z ↑w / (2 * √(z.im * w.im))",
" (dist z w / 2).cosh = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))",
" (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =\n dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))",
" 2 ^ 2 * (z.im * w.im) + dist ... | [
" (dist z w / 2).sinh = dist ↑z ↑w / (2 * √(z.im * w.im))",
" (dist z w / 2).cosh = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))",
" (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =\n dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))",
" 2 ^ 2 * (z.im * w.im) + dist ... |
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Function PNat
namespace PNat
variable (a b : ℕ+)
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ+ := Subtype.... | Mathlib/Data/PNat/Interval.lean | 108 | 109 | theorem card_fintype_Icc : Fintype.card (Set.Icc a b) = b + 1 - a := by |
rw [← card_Icc, Fintype.card_ofFinset]
| [
" (Icc a b).card = ↑b + 1 - ↑a",
" (Icc a b).card = (Icc ↑a ↑b).card",
" (Icc a b).card = (map (Embedding.subtype fun n => 0 < n) (Icc a b)).card",
" (Ico a b).card = ↑b - ↑a",
" (Ico a b).card = (Ico ↑a ↑b).card",
" (Ico a b).card = (map (Embedding.subtype fun n => 0 < n) (Ico a b)).card",
" (Ioc a b).... | [
" (Icc a b).card = ↑b + 1 - ↑a",
" (Icc a b).card = (Icc ↑a ↑b).card",
" (Icc a b).card = (map (Embedding.subtype fun n => 0 < n) (Icc a b)).card",
" (Ico a b).card = ↑b - ↑a",
" (Ico a b).card = (Ico ↑a ↑b).card",
" (Ico a b).card = (map (Embedding.subtype fun n => 0 < n) (Ico a b)).card",
" (Ioc a b).... |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 141 | 148 | theorem range_circleMap (c : ℂ) (R : ℝ) : range (circleMap c R) = sphere c |R| :=
calc
range (circleMap c R) = c +ᵥ R • range fun θ : ℝ => exp (θ * I) := by |
simp (config := { unfoldPartialApp := true }) only [← image_vadd, ← image_smul, ← range_comp,
vadd_eq_add, circleMap, Function.comp_def, real_smul]
_ = sphere c |R| := by
rw [Complex.range_exp_mul_I, smul_sphere R 0 zero_le_one]
simp
| [
" circleMap c R (θ + 2 * π) = circleMap c R θ",
" circleMap c R θ - c = circleMap 0 R θ",
" Complex.abs (circleMap 0 R θ) = |R|",
" circleMap c R θ ∈ sphere c |R|",
" circleMap c R θ ∈ sphere c R",
" circleMap c R θ ∉ ball c R",
" range (circleMap c R) = c +ᵥ R • range fun θ => cexp (↑θ * I)",
" (c +ᵥ... | [
" circleMap c R (θ + 2 * π) = circleMap c R θ",
" circleMap c R θ - c = circleMap 0 R θ",
" Complex.abs (circleMap 0 R θ) = |R|",
" circleMap c R θ ∈ sphere c |R|",
" circleMap c R θ ∈ sphere c R",
" circleMap c R θ ∉ ball c R"
] |
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 199 | 200 | theorem realize_one (v : α → R) : Term.realize v (1 : ring.Term α) = 1 := by |
simp [one_def, funMap_one, constantMap]
| [
" DecidableEq (ring.Functions n)",
" DecidableEq (ringFunc n)",
" DecidableEq (ring.Relations n)",
" DecidableEq Empty",
" (↑[Sum.inl ⟨2, add⟩, Sum.inl ⟨2, mul⟩, Sum.inl ⟨1, neg⟩, Sum.inl ⟨0, zero⟩, Sum.inl ⟨0, one⟩]).Nodup",
" ∀ (x : ring.Symbols),\n x ∈\n { val := ↑[Sum.inl ⟨2, add⟩, Sum.inl ⟨2,... | [
" DecidableEq (ring.Functions n)",
" DecidableEq (ringFunc n)",
" DecidableEq (ring.Relations n)",
" DecidableEq Empty",
" (↑[Sum.inl ⟨2, add⟩, Sum.inl ⟨2, mul⟩, Sum.inl ⟨1, neg⟩, Sum.inl ⟨0, zero⟩, Sum.inl ⟨0, one⟩]).Nodup",
" ∀ (x : ring.Symbols),\n x ∈\n { val := ↑[Sum.inl ⟨2, add⟩, Sum.inl ⟨2,... |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 103 | 103 | theorem birthday_zero : birthday 0 = 0 := by | simp [inferInstanceAs (IsEmpty PEmpty)]
| [
" x.birthday = max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)",
" (mk α✝ β✝ a✝¹ a✝).birthday =\n max (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday)",
" max (lsub fun i => (a✝¹ i).birthday) (lsub fun i => (... | [
" x.birthday = max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)",
" (mk α✝ β✝ a✝¹ a✝).birthday =\n max (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday) (lsub fun i => ((mk α✝ β✝ a✝¹ a✝).moveRight i).birthday)",
" max (lsub fun i => (a✝¹ i).birthday) (lsub fun i => (... |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {𝕜 : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 46 | 48 | theorem contDiff_clm_apply_iff {n : ℕ∞} {f : E → F →L[𝕜] G} [FiniteDimensional 𝕜 F] :
ContDiff 𝕜 n f ↔ ∀ y, ContDiff 𝕜 n fun x => f x y := by |
simp_rw [← contDiffOn_univ, contDiffOn_clm_apply]
| [
" ContDiffOn 𝕜 n f s ↔ ∀ (y : F), ContDiffOn 𝕜 n (fun x => (f x) y) s",
" ContDiffOn 𝕜 n f s",
" ContDiffOn 𝕜 n ((⇑e₂.symm ∘ ⇑e₂) ∘ f) s",
" ContDiff 𝕜 n f ↔ ∀ (y : F), ContDiff 𝕜 n fun x => (f x) y"
] | [
" ContDiffOn 𝕜 n f s ↔ ∀ (y : F), ContDiffOn 𝕜 n (fun x => (f x) y) s",
" ContDiffOn 𝕜 n f s",
" ContDiffOn 𝕜 n ((⇑e₂.symm ∘ ⇑e₂) ∘ f) s"
] |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 120 | 123 | theorem toDFA_correct : M.toDFA.accepts = M.accepts := by |
ext x
rw [mem_accepts, DFA.mem_accepts]
constructor <;> · exact fun ⟨w, h2, h3⟩ => ⟨w, h3, h2⟩
| [
" s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a",
" M.stepSet ∅ a = ∅",
" M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a",
" x ∈ M.accepts ↔ ∃ S ∈ M.accept, S ∈ M.evalFrom M.start x",
" M.toDFA.accepts = M.accepts",
" x ∈ M.toDFA.accepts ↔ x ∈ M.accepts",
" M.toDFA.evalFrom M.toDFA.start x ∈ M.toD... | [
" s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a",
" M.stepSet ∅ a = ∅",
" M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a",
" x ∈ M.accepts ↔ ∃ S ∈ M.accept, S ∈ M.evalFrom M.start x"
] |
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 91 | 102 | theorem Pairwise.forall_of_forall_of_flip (h₁ : ∀ x ∈ l, R x x) (h₂ : Pairwise R l)
(h₃ : l.Pairwise (flip R)) : ∀ ⦃x⦄, x ∈ l → ∀ ⦃y⦄, y ∈ l → R x y := by |
induction l with
| nil => exact forall_mem_nil _
| cons a l ih =>
rw [pairwise_cons] at h₂ h₃
simp only [mem_cons]
rintro x (rfl | hx) y (rfl | hy)
· exact h₁ _ (l.mem_cons_self _)
· exact h₂.1 _ hy
· exact h₃.1 _ hx
· exact ih (fun x hx => h₁ _ <| mem_cons_of_mem _ hx) h₂.2 h₃.2 hx h... | [
" Pairwise S l",
" Pairwise S []",
" Pairwise S (a :: l)",
" ∀ (a' : α), a' ∈ l → S a a'",
" Pairwise (fun a b => R a b ∧ S a b) l",
" Pairwise (fun a b => R a b ∧ S a b) []",
" Pairwise (fun a b => R a b ∧ S a b) (a✝¹ :: l✝)",
" (∀ (a' : α✝), a' ∈ l✝ → R a✝¹ a' ∧ S a✝¹ a') ∧ Pairwise (fun a b => R a ... | [
" Pairwise S l",
" Pairwise S []",
" Pairwise S (a :: l)",
" ∀ (a' : α), a' ∈ l → S a a'",
" Pairwise (fun a b => R a b ∧ S a b) l",
" Pairwise (fun a b => R a b ∧ S a b) []",
" Pairwise (fun a b => R a b ∧ S a b) (a✝¹ :: l✝)",
" (∀ (a' : α✝), a' ∈ l✝ → R a✝¹ a' ∧ S a✝¹ a') ∧ Pairwise (fun a b => R a ... |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 134 | 135 | theorem nhds_pure (x : α) : 𝓝 (pure x : Filter α) = 𝓟 {⊥, pure x} := by |
rw [← principal_singleton, nhds_principal, principal_singleton, Iic_pure]
| [
" IsOpen {l | s ∈ l}",
" ∀ t₁ ∈ range (Iic ∘ 𝓟), ∀ t₂ ∈ range (Iic ∘ 𝓟), ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ range (Iic ∘ 𝓟), x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂",
" ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t",
" (Iic ∘ 𝓟) (s ∩ t) = Iic (𝓟 s) ∩ Iic (𝓟 t)",
" (∃ S ⊆ range (Iic ∘ 𝓟), s = ⋃₀ S) ↔ ∃ T, s = ⋃... | [
" IsOpen {l | s ∈ l}",
" ∀ t₁ ∈ range (Iic ∘ 𝓟), ∀ t₂ ∈ range (Iic ∘ 𝓟), ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ range (Iic ∘ 𝓟), x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂",
" ∃ t₃ ∈ range (Iic ∘ 𝓟), l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t",
" (Iic ∘ 𝓟) (s ∩ t) = Iic (𝓟 s) ∩ Iic (𝓟 t)",
" (∃ S ⊆ range (Iic ∘ 𝓟), s = ⋃₀ S) ↔ ∃ T, s = ⋃... |
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Tactic.ClearExclamation
import Mathlib.Data.Nat.Cast.Basic
set_option autoImplicit true
namespace Mathlib
open Lean hidi... | Mathlib/Tactic/NormNum/Basic.lean | 119 | 120 | theorem isNat_intCast {R} [Ring R] (n : ℤ) (m : ℕ) :
IsNat n m → IsNat (n : R) m := by | rintro ⟨⟨⟩⟩; exact ⟨by simp⟩
| [
" (↑(Int.negOfNat a✝)).natAbs = ↑a✝",
" IsNat n m → IsNat (↑n) m",
" IsNat (↑↑m) m",
" ↑↑m = ↑m"
] | [
" (↑(Int.negOfNat a✝)).natAbs = ↑a✝",
" IsNat n m → IsNat (↑n) m",
" IsNat (↑↑m) m"
] |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
| Mathlib/Data/Nat/Factorization/PrimePow.lean | 20 | 24 | theorem IsPrimePow.minFac_pow_factorization_eq {n : ℕ} (hn : IsPrimePow n) :
n.minFac ^ n.factorization n.minFac = n := by |
obtain ⟨p, k, hp, hk, rfl⟩ := hn
rw [← Nat.prime_iff] at hp
rw [hp.pow_minFac hk.ne', hp.factorization_pow, Finsupp.single_eq_same]
| [
" n.minFac ^ n.factorization n.minFac = n",
" (p ^ k).minFac ^ (p ^ k).factorization (p ^ k).minFac = p ^ k"
] | [] |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 81 | 92 | theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by |
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl ... | [
" (trinomial k m n u v w).coeff n = w",
" (trinomial k m n u v w).coeff m = v",
" (trinomial k m n u v w).coeff k = u",
" (trinomial k m n u v w).natDegree = n",
" (trinomial k m n u v w).coeff n ≠ 0",
" ↑i ≤ ↑n",
" ↑i ≤ ↑i",
" (trinomial k m n u v w).natTrailingDegree = k",
" (trinomial k m n u v w... | [
" (trinomial k m n u v w).coeff n = w",
" (trinomial k m n u v w).coeff m = v",
" (trinomial k m n u v w).coeff k = u",
" (trinomial k m n u v w).natDegree = n",
" (trinomial k m n u v w).coeff n ≠ 0",
" ↑i ≤ ↑n",
" ↑i ≤ ↑i"
] |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
theorem sum_conjClasses_card_eq_card [Fintype <| Conj... | Mathlib/GroupTheory/ClassEquation.lean | 47 | 70 | theorem Group.nat_card_center_add_sum_card_noncenter_eq_card [Finite G] :
Nat.card (Subgroup.center G) + ∑ᶠ x ∈ noncenter G, Nat.card x.carrier = Nat.card G := by |
classical
cases nonempty_fintype G
rw [@Nat.card_eq_fintype_card G, ← sum_conjClasses_card_eq_card, ←
Finset.sum_sdiff (ConjClasses.noncenter G).toFinset.subset_univ]
simp only [Nat.card_eq_fintype_card, Set.toFinset_card]
congr 1
swap
· convert finsum_cond_eq_sum_of_cond_iff _ _
simp [Set.mem_to... | [
" ∑ x : ConjClasses G, x.carrier.toFinset.card = Fintype.card G",
" (x : ConjClasses G) × ↑x.carrier ≃ G",
" ∑ᶠ (x : ConjClasses G), x.carrier.ncard = Nat.card G",
" ∑ i : ConjClasses G, i.carrier.ncard = ∑ x : ConjClasses G, x.carrier.toFinset.card",
" Nat.card ↥(Subgroup.center G) + ∑ᶠ (x : ConjClasses G)... | [
" ∑ x : ConjClasses G, x.carrier.toFinset.card = Fintype.card G",
" (x : ConjClasses G) × ↑x.carrier ≃ G",
" ∑ᶠ (x : ConjClasses G), x.carrier.ncard = Nat.card G",
" ∑ i : ConjClasses G, i.carrier.ncard = ∑ x : ConjClasses G, x.carrier.toFinset.card"
] |
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 52 | 56 | theorem kernel.measure_eq_zero_or_one_of_indepSet_self [∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
(h_indep : IndepSet t t κ μα) :
∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by |
filter_upwards [measure_eq_zero_or_one_or_top_of_indepSet_self h_indep] with a h_0_1_top
simpa only [measure_ne_top (κ a), or_false] using h_0_1_top
| [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤",
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1",
" (κ a) t = 0 ∨ (κ a) t = 1"
] | [
" ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤",
" μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤"
] |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-community/mathlib"@"ba2245edf0c8bb155f1569fd9b9492a9b384cde6"
namespace OrderHom
variable {α β : Type*}
section Preorder
variable [Preorder α]
instance [Sem... | Mathlib/Order/Hom/Order.lean | 97 | 99 | theorem coe_iInf {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) :
((⨅ i, f i : α →o β) : α → β) = ⨅ i, (f i : α → β) := by |
funext x; simp [iInf_apply]
| [
" ⇑(⨅ i, f i) = ⨅ i, ⇑(f i)",
" (⨅ i, f i) x = (⨅ i, ⇑(f i)) x"
] | [] |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 319 | 320 | theorem average_zero_measure (f : α → E) : ⨍ x, f x ∂(0 : Measure α) = 0 := by |
rw [average, smul_zero, integral_zero_measure]
| [
" ⨍ (x : α), 0 ∂μ = 0",
" ⨍ (x : α), f x ∂0 = 0"
] | [
" ⨍ (x : α), 0 ∂μ = 0"
] |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 112 | 133 | theorem coeff_add_of_disjoint (x y : 𝕎 R) (h : ∀ n, x.coeff n = 0 ∨ y.coeff n = 0) :
(x + y).coeff n = x.coeff n + y.coeff n := by |
let P : ℕ → Prop := fun n => y.coeff n = 0
haveI : DecidablePred P := Classical.decPred P
set z := mk p fun n => if P n then x.coeff n else y.coeff n
have hx : select P z = x := by
ext1 n; rw [select, coeff_mk, coeff_mk]
split_ifs with hn
· rfl
· rw [(h n).resolve_right hn]
have hy : select (... | [
" (select P x).coeff n = (aeval x.coeff) (selectPoly P n)",
" (mk p fun n => if P n then x.coeff n else 0).coeff n = (aeval x.coeff) (if P n then X n else 0)",
" (mk p fun n => if P n then x.coeff n else 0).coeff n = (aeval x.coeff) (X n)",
" { coeff := fun n => if P n then x.coeff n else 0 }.coeff n = x.coef... | [
" (select P x).coeff n = (aeval x.coeff) (selectPoly P n)",
" (mk p fun n => if P n then x.coeff n else 0).coeff n = (aeval x.coeff) (if P n then X n else 0)",
" (mk p fun n => if P n then x.coeff n else 0).coeff n = (aeval x.coeff) (X n)",
" { coeff := fun n => if P n then x.coeff n else 0 }.coeff n = x.coef... |
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
(f : 𝕜 →ᵃ[𝕜] E) {a b : E} {L : Filter 𝕜} {s : Set 𝕜} {x : 𝕜}
n... | Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 64 | 65 | theorem hasStrictDerivAt_lineMap : HasStrictDerivAt (lineMap a b) (b - a) x := by |
simpa using (lineMap a b : 𝕜 →ᵃ[𝕜] E).hasStrictDerivAt
| [
" HasStrictDerivAt (⇑f) (f.linear 1) x",
" HasStrictDerivAt (⇑f.linear + fun x => f 0) (f.linear 1) x",
" HasDerivAtFilter (⇑f) (f.linear 1) x L",
" HasDerivAtFilter (⇑f.linear + fun x => f 0) (f.linear 1) x L",
" HasStrictDerivAt (⇑(lineMap a b)) (b - a) x"
] | [
" HasStrictDerivAt (⇑f) (f.linear 1) x",
" HasStrictDerivAt (⇑f.linear + fun x => f 0) (f.linear 1) x",
" HasDerivAtFilter (⇑f) (f.linear 1) x L",
" HasDerivAtFilter (⇑f.linear + fun x => f 0) (f.linear 1) x L"
] |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 135 | 137 | theorem _root_.MeasureTheory.AEStronglyMeasurable.integrable_truncation [IsFiniteMeasure μ]
(hf : AEStronglyMeasurable f μ) {A : ℝ} : Integrable (truncation f A) μ := by |
rw [← memℒp_one_iff_integrable]; exact hf.memℒp_truncation
| [
" AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ",
" AEStronglyMeasurable ((Set.Ioc (-A) A).indicator id) (Measure.map f μ)",
" |truncation f A x| ≤ |A|",
" |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |A|",
" |f x| ≤ |A|",
" |0| ≤ |A|",
" truncation f 0 = 0",
" (fun x => 0) ∘ f = 0",
" |t... | [
" AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ",
" AEStronglyMeasurable ((Set.Ioc (-A) A).indicator id) (Measure.map f μ)",
" |truncation f A x| ≤ |A|",
" |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |A|",
" |f x| ≤ |A|",
" |0| ≤ |A|",
" truncation f 0 = 0",
" (fun x => 0) ∘ f = 0",
" |t... |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame → Prop
| ⟨_, _, L, R⟩ => (... | Mathlib/SetTheory/Surreal/Basic.lean | 89 | 90 | theorem moveLeft {x : PGame} (o : Numeric x) (i : x.LeftMoves) : Numeric (x.moveLeft i) := by |
cases x; exact o.2.1 i
| [
" x.Numeric ↔\n (∀ (i : x.LeftMoves) (j : x.RightMoves), x.moveLeft i < x.moveRight j) ∧\n (∀ (i : x.LeftMoves), (x.moveLeft i).Numeric) ∧ ∀ (j : x.RightMoves), (x.moveRight j).Numeric",
" (mk α✝ β✝ a✝¹ a✝).Numeric ↔\n (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).RightMoves),\n (m... | [
" x.Numeric ↔\n (∀ (i : x.LeftMoves) (j : x.RightMoves), x.moveLeft i < x.moveRight j) ∧\n (∀ (i : x.LeftMoves), (x.moveLeft i).Numeric) ∧ ∀ (j : x.RightMoves), (x.moveRight j).Numeric",
" (mk α✝ β✝ a✝¹ a✝).Numeric ↔\n (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).RightMoves),\n (m... |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 159 | 164 | theorem norm_max_aux₃ {f : ℂ → F} {z w : ℂ} {r : ℝ} (hr : dist w z = r)
(hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) : ‖f w‖ = ‖f z‖ := by |
subst r
rcases eq_or_ne w z with (rfl | hne); · rfl
rw [← dist_ne_zero] at hne
exact norm_max_aux₂ hd (closure_ball z hne ▸ hz.closure hd.continuousOn.norm)
| [
" ‖f w‖ = ‖f z‖",
" ¬(norm ∘ f) w < (norm ∘ f) z",
" False",
" ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ = 2 * π * ‖f z‖",
" ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖",
" ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * r * (‖f z‖ / r)",
" ContinuousOn (fun ζ => (ζ - z)⁻¹ • f ζ) (sphere z r... | [
" ‖f w‖ = ‖f z‖",
" ¬(norm ∘ f) w < (norm ∘ f) z",
" False",
" ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ = 2 * π * ‖f z‖",
" ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖",
" ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * r * (‖f z‖ / r)",
" ContinuousOn (fun ζ => (ζ - z)⁻¹ • f ζ) (sphere z r... |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 62 | 63 | theorem ball_eq_Ioo (x : ℤ) (r : ℝ) : ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉ := by |
rw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]
| [
" dist m n = ↑|m - n|",
" |↑m - ↑n| = ↑|m - n|",
" Pairwise fun m n => 1 ≤ dist m n",
" 1 ≤ dist m n",
" 1 ≤ |↑m - ↑n|",
" 1 ≤ |m - n|",
" ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉"
] | [
" dist m n = ↑|m - n|",
" |↑m - ↑n| = ↑|m - n|",
" Pairwise fun m n => 1 ≤ dist m n",
" 1 ≤ dist m n",
" 1 ≤ |↑m - ↑n|",
" 1 ≤ |m - n|"
] |
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 | 150 | 156 | theorem isSpecial_iff : u.IsSpecial ↔ u.IsSpecial' := by |
dsimp [IsSpecial, IsSpecial']
let ⟨wp, x, y, zp, ap, bp⟩ := u
constructor <;> intro h <;> simp [w, z, succPNat] at * <;>
simp only [← coe_inj, mul_coe, mk_coe] at *
· simp_all [← h, Nat.mul, Nat.succ_eq_add_one]; ring
· simp [Nat.succ_eq_add_one, Nat.mul_add, Nat.add_mul, ← Nat.add_assoc] at h; rw [← h];... | [
" 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"
] |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame → Prop
| ⟨_, _, L, R⟩ => (... | Mathlib/SetTheory/Surreal/Basic.lean | 93 | 94 | theorem moveRight {x : PGame} (o : Numeric x) (j : x.RightMoves) : Numeric (x.moveRight j) := by |
cases x; exact o.2.2 j
| [
" x.Numeric ↔\n (∀ (i : x.LeftMoves) (j : x.RightMoves), x.moveLeft i < x.moveRight j) ∧\n (∀ (i : x.LeftMoves), (x.moveLeft i).Numeric) ∧ ∀ (j : x.RightMoves), (x.moveRight j).Numeric",
" (mk α✝ β✝ a✝¹ a✝).Numeric ↔\n (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).RightMoves),\n (m... | [
" x.Numeric ↔\n (∀ (i : x.LeftMoves) (j : x.RightMoves), x.moveLeft i < x.moveRight j) ∧\n (∀ (i : x.LeftMoves), (x.moveLeft i).Numeric) ∧ ∀ (j : x.RightMoves), (x.moveRight j).Numeric",
" (mk α✝ β✝ a✝¹ a✝).Numeric ↔\n (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).RightMoves),\n (m... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.add from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 102 | 103 | theorem deriv_add_const (c : F) : deriv (fun y => f y + c) x = deriv f x := by |
simp only [deriv, fderiv_add_const]
| [
" HasDerivAtFilter (fun y => f y + g y) (f' + g') x L",
" HasStrictDerivAt (fun y => f y + g y) (f' + g') x",
" derivWithin (fun y => f y + c) s x = derivWithin f s x",
" deriv (fun y => f y + c) x = deriv f x"
] | [
" HasDerivAtFilter (fun y => f y + g y) (f' + g') x L",
" HasStrictDerivAt (fun y => f y + g y) (f' + g') x",
" derivWithin (fun y => f y + c) s x = derivWithin f s x"
] |
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 | 73 | 79 | theorem oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arctan (‖y‖ / ‖x‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h) (o.left_ne_zero_of_oangle_eq_pi_div_two h)]
| [
" o.oangle x (x + y) = ↑(‖x‖ / ‖x + y‖).arccos",
" (o.oangle x (x + y)).sign = 1",
" o.oangle (x + y) y = ↑(‖y‖ / ‖x + y‖).arccos",
" (-o).oangle y (x + y) = ↑(‖y‖ / ‖x + y‖).arccos",
" (-o).oangle y (y + x) = ↑(‖y‖ / ‖y + x‖).arccos",
" o.oangle x (x + y) = ↑(‖y‖ / ‖x + y‖).arcsin",
" o.oangle (x + y) ... | [
" o.oangle x (x + y) = ↑(‖x‖ / ‖x + y‖).arccos",
" (o.oangle x (x + y)).sign = 1",
" o.oangle (x + y) y = ↑(‖y‖ / ‖x + y‖).arccos",
" (-o).oangle y (x + y) = ↑(‖y‖ / ‖x + y‖).arccos",
" (-o).oangle y (y + x) = ↑(‖y‖ / ‖y + x‖).arccos",
" o.oangle x (x + y) = ↑(‖y‖ / ‖x + y‖).arcsin",
" o.oangle (x + y) ... |
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 | 69 | 72 | theorem Hom.eq_cast_iff_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) (e' : u' ⟶ v') :
e' = e.cast hu hv ↔ HEq e' e := by |
rw [eq_comm, Hom.cast_eq_iff_heq]
exact ⟨HEq.symm, HEq.symm⟩
| [
" (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'"
] |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 140 | 142 | theorem count_le_card (hp : (setOf p).Finite) (n : ℕ) : count p n ≤ hp.toFinset.card := by |
rw [count_eq_card_filter_range]
exact Finset.card_mono fun x hx ↦ hp.mem_toFinset.2 (mem_filter.1 hx).2
| [
" count p 0 = 0",
" Fintype { i // i < n ∧ p i }",
" ∀ (x : ℕ), x ∈ filter p (range n) ↔ x ∈ fun x => x < n ∧ p x",
" x ∈ filter p (range n) ↔ x ∈ fun x => x < n ∧ p x",
" x < n ∧ p x ↔ x ∈ fun x => x < n ∧ p x",
" count p n = (filter p (range n)).card",
" (List.filter (fun b => decide (p b)) (List.rang... | [
" count p 0 = 0",
" Fintype { i // i < n ∧ p i }",
" ∀ (x : ℕ), x ∈ filter p (range n) ↔ x ∈ fun x => x < n ∧ p x",
" x ∈ filter p (range n) ↔ x ∈ fun x => x < n ∧ p x",
" x < n ∧ p x ↔ x ∈ fun x => x < n ∧ p x",
" count p n = (filter p (range n)).card",
" (List.filter (fun b => decide (p b)) (List.rang... |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 128 | 130 | theorem lcm_pos {m n : ℕ} : 0 < m → 0 < n → 0 < m.lcm n := by |
simp_rw [pos_iff_ne_zero]
exact lcm_ne_zero
| [
" m.gcd (n + k * m) = m.gcd n",
" m.gcd (n + m * k) = m.gcd n",
" m.gcd (k * m + n) = m.gcd n",
" m.gcd (m * k + n) = m.gcd n",
" (m + k * n).gcd n = m.gcd n",
" (m + n * k).gcd n = m.gcd n",
" (k * n + m).gcd n = m.gcd n",
" (n * k + m).gcd n = m.gcd n",
" m.gcd (n + m) = m.gcd (n + 1 * m)",
" (m... | [
" m.gcd (n + k * m) = m.gcd n",
" m.gcd (n + m * k) = m.gcd n",
" m.gcd (k * m + n) = m.gcd n",
" m.gcd (m * k + n) = m.gcd n",
" (m + k * n).gcd n = m.gcd n",
" (m + n * k).gcd n = m.gcd n",
" (k * n + m).gcd n = m.gcd n",
" (n * k + m).gcd n = m.gcd n",
" m.gcd (n + m) = m.gcd (n + 1 * m)",
" (m... |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 61 | 64 | theorem AffineMap.restrict.linear_aux {φ : P₁ →ᵃ[k] P₂} {E : AffineSubspace k P₁}
{F : AffineSubspace k P₂} (hEF : E.map φ ≤ F) : E.direction ≤ F.direction.comap φ.linear := by |
rw [← Submodule.map_le_iff_le_comap, ← AffineSubspace.map_direction]
exact AffineSubspace.direction_le hEF
| [
" Nonempty ↥(map φ E)",
" ↥E →ᵃ[k] ↥F",
" ↥E → ↥F",
" ↥E.direction →ₗ[k] ↥F.direction",
" E.direction ≤ Submodule.comap φ.linear F.direction",
" (AffineSubspace.map φ E).direction ≤ F.direction",
" ∀ (p : ↥E) (v : ↥E.direction), ⟨φ ↑(v +ᵥ p), ⋯⟩ = (φ.linear.restrict ⋯) v +ᵥ ⟨φ ↑p, ⋯⟩",
" ⟨φ ↑(v +ᵥ p),... | [
" Nonempty ↥(map φ E)",
" ↥E →ᵃ[k] ↥F",
" ↥E → ↥F",
" ↥E.direction →ₗ[k] ↥F.direction",
" E.direction ≤ Submodule.comap φ.linear F.direction",
" (AffineSubspace.map φ E).direction ≤ F.direction",
" ∀ (p : ↥E) (v : ↥E.direction), ⟨φ ↑(v +ᵥ p), ⋯⟩ = (φ.linear.restrict ⋯) v +ᵥ ⟨φ ↑p, ⋯⟩",
" ⟨φ ↑(v +ᵥ p),... |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {σ R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 260 | 263 | theorem X_dvd_monomial {i : σ} {j : σ →₀ ℕ} {r : R} :
(X i : MvPolynomial σ R) ∣ monomial j r ↔ r = 0 ∨ j i ≠ 0 := by |
refine monomial_dvd_monomial.trans ?_
simp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
| [
" (monomial i) r ∣ (monomial j) s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s",
" (monomial i) r ∣ (monomial j) (r * d)",
" (monomial j) (r * d) = (monomial i) r * (monomia... | [
" (monomial i) r ∣ (monomial j) s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s",
" (s = 0 ∨ i ≤ j) ∧ r ∣ s → (monomial i) r ∣ (monomial j) s",
" (monomial i) r ∣ (monomial j) (r * d)",
" (monomial j) (r * d) = (monomial i) r * (monomia... |
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