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.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 116 | 125 | theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
(hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)] <;>
simp_rw [hXint, sub_zero]
· rfl
· convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top with ω
simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toRe... | [
" evariance X μ < ⊤",
" evariance X μ = ⊤",
" False",
" Memℒp (fun ω => X ω - ∫ (x : Ω), X x ∂μ) 2 μ",
" snorm (fun ω => X ω - ∫ (x : Ω), X x ∂μ) 2 μ < ⊤",
" (∫⁻ (x : Ω), ↑‖X x - ∫ (x : Ω), X x ∂μ‖₊ ^ ENNReal.toReal 2 ∂μ) ^ (1 / ENNReal.toReal 2) < ⊤",
" (∫⁻ (x : Ω), ↑‖X x - ∫ (x : Ω), X x ∂μ‖₊ ^ 2 ∂μ) ... | [
" evariance X μ < ⊤",
" evariance X μ = ⊤",
" False",
" Memℒp (fun ω => X ω - ∫ (x : Ω), X x ∂μ) 2 μ",
" snorm (fun ω => X ω - ∫ (x : Ω), X x ∂μ) 2 μ < ⊤",
" (∫⁻ (x : Ω), ↑‖X x - ∫ (x : Ω), X x ∂μ‖₊ ^ ENNReal.toReal 2 ∂μ) ^ (1 / ENNReal.toReal 2) < ⊤",
" (∫⁻ (x : Ω), ↑‖X x - ∫ (x : Ω), X x ∂μ‖₊ ^ 2 ∂μ) ... |
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 | 196 | 197 | theorem mul_mul_mul_comm (a b c d : G) : a * b * (c * d) = a * c * (b * d) := by |
simp only [mul_left_comm, mul_assoc]
| [
" a * b * (c * d) = a * c * (b * d)"
] | [] |
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVe... | Mathlib/RingTheory/WittVector/Domain.lean | 69 | 76 | theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by |
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
| [
" verschiebung (x.shift k.succ) = x.shift k",
" (verschiebung (x.shift k.succ)).coeff 0 = (x.shift k).coeff 0",
" k + 0 < k + 1",
" (verschiebung (x.shift k.succ)).coeff (n✝ + 1) = (x.shift k).coeff (n✝ + 1)",
" x.coeff (k.succ + n✝) = x.coeff (k + (n✝ + 1))",
" k.succ + n✝ = k + (n✝ + 1)"
] | [] |
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 | 167 | 167 | theorem one_minus_nonneg (x : I) : 0 ≤ 1 - (x : ℝ) := by | simpa using x.2.2
| [
" 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",
" 1 ∈ I",
" 0 ≤ 1",
" 1 ≤ 1",
" ↑(σ 0) = ↑1",
" ... | [
" 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",
" 1 ∈ I",
" 0 ≤ 1",
" 1 ≤ 1",
" ↑(σ 0) = ↑1",
" ... |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 75 | 78 | theorem rank_range_add_rank_ker (f : M →ₗ[R] M₁) :
Module.rank R (LinearMap.range f) + Module.rank R (LinearMap.ker f) = Module.rank R M := by |
haveI := fun p : Submodule R M => Classical.decEq (M ⧸ p)
rw [← f.quotKerEquivRange.rank_eq, rank_quotient_add_rank]
| [
" Nontrivial R",
" False",
" lift.{u, v} (Module.rank R ↥(LinearMap.range f)) + lift.{v, u} (Module.rank R ↥(LinearMap.ker f)) =\n lift.{v, u} (Module.rank R M)",
" Module.rank R ↥(LinearMap.range f) + Module.rank R ↥(LinearMap.ker f) = Module.rank R M"
] | [
" Nontrivial R",
" False",
" lift.{u, v} (Module.rank R ↥(LinearMap.range f)) + lift.{v, u} (Module.rank R ↥(LinearMap.ker f)) =\n lift.{v, u} (Module.rank R M)"
] |
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
op... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 107 | 120 | theorem hasFiniteIntegral_compProd_iff' ⦃f : β × γ → E⦄
(h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by |
rw [hasFiniteIntegral_congr h1f.ae_eq_mk,
hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk]
apply and_congr
· apply eventually_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with x hx using
hasFiniteIntegral_congr hx
· apply hasFiniteIntegral_congr
filter_upwards [ae_ae... | [
" HasFiniteIntegral (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)",
" ∫⁻ (a_1 : β), ENNReal.ofReal ((η (a, a_1)) (Prod.mk a_1 ⁻¹' s)).toReal ∂κ a < ⊤",
" ∫⁻ (b : β), ENNReal.ofReal ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal ∂κ a ≤ ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' t) ∂κ a",
" ∀ᵐ (a_1 : β) ∂κ a, ENNRea... | [
" HasFiniteIntegral (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)",
" ∫⁻ (a_1 : β), ENNReal.ofReal ((η (a, a_1)) (Prod.mk a_1 ⁻¹' s)).toReal ∂κ a < ⊤",
" ∫⁻ (b : β), ENNReal.ofReal ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal ∂κ a ≤ ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' t) ∂κ a",
" ∀ᵐ (a_1 : β) ∂κ a, ENNRea... |
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 68 | 73 | theorem mem_reesAlgebra_iff_support (f : R[X]) :
f ∈ reesAlgebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i := by |
apply forall_congr'
intro a
rw [mem_support_iff, Iff.comm, Classical.imp_iff_right_iff, Ne, ← imp_iff_not_or]
exact fun e => e.symm ▸ (I ^ a).zero_mem
| [
" (a✝ * b✝).coeff i ∈ I ^ i",
" ∑ x ∈ Finset.antidiagonal i, a✝.coeff x.1 * b✝.coeff x.2 ∈ I ^ i",
" ∀ c ∈ Finset.antidiagonal i, a✝.coeff c.1 * b✝.coeff c.2 ∈ I ^ i",
" a✝.coeff (j, k).1 * b✝.coeff (j, k).2 ∈ I ^ i",
" a✝.coeff (j, k).1 * b✝.coeff (j, k).2 ∈ I ^ (j, k).1 * I ^ (j, k).2",
" coeff 1 i ∈ I ... | [
" (a✝ * b✝).coeff i ∈ I ^ i",
" ∑ x ∈ Finset.antidiagonal i, a✝.coeff x.1 * b✝.coeff x.2 ∈ I ^ i",
" ∀ c ∈ Finset.antidiagonal i, a✝.coeff c.1 * b✝.coeff c.2 ∈ I ^ i",
" a✝.coeff (j, k).1 * b✝.coeff (j, k).2 ∈ I ^ i",
" a✝.coeff (j, k).1 * b✝.coeff (j, k).2 ∈ I ^ (j, k).1 * I ^ (j, k).2",
" coeff 1 i ∈ I ... |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.measure.lebesgue.complex from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
noncomputable section
namespace Complex... | Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean | 53 | 59 | theorem volume_preserving_equiv_pi : MeasurePreserving measurableEquivPi := by |
convert (measurableEquivPi.symm.measurable.measurePreserving volume).symm
rw [← addHaarMeasure_eq_volume_pi, ← Basis.parallelepiped_basisFun, ← Basis.addHaar,
measurableEquivPi, Homeomorph.toMeasurableEquiv_symm_coe,
ContinuousLinearEquiv.symm_toHomeomorph, ContinuousLinearEquiv.coe_toHomeomorph,
Basis... | [
" MeasurePreserving (⇑measurableEquivPi) volume volume",
" volume = Measure.map (⇑measurableEquivPi.symm) volume",
" ((Pi.basisFun ℝ (Fin 2)).map basisOneI.equivFun.toContinuousLinearEquiv.symm.toLinearEquiv).addHaar = volume"
] | [] |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 169 | 186 | theorem convexBodyLT'_mem {x : K} :
mixedEmbedding K x ∈ convexBodyLT' K f w₀ ↔
(∀ w : InfinitePlace K, w ≠ w₀ → w x < f w) ∧
|(w₀.val.embedding x).re| < 1 ∧ |(w₀.val.embedding x).im| < (f w₀: ℝ) ^ 2 := by |
simp_rw [mixedEmbedding, RingHom.prod_apply, Set.mem_prod, Set.mem_pi, Set.mem_univ,
forall_true_left, Pi.ringHom_apply, apply_ite, mem_ball_zero_iff, ← Complex.norm_real,
embedding_of_isReal_apply, norm_embedding_eq, Subtype.forall, Set.mem_setOf_eq]
refine ⟨fun ⟨h₁, h₂⟩ ↦ ⟨fun w h_ne ↦ ?_, ?_⟩, fun ⟨h₁, ... | [
" (mixedEmbedding K) x ∈ convexBodyLT' K f w₀ ↔\n (∀ (w : InfinitePlace K), w ≠ ↑w₀ → w x < ↑(f w)) ∧\n |((↑w₀).embedding x).re| < 1 ∧ |((↑w₀).embedding x).im| < ↑(f ↑w₀) ^ 2",
" ((∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n if ⟨a, b⟩ = w₀... | [] |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 65 | 67 | theorem continuous_cos : Continuous cos := by |
change Continuous fun z => (exp (z * I) + exp (-z * I)) / 2
continuity
| [
" Continuous sin",
" Continuous fun z => (cexp (-z * I) - cexp (z * I)) * I / 2",
" Continuous cos",
" Continuous fun z => (cexp (z * I) + cexp (-z * I)) / 2"
] | [
" Continuous sin",
" Continuous fun z => (cexp (-z * I) - cexp (z * I)) * I / 2"
] |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 178 | 181 | theorem fibRec_charPoly_eq {β : Type*} [CommRing β] :
fibRec.charPoly = X ^ 2 - (X + (1 : β[X])) := by |
rw [fibRec, LinearRecurrence.charPoly]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ', ← smul_X_eq_monomial]
| [
" φ⁻¹ = -ψ",
" 0 < 1",
" 0 < 5",
" 2 * 2 = 5 - 1",
" ψ⁻¹ = -φ",
" -ψ = φ⁻¹",
" φ * ψ = -1",
" (1 + √5) * (1 - √5) = -(2 * 2)",
" 1 ^ 2 - √5 ^ 2 = -(2 * 2)",
" ψ * φ = -1",
" φ + ψ = 1",
" (1 + √5) / 2 + (1 - √5) / 2 = 1",
" 1 - φ = ψ",
" 1 - ψ = φ",
" φ - ψ = √5",
" φ ^ (n + 2) - φ ^ (... | [
" φ⁻¹ = -ψ",
" 0 < 1",
" 0 < 5",
" 2 * 2 = 5 - 1",
" ψ⁻¹ = -φ",
" -ψ = φ⁻¹",
" φ * ψ = -1",
" (1 + √5) * (1 - √5) = -(2 * 2)",
" 1 ^ 2 - √5 ^ 2 = -(2 * 2)",
" ψ * φ = -1",
" φ + ψ = 1",
" (1 + √5) / 2 + (1 - √5) / 2 = 1",
" 1 - φ = ψ",
" 1 - ψ = φ",
" φ - ψ = √5",
" φ ^ (n + 2) - φ ^ (... |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.Cyclotomic.Rat
section case1
open ZMod
private lemma cube_of_castHom_ne_zero {n : ZMod 9} :
castHom (show 3 ∣ 9 by norm_num) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 := by
revert n; decide
private lemma cube_of_n... | Mathlib/NumberTheory/FLT/Three.lean | 36 | 44 | theorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :
a ^ 3 + b ^ 3 ≠ c ^ 3 := by |
simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd
apply mt (congrArg (Int.cast : ℤ → ZMod 9))
simp_rw [Int.cast_add, Int.cast_pow]
rcases cube_of_not_dvd hdvd.1.1 with ha | ha <;>
rcases cube_of_not_dvd hdvd.1.2 with hb | hb <;>
rcases cube_of_not_dvd hdvd.2 with hc | hc <;>
rw [ha, hb, hc] <;> decide
| [
" 3 ∣ 9",
" (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8",
" ∀ {n : ZMod 9}, (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8",
" ↑n ^ 3 = 1 ∨ ↑n ^ 3 = 8",
" (castHom ⋯ (ZMod 3)) ↑n ≠ 0",
" a ^ 3 + b ^ 3 ≠ c ^ 3",
" ¬↑(a ^ 3 + b ^ 3) = ↑(c ^ 3)",
" ¬↑a ^ 3 + ↑b ^ 3 = ↑c ^ 3",
" ¬1 + 1 = 1",
... | [
" 3 ∣ 9",
" (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8",
" ∀ {n : ZMod 9}, (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8",
" ↑n ^ 3 = 1 ∨ ↑n ^ 3 = 8",
" (castHom ⋯ (ZMod 3)) ↑n ≠ 0"
] |
import Mathlib.CategoryTheory.Idempotents.Karoubi
#align_import category_theory.idempotents.functor_extension from "leanprover-community/mathlib"@"5f68029a863bdf76029fa0f7a519e6163c14152e"
namespace CategoryTheory
namespace Idempotents
open Category Karoubi
variable {C D E : Type*} [Category C] [Category D] [Ca... | Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | 35 | 40 | theorem natTrans_eq {F G : Karoubi C ⥤ D} (φ : F ⟶ G) (P : Karoubi C) :
φ.app P = F.map (decompId_i P) ≫ φ.app P.X ≫ G.map (decompId_p P) := by |
rw [← φ.naturality, ← assoc, ← F.map_comp]
conv_lhs => rw [← id_comp (φ.app P), ← F.map_id]
congr
apply decompId
| [
" φ.app P = F.map P.decompId_i ≫ φ.app { X := P.X, p := 𝟙 P.X, idem := ⋯ } ≫ G.map P.decompId_p",
" φ.app P = F.map (P.decompId_i ≫ P.decompId_p) ≫ φ.app P",
"C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{u_4, u_1} C\ninst✝¹ : Category.{u_5, u_2} D\ninst✝ : Category.{?u.41, u_3} E\nF G : Karoubi... | [] |
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y X₁... | Mathlib/CategoryTheory/Conj.lean | 50 | 52 | theorem homCongr_apply {X Y X₁ Y₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) (f : X ⟶ Y) :
α.homCongr β f = α.inv ≫ f ≫ β.hom := by |
rfl
| [
" α.hom ≫ (α.inv ≫ f ≫ β.hom) ≫ β.inv = f",
" α.inv ≫ (α.hom ≫ f ≫ β.inv) ≫ β.hom = f",
" (α.homCongr β) f = α.inv ≫ f ≫ β.hom"
] | [
" α.hom ≫ (α.inv ≫ f ≫ β.hom) ≫ β.inv = f",
" α.inv ≫ (α.hom ≫ f ≫ β.inv) ≫ β.hom = f"
] |
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {𝕜 E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 46 | 52 | theorem ContinuousLinearMap.dslope_comp {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
(f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a) :
dslope (f ∘ g) a b = f (dslope g a b) := by |
rcases eq_or_ne b a with (rfl | hne)
· simp only [dslope_same]
exact (f.hasFDerivAt.comp_hasDerivAt b (H rfl).hasDerivAt).deriv
· simpa only [dslope_of_ne _ hne] using f.toLinearMap.slope_comp g a b
| [
" dslope (⇑f ∘ g) a b = f (dslope g a b)",
" dslope (⇑f ∘ g) b b = f (dslope g b b)",
" deriv (⇑f ∘ g) b = f (deriv g b)"
] | [] |
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 95 | 103 | theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
HasFDerivAt polarCoord.symm
(LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
!![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])) p := by |
rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
convert HasFDerivAt.prod (𝕜 := ℝ)
(hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
(hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>
simp [smul_smul, add_comm, neg_mul, smul_neg, n... | [
" ∀ ⦃x : ℝ × ℝ⦄,\n x ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0} →\n (fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) x ∈ Ioi 0 ×ˢ Ioo (-π) π",
" (fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y) ∈ Ioi 0 ×ˢ Ioo (-π) π",
" 0 < x ^ 2 + y ^ 2 ∧ (0 ≤ (Complex.equivRealPro... | [
" ∀ ⦃x : ℝ × ℝ⦄,\n x ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0} →\n (fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) x ∈ Ioi 0 ×ˢ Ioo (-π) π",
" (fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y) ∈ Ioi 0 ×ˢ Ioo (-π) π",
" 0 < x ^ 2 + y ^ 2 ∧ (0 ≤ (Complex.equivRealPro... |
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 41 | 46 | theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c := by |
refine
tendsto_nhds_unique
(intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_
simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
exact tendsto_exp_atBot.const_sub _
| [
" IntegrableOn rexp (Iic c) volume",
" ∫ (x : ℝ) in id y..c, ‖rexp x‖ ≤ rexp c",
" 0 ≤ rexp (id y)",
" ∫ (x : ℝ) in Iic c, rexp x = rexp c",
" Tendsto (fun i => ∫ (x : ℝ) in id i..c, rexp x) atBot (𝓝 (rexp c))",
" 𝓝 (rexp c) = 𝓝 (rexp c - 0)",
" Tendsto (fun i => rexp c - rexp (id i)) atBot (𝓝 (rexp... | [
" IntegrableOn rexp (Iic c) volume",
" ∫ (x : ℝ) in id y..c, ‖rexp x‖ ≤ rexp c",
" 0 ≤ rexp (id y)"
] |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 36 | 41 | theorem AffineSubspace.isClosed_direction_iff (s : AffineSubspace 𝕜 Q) :
IsClosed (s.direction : Set W) ↔ IsClosed (s : Set Q) := by |
rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩); · simp [isClosed_singleton]
rw [← (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image,
AffineSubspace.coe_direction_eq_vsub_set_right hx]
rfl
| [
" IsClosed ↑s.direction ↔ IsClosed ↑s",
" IsClosed ↑⊥.direction ↔ IsClosed ↑⊥",
" IsClosed ((fun x_1 => x_1 -ᵥ x) '' ↑s) ↔ IsClosed (⇑(IsometryEquiv.vaddConst x).toHomeomorph.symm '' ↑s)"
] | [] |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 254 | 257 | theorem smul (H : InnerRegularWRT μ p q) (c : ℝ≥0∞) : InnerRegularWRT (c • μ) p q := by |
intro U hU r hr
rw [smul_apply, H.measure_eq_iSup hU, smul_eq_mul] at hr
simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
| [
" μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K",
" r < ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K",
" ∃ K ⊆ U, p K ∧ μ U < μ K + ε",
" μ U < μ ∅ + ε",
" (map f μ).InnerRegularWRT pb qb",
" ∃ K ⊆ U, pb K ∧ r < (map f μ) K",
" r < (map f μ) (f '' K)",
" (map (⇑f) μ).InnerRegularWRT pb qb",
" ∃ K ⊆ U, pb K ∧ r... | [
" μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K",
" r < ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K",
" ∃ K ⊆ U, p K ∧ μ U < μ K + ε",
" μ U < μ ∅ + ε",
" (map f μ).InnerRegularWRT pb qb",
" ∃ K ⊆ U, pb K ∧ r < (map f μ) K",
" r < (map f μ) (f '' K)",
" (map (⇑f) μ).InnerRegularWRT pb qb",
" ∃ K ⊆ U, pb K ∧ r... |
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 60 | 61 | theorem ker_pi (f : (i : ι) → M₂ →ₗ[R] φ i) : ker (pi f) = ⨅ i : ι, ker (f i) := by |
ext c; simp [funext_iff]
| [
" ker (pi f) = ⨅ i, ker (f i)",
" c ∈ ker (pi f) ↔ c ∈ ⨅ i, ker (f i)"
] | [] |
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegrallyClosed
#align_import ring_theory.polynomial.gauss_lemma from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
open... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 124 | 130 | theorem IsPrimitive.irreducible_of_irreducible_map_of_injective (h_irr : Irreducible (map φ f)) :
Irreducible f := by |
refine
⟨fun h => h_irr.not_unit (IsUnit.map (mapRingHom φ) h), fun a b h =>
(h_irr.isUnit_or_isUnit <| by rw [h, Polynomial.map_mul]).imp ?_ ?_⟩
all_goals apply ((isPrimitive_of_dvd hf _).isUnit_iff_isUnit_map_of_injective hinj).mpr
exacts [Dvd.intro _ h.symm, Dvd.intro_left _ h.symm]
| [
" IsUnit f ↔ IsUnit (map φ f)",
" IsUnit f",
" IsUnit (C (f.coeff 0))",
" Irreducible f",
" map φ f = ?m.6631 a b h * ?m.6632 a b h",
" IsUnit (map φ b) → IsUnit b",
" IsUnit (map φ a) → IsUnit a",
" b ∣ f"
] | [
" IsUnit f ↔ IsUnit (map φ f)",
" IsUnit f",
" IsUnit (C (f.coeff 0))"
] |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 171 | 178 | theorem trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)
(hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
IntervalIntegrable f μ (a m) (a n) := by |
revert hint
refine Nat.le_induction ?_ ?_ n hmn
· simp
· intro p hp IH h
exact (IH fun k hk => h k (Ico_subset_Ico_right p.le_succ hk)).trans (h p (by simp [hp]))
| [
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f [[a, b]] μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Icc a b) μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ",
" Interva... | [
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f [[a, b]] μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Icc a b) μ",
" IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ",
" Interva... |
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 | 158 | 159 | theorem circleMap_eq_center_iff {c : ℂ} {R : ℝ} {θ : ℝ} : circleMap c R θ = c ↔ R = 0 := by |
simp [circleMap, exp_ne_zero]
| [
" 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",
" range (circleMap c R) = c +ᵥ R • range fun θ => cexp (↑θ * I)",
" (c +ᵥ... |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 126 | 128 | theorem dedup_nsmul {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : (n • s).dedup = s.dedup := by |
ext a
by_cases h : a ∈ s <;> simp [h, h0]
| [
" count a (dedup (Quot.mk Setoid.r x✝)) = if a ∈ Quot.mk Setoid.r x✝ then 1 else 0",
" List.count a x✝.dedup = if a ∈ x✝ then 1 else 0",
" s ≤ s.dedup ↔ s.Nodup",
" s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t",
" (map f s.dedup).dedup = (map f s).dedup",
" (n • s).dedup = s.dedup",
" count a (n • s).de... | [
" count a (dedup (Quot.mk Setoid.r x✝)) = if a ∈ Quot.mk Setoid.r x✝ then 1 else 0",
" List.count a x✝.dedup = if a ∈ x✝ then 1 else 0",
" s ≤ s.dedup ↔ s.Nodup",
" s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t",
" (map f s.dedup).dedup = (map f s).dedup"
] |
import Mathlib.CategoryTheory.Limits.Preserves.Opposites
import Mathlib.Topology.Category.TopCat.Yoneda
import Mathlib.Condensed.Explicit
universe w w' v u
open CategoryTheory Opposite Limits regularTopology ContinuousMap
variable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})
(X : Type w') [TopologicalSpac... | Mathlib/Condensed/TopComparison.lean | 40 | 58 | theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
[PreservesLimit (cospan π π) G]
{a : C(G.obj Z, X)}
(ha : a ∘ (G.map pullback.fst) = a ∘ (G.map (pullback.snd (f := π) (g := π)))) :
Function.FactorsThrough a (G.map π) := by |
intro x y hxy
let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|
(TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩
have ha' := congr_fun ha xy
dsimp at ha'
have h₁ : ∀ y, G.map pullback.fst ((PreservesPullback.iso G π π).inv y) =
pullback.fst (f := G.map π) (g... | [
" Function.FactorsThrough ⇑a ⇑(G.map π)",
" a x = a y",
" ∀ (y : ↑(pullback (G.map π) (G.map π))), (G.map pullback.fst) ((PreservesPullback.iso G π π).inv y) = pullback.fst y",
" ∀ (y : ↑(pullback (G.map π) (G.map π))),\n (G.map pullback.fst) ((PreservesPullback.iso G π π).inv y) =\n ((PreservesPullba... | [] |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 141 | 145 | theorem StrictConvexSpace.of_norm_add
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y) : StrictConvexSpace ℝ E := by |
refine StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun h₂ => ?_
rw [mem_sphere_zero_iff_norm] at hx hy
exact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h₂)
| [
" StrictConvex 𝕜 (closedBall x r)",
" StrictConvex 𝕜 (x +ᵥ closedBall 0 r)",
" StrictConvex 𝕜 (closedBall 0 r)",
" StrictConvexSpace ℝ E",
" (fun x y => ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)) x y",
" ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)",
" (AffineMap.lineMa... | [
" StrictConvex 𝕜 (closedBall x r)",
" StrictConvex 𝕜 (x +ᵥ closedBall 0 r)",
" StrictConvex 𝕜 (closedBall 0 r)",
" StrictConvexSpace ℝ E",
" (fun x y => ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)) x y",
" ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)",
" (AffineMap.lineMa... |
import Mathlib.Data.Matroid.Dual
open Set
namespace Matroid
variable {α : Type*} {M : Matroid α} {R I J X Y : Set α}
section restrict
@[simps] def restrictIndepMatroid (M : Matroid α) (R : Set α) : IndepMatroid α where
E := R
Indep I := M.Indep I ∧ I ⊆ R
indep_empty := ⟨M.empty_indep, empty_subset _⟩
i... | Mathlib/Data/Matroid/Restrict.lean | 156 | 157 | theorem base_restrict_iff' : (M ↾ X).Base I ↔ M.Basis' I X := by |
simp_rw [Basis', base_iff_maximal_indep, mem_maximals_setOf_iff, restrict_indep_iff]
| [
" ∀ ⦃I B : Set α⦄,\n (fun I => M.Indep I ∧ I ⊆ R) I →\n I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | (fun I => M.Indep I ∧ I ⊆ R) I} →\n B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | (fun I => M.Indep I ∧ I ⊆ R) I} →\n ∃ x ∈ B \\ I, (fun I => M.Indep I ∧ I ⊆ R) (insert x I)",
" ∃ x ∈ I' \\ I, (fun ... | [
" ∀ ⦃I B : Set α⦄,\n (fun I => M.Indep I ∧ I ⊆ R) I →\n I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | (fun I => M.Indep I ∧ I ⊆ R) I} →\n B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | (fun I => M.Indep I ∧ I ⊆ R) I} →\n ∃ x ∈ B \\ I, (fun I => M.Indep I ∧ I ⊆ R) (insert x I)",
" ∃ x ∈ I' \\ I, (fun ... |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 175 | 175 | theorem add_top (x : PartENat) : x + ⊤ = ⊤ := by | rw [add_comm, top_add]
| [
" ∀ (a : PartENat), P ⊤ → (∀ (n : ℕ), P ↑n) → P a",
" x + ⊤ = ⊤"
] | [
" ∀ (a : PartENat), P ⊤ → (∀ (n : ℕ), P ↑n) → P a"
] |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.IntervalCases
#align_import number_theory.ADE_inequality from "leanprover-community/math... | Mathlib/NumberTheory/ADEInequality.lean | 117 | 119 | theorem sumInv_pqr (p q r : ℕ+) : sumInv {p, q, r} = (p : ℚ)⁻¹ + (q : ℚ)⁻¹ + (r : ℚ)⁻¹ := by |
simp only [sumInv, add_zero, insert_eq_cons, add_assoc, map_cons, sum_cons,
map_singleton, sum_singleton]
| [
" sumInv {p, q, r} = (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹"
] | [] |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
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
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 80 | 80 | theorem isReflection_inv : cs.IsReflection t⁻¹ := by | rwa [ht.inv]
| [
" cs.IsReflection (cs.simple i)",
" cs.simple i = 1 * cs.simple i * 1⁻¹",
" t ^ 2 = 1",
" (w * cs.simple i * w⁻¹) ^ 2 = 1",
" t * t = 1",
" w * cs.simple i * w⁻¹ * (w * cs.simple i * w⁻¹) = 1",
" t⁻¹ = t",
" (w * cs.simple i * w⁻¹)⁻¹ = w * cs.simple i * w⁻¹",
" cs.IsReflection t⁻¹"
] | [
" cs.IsReflection (cs.simple i)",
" cs.simple i = 1 * cs.simple i * 1⁻¹",
" t ^ 2 = 1",
" (w * cs.simple i * w⁻¹) ^ 2 = 1",
" t * t = 1",
" w * cs.simple i * w⁻¹ * (w * cs.simple i * w⁻¹) = 1",
" t⁻¹ = t",
" (w * cs.simple i * w⁻¹)⁻¹ = w * cs.simple i * w⁻¹"
] |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 125 | 130 | theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by |
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
| [
" A.Definable L' s",
" A.Definable L' (setOf ψ.Realize)",
" setOf ψ.Realize = setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" x ∈ setOf ψ.Realize ↔ x ∈ setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" A.Definable L s ↔ ∃ φ, s = {v | φ.Realize (Sum.elim Subtype.val v)}",
" (∃ b, s = se... | [
" A.Definable L' s",
" A.Definable L' (setOf ψ.Realize)",
" setOf ψ.Realize = setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" x ∈ setOf ψ.Realize ↔ x ∈ setOf ((LHom.addConstants (↑A) φ).onFormula ψ).Realize",
" A.Definable L s ↔ ∃ φ, s = {v | φ.Realize (Sum.elim Subtype.val v)}",
" (∃ b, s = se... |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 150 | 152 | theorem second_continuant_aux_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.continuantsAux 2 = ⟨gp.b * g.h + gp.a, gp.b⟩ := by |
simp [zeroth_s_eq, continuantsAux, nextContinuants, nextDenominator, nextNumerator]
| [
" ∃ conts, g.continuants n = conts ∧ conts.a = A",
" ∃ conts, g.continuants n = conts ∧ conts.b = B",
" g.convergents 0 = g.h",
" g.continuantsAux 2 = { a := gp.b * g.h + gp.a, b := gp.b }"
] | [
" ∃ conts, g.continuants n = conts ∧ conts.a = A",
" ∃ conts, g.continuants n = conts ∧ conts.b = B",
" g.convergents 0 = g.h"
] |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 51 | 53 | theorem pairwise_lt (n m : ℕ) : Pairwise (· < ·) (Ico n m) := by |
dsimp [Ico]
simp [pairwise_lt_range', autoParam]
| [
" Ico 0 n = range n",
" (Ico n m).length = m - n",
" (range' n (m - n)).length = m - n",
" Pairwise (fun x x_1 => x < x_1) (Ico n m)",
" Pairwise (fun x x_1 => x < x_1) (range' n (m - n))"
] | [
" Ico 0 n = range n",
" (Ico n m).length = m - n",
" (range' n (m - n)).length = m - n"
] |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 170 | 171 | theorem lineDerivWithin_univ : lineDerivWithin 𝕜 f univ x v = lineDeriv 𝕜 f x v := by |
simp [lineDerivWithin, lineDeriv]
| [
" HasLineDerivWithinAt 𝕜 f f' univ x v ↔ HasLineDerivAt 𝕜 f f' x v",
" HasLineDerivAt 𝕜 f f' x v ↔ (fun t => f (x + t • v) - f x - t • f') =o[𝓝 0] fun t => t",
" lineDeriv 𝕜 f x v = f'",
" LineDifferentiableWithinAt 𝕜 f univ x v ↔ LineDifferentiableAt 𝕜 f x v",
" lineDerivWithin 𝕜 f univ x v = lineD... | [
" HasLineDerivWithinAt 𝕜 f f' univ x v ↔ HasLineDerivAt 𝕜 f f' x v",
" HasLineDerivAt 𝕜 f f' x v ↔ (fun t => f (x + t • v) - f x - t • f') =o[𝓝 0] fun t => t",
" lineDeriv 𝕜 f x v = f'",
" LineDifferentiableWithinAt 𝕜 f univ x v ↔ LineDifferentiableAt 𝕜 f x v"
] |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 38 | 41 | theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by |
rw [← Functor.map_comp, prod_comp, Category.comp_id]
| [
" F.map (𝟙 W, f ≫ g) = F.map (𝟙 W, f) ≫ F.map (𝟙 W, g)",
" F.map (f ≫ g, 𝟙 W) = F.map (f, 𝟙 W) ≫ F.map (g, 𝟙 W)"
] | [
" F.map (𝟙 W, f ≫ g) = F.map (𝟙 W, f) ≫ F.map (𝟙 W, g)"
] |
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 78 | 81 | theorem isOpen_connectedComponent [LocallyConnectedSpace α] {x : α} :
IsOpen (connectedComponent x) := by |
rw [← connectedComponentIn_univ]
exact isOpen_univ.connectedComponentIn
| [
" LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V",
" (∀ (x : α), (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id) ↔\n ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V",
" (𝓝 x✝).HasBasis (fun s => IsOpen s ∧ x✝ ∈ s ∧ IsConnected s) i... | [
" LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V",
" (∀ (x : α), (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id) ↔\n ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V",
" (𝓝 x✝).HasBasis (fun s => IsOpen s ∧ x✝ ∈ s ∧ IsConnected s) i... |
import Mathlib.Logic.Basic
import Mathlib.Init.ZeroOne
import Mathlib.Init.Order.Defs
#align_import algebra.ne_zero from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
variable {R : Type*} [Zero R]
class NeZero (n : R) : Prop where
out : n ≠ 0
#align ne_zero NeZero
theorem NeZero... | Mathlib/Algebra/NeZero.lean | 45 | 45 | theorem not_neZero {n : R} : ¬NeZero n ↔ n = 0 := by | simp [neZero_iff]
| [
" ¬NeZero n ↔ n = 0"
] | [] |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 110 | 121 | theorem nhds_mkOfNhds_of_hasBasis {n : α → Filter α} {ι : α → Sort*} {p : ∀ a, ι a → Prop}
{s : ∀ a, ι a → Set α} (hb : ∀ a, (n a).HasBasis (p a) (s a))
(hpure : ∀ a i, p a i → a ∈ s a i) (hopen : ∀ a i, p a i → ∀ᶠ x in n a, s a i ∈ n x) (a : α) :
@nhds α (.mkOfNhds n) a = n a := by |
let t : TopologicalSpace α := .mkOfNhds n
apply le_antisymm
· intro U hU
replace hpure : pure ≤ n := fun x ↦ (hb x).ge_iff.2 (hpure x)
refine mem_nhds_iff.2 ⟨{x | U ∈ n x}, fun x hx ↦ hpure x hx, fun x hx ↦ ?_, hU⟩
rcases (hb x).mem_iff.1 hx with ⟨i, hpi, hi⟩
exact (hopen x i hpi).mono fun y hy ↦... | [
" 𝓝 a = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s",
" ⨅ s ∈ {s | a ∈ s ∧ IsOpen s}, 𝓟 s = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s",
" ∀ i ∈ {s | a ∈ s ∧ IsOpen s}, ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 i",
" ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 s",
" ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 s✝",
" ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, �... | [
" 𝓝 a = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s",
" ⨅ s ∈ {s | a ∈ s ∧ IsOpen s}, 𝓟 s = ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s",
" ∀ i ∈ {s | a ∈ s ∧ IsOpen s}, ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 i",
" ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 s",
" ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, 𝓟 s ≤ 𝓟 s✝",
" ⨅ s ∈ {s | a ∈ s ∧ s ∈ g}, �... |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 84 | 87 | theorem natDegree_add_le_iff_right {n : ℕ} (p q : R[X]) (pn : p.natDegree ≤ n) :
(p + q).natDegree ≤ n ↔ q.natDegree ≤ n := by |
rw [add_comm]
exact natDegree_add_le_iff_left _ _ pn
| [
" (p.comp q).natDegree ≤ p.natDegree * q.natDegree",
" 0 ≤ p.natDegree * q.natDegree",
" ↑(C (p.coeff n)).natDegree + n • ↑q.natDegree = ↑(n * q.natDegree)",
" ↑n * ↑q.natDegree = ↑(n * q.natDegree)",
" False",
" p.natDegree ≤ n ↔ ∀ (N : ℕ), n < N → p.coeff N = 0",
" (p + q).natDegree ≤ n ↔ p.natDegree ... | [
" (p.comp q).natDegree ≤ p.natDegree * q.natDegree",
" 0 ≤ p.natDegree * q.natDegree",
" ↑(C (p.coeff n)).natDegree + n • ↑q.natDegree = ↑(n * q.natDegree)",
" ↑n * ↑q.natDegree = ↑(n * q.natDegree)",
" False",
" p.natDegree ≤ n ↔ ∀ (N : ℕ), n < N → p.coeff N = 0",
" (p + q).natDegree ≤ n ↔ p.natDegree ... |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 151 | 153 | theorem mirror_trailingCoeff : p.mirror.trailingCoeff = p.leadingCoeff := by |
rw [leadingCoeff, trailingCoeff, mirror_natTrailingDegree, coeff_mirror,
revAt_le (Nat.le_add_left _ _), add_tsub_cancel_right]
| [
" mirror 0 = 0",
" ((monomial n) a).mirror = (monomial n) a",
" p.mirror.natDegree = p.natDegree",
" p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0",
" p.mirror.natTrailingDegree = p.natTrailingDegree",
" p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)",
"... | [
" mirror 0 = 0",
" ((monomial n) a).mirror = (monomial n) a",
" p.mirror.natDegree = p.natDegree",
" p.reverse.leadingCoeff * (X ^ p.natTrailingDegree).leadingCoeff ≠ 0",
" p.mirror.natTrailingDegree = p.natTrailingDegree",
" p.mirror.coeff n = p.coeff ((revAt (p.natDegree + p.natTrailingDegree)) n)",
"... |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 293 | 295 | theorem finsuppTensorFinsuppLid_apply_apply (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ) :
finsuppTensorFinsuppLid R N ι κ (f ⊗ₜ[R] g) (a, b) = f a • g b := by |
simp [finsuppTensorFinsuppLid]
| [
" (finsuppTensorFinsupp R S M N ι κ) (Finsupp.single i m ⊗ₜ[R] Finsupp.single k n) = Finsupp.single (i, k) (m ⊗ₜ[R] n)",
" ((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k",
" ((finsuppTensorFinsupp R S M N ι κ) (0 ⊗ₜ[R] g)) (i, k) = 0 i ⊗ₜ[R] g k",
" ∀ (f g_1 : ι →₀ M),\n ((finsuppT... | [
" (finsuppTensorFinsupp R S M N ι κ) (Finsupp.single i m ⊗ₜ[R] Finsupp.single k n) = Finsupp.single (i, k) (m ⊗ₜ[R] n)",
" ((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k",
" ((finsuppTensorFinsupp R S M N ι κ) (0 ⊗ₜ[R] g)) (i, k) = 0 i ⊗ₜ[R] g k",
" ∀ (f g_1 : ι →₀ M),\n ((finsuppT... |
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 64 | 66 | theorem toNat_strictMonoOn : StrictMonoOn toNat (Iio ℵ₀) := by |
simp only [← range_natCast, StrictMonoOn, forall_mem_range, toNat_natCast, Nat.cast_lt]
exact fun _ _ ↦ id
| [
" toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c",
" toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀",
" ↑(toNat c) = c",
" ↑(toNat ↑c) = ↑c",
" ↑(toNat c) = ↑(Classical.choose ⋯)",
" toNat c = 0",
" ↑(toNat c) = 0",
" StrictMonoOn (⇑toNat) (Iio ℵ₀)",
" ∀ (i i_1 : ℕ), i < i_1 → i < i_1"
] | [
" toNat c = 0 ↔ c = 0 ∨ ℵ₀ ≤ c",
" toNat c ≠ 0 ↔ c ≠ 0 ∧ c < ℵ₀",
" ↑(toNat c) = c",
" ↑(toNat ↑c) = ↑c",
" ↑(toNat c) = ↑(Classical.choose ⋯)",
" toNat c = 0",
" ↑(toNat c) = 0"
] |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 68 | 73 | theorem inr_mem_sumLift₂ {c₂ : γ₂} :
inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by |
rw [mem_sumLift₂, or_iff_right]
· simp only [inr.injEq, exists_and_left, exists_eq_left']
rintro ⟨_, _, c₂, _, _, h, _⟩
exact inr_ne_inl h
| [
" c ∈ sumLift₂ f g a b ↔\n (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂",
" c ∈ sumLift₂ f g a b →\n (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ ... | [
" c ∈ sumLift₂ f g a b ↔\n (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂",
" c ∈ sumLift₂ f g a b →\n (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ ... |
import Mathlib.Analysis.SpecialFunctions.Bernstein
import Mathlib.Topology.Algebra.Algebra
#align_import topology.continuous_function.weierstrass from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open ContinuousMap Filter
open scoped unitInterval
| Mathlib/Topology/ContinuousFunction/Weierstrass.lean | 32 | 44 | theorem polynomialFunctions_closure_eq_top' : (polynomialFunctions I).topologicalClosure = ⊤ := by |
rw [eq_top_iff]
rintro f -
refine Filter.Frequently.mem_closure ?_
refine Filter.Tendsto.frequently (bernsteinApproximation_uniform f) ?_
apply frequently_of_forall
intro n
simp only [SetLike.mem_coe]
apply Subalgebra.sum_mem
rintro n -
apply Subalgebra.smul_mem
dsimp [bernstein, polynomialFuncti... | [
" (polynomialFunctions I).topologicalClosure = ⊤",
" ⊤ ≤ (polynomialFunctions I).topologicalClosure",
" f ∈ (polynomialFunctions I).topologicalClosure",
" ∃ᶠ (x : C(↑I, ℝ)) in nhds f, x ∈ ↑(polynomialFunctions I)",
" ∃ᶠ (x : ℕ) in atTop, bernsteinApproximation x f ∈ ↑(polynomialFunctions I)",
" ∀ (x : ℕ),... | [] |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 102 | 103 | theorem supported_univ : supported R (Set.univ : Set σ) = ⊤ := by |
simp [Algebra.eq_top_iff, mem_supported]
| [
" supported R s = (rename Subtype.val).range",
" (aeval fun x => X ↑x).range = (aeval (X ∘ Subtype.val)).range",
" (supportedEquivMvPolynomial s).symm (C x) = (algebraMap R ↥(supported R s)) x",
" ↑((supportedEquivMvPolynomial s).symm (C x)) = ↑((algebraMap R ↥(supported R s)) x)",
" ↑((supportedEquivMvPoly... | [
" supported R s = (rename Subtype.val).range",
" (aeval fun x => X ↑x).range = (aeval (X ∘ Subtype.val)).range",
" (supportedEquivMvPolynomial s).symm (C x) = (algebraMap R ↥(supported R s)) x",
" ↑((supportedEquivMvPolynomial s).symm (C x)) = ↑((algebraMap R ↥(supported R s)) x)",
" ↑((supportedEquivMvPoly... |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coef... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 127 | 133 | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by |
cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative]
| [
" (coeff R n) f.derivativeFun = (coeff R (n + 1)) f * (↑n + 1)",
" (↑f).derivativeFun = ↑(derivative f)",
" (coeff R n✝) (↑f).derivativeFun = (coeff R n✝) ↑(derivative f)",
" (f + g).derivativeFun = f.derivativeFun + g.derivativeFun",
" (coeff R n✝) (f + g).derivativeFun = (coeff R n✝) (f.derivativeFun + g.... | [
" (coeff R n) f.derivativeFun = (coeff R (n + 1)) f * (↑n + 1)",
" (↑f).derivativeFun = ↑(derivative f)",
" (coeff R n✝) (↑f).derivativeFun = (coeff R n✝) ↑(derivative f)",
" (f + g).derivativeFun = f.derivativeFun + g.derivativeFun",
" (coeff R n✝) (f + g).derivativeFun = (coeff R n✝) (f.derivativeFun + g.... |
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 | 108 | 112 | theorem vars_add_subset [DecidableEq σ] (p q : MvPolynomial σ R) :
(p + q).vars ⊆ p.vars ∪ q.vars := by |
intro x hx
simp only [vars_def, Finset.mem_union, Multiset.mem_toFinset] at hx ⊢
simpa using Multiset.mem_of_le (degrees_add _ _) hx
| [
" 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"
] |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 95 | 97 | theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by | simp [hfp.2]
| [
" snorm (↑(AEEqFun.mk f ⋯)) p μ < ⊤"
] | [] |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
variable {K n : Type*} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K}
| Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 26 | 56 | theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) := by |
cases isEmpty_or_nonempty n
· exfalso
exact hμ Submodule.eq_bot_of_subsingleton
· obtain ⟨v, h_eg, h_nz⟩ := hμ.exists_hasEigenvector
obtain ⟨i, -, h_i⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun i => ‖v i‖)
have h_nz : v i ≠ 0 := by
contrapose! h_nz
ext j
rw [Pi.zero_app... | [
" ∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖)",
" False",
" v i ≠ 0",
" v = 0",
" v j = 0 j",
" ‖v j‖ ≤ 0",
" ‖v i‖ ≤ 0",
" ‖v j * (v i)⁻¹‖ ≤ 1",
" ‖v j‖ ≤ ‖v i‖",
" ∃ k, ‖A k k - μ‖ ≤ ∑ j ∈ Finset.univ.erase k, ‖A k j‖",
" ‖A i i - μ‖ ≤ ∑ j ∈ Finset.univ.erase i, ‖A ... | [] |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 93 | 102 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by |
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul,... | [
" IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) volume 0 (1 / 2)",
" IntervalIntegrable (fun x => ↑x ^ (u - 1)) volume 0 (1 / 2)",
" -1 < (u - 1).re",
" ContinuousOn (fun x => (1 - ↑x) ^ (v - 1)) (uIcc 0 (1 / 2))",
" ∀ x ∈ uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - ↑x) ^ (v - 1)) x",
" ... | [
" IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) volume 0 (1 / 2)",
" IntervalIntegrable (fun x => ↑x ^ (u - 1)) volume 0 (1 / 2)",
" -1 < (u - 1).re",
" ContinuousOn (fun x => (1 - ↑x) ^ (v - 1)) (uIcc 0 (1 / 2))",
" ∀ x ∈ uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - ↑x) ^ (v - 1)) x",
" ... |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 91 | 101 | theorem Ideal.sInf_minimalPrimes : sInf I.minimalPrimes = I.radical := by |
rw [I.radical_eq_sInf]
apply le_antisymm
· intro x hx
rw [Ideal.mem_sInf] at hx ⊢
rintro J ⟨e, hJ⟩
obtain ⟨p, hp, hp'⟩ := Ideal.exists_minimalPrimes_le e
exact hp' (hx hp)
· apply sInf_le_sInf _
intro I hI
exact hI.1.symm
| [
" {p | p.IsPrime ∧ ⊥ ≤ p} = setOf Ideal.IsPrime",
" ∃ p ∈ I.minimalPrimes, p ≤ J",
" ∃ m ∈ {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}, OrderDual.toDual J ≤ m ∧ ∀ z ∈ {p | IsPrime p ∧ I ≤ p}, m ≤ z → z = m",
" OrderDual.toDual J ∈ {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}",
" J.IsPrime",
" ∀ c ⊆ {p | IsPrime ... | [
" {p | p.IsPrime ∧ ⊥ ≤ p} = setOf Ideal.IsPrime",
" ∃ p ∈ I.minimalPrimes, p ≤ J",
" ∃ m ∈ {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}, OrderDual.toDual J ≤ m ∧ ∀ z ∈ {p | IsPrime p ∧ I ≤ p}, m ≤ z → z = m",
" OrderDual.toDual J ∈ {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}",
" J.IsPrime",
" ∀ c ⊆ {p | IsPrime ... |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 87 | 92 | theorem Invariant.comp [IsSFiniteKernel κ] (hκ : Invariant κ μ) (hη : Invariant η μ) :
Invariant (κ ∘ₖ η) μ := by |
cases' isEmpty_or_nonempty α with _ hα
· exact Subsingleton.elim _ _
· simp_rw [Invariant, ← comp_const_apply_eq_bind (κ ∘ₖ η) μ hα.some, comp_assoc, hη.comp_const,
hκ.comp_const, const_apply]
| [
" (μ + ν).bind ⇑κ = μ.bind ⇑κ + ν.bind ⇑κ",
" ((μ + ν).bind ⇑κ) s = (μ.bind ⇑κ + ν.bind ⇑κ) s",
" (r • μ).bind ⇑κ = r • μ.bind ⇑κ",
" ((r • μ).bind ⇑κ) s = (r • μ.bind ⇑κ) s",
" const α (μ.bind ⇑κ) = κ ∘ₖ const α μ",
" ((const α (μ.bind ⇑κ)) a) s = ((κ ∘ₖ const α μ) a) s",
" (κ ∘ₖ const α μ) a = μ.bind ... | [
" (μ + ν).bind ⇑κ = μ.bind ⇑κ + ν.bind ⇑κ",
" ((μ + ν).bind ⇑κ) s = (μ.bind ⇑κ + ν.bind ⇑κ) s",
" (r • μ).bind ⇑κ = r • μ.bind ⇑κ",
" ((r • μ).bind ⇑κ) s = (r • μ.bind ⇑κ) s",
" const α (μ.bind ⇑κ) = κ ∘ₖ const α μ",
" ((const α (μ.bind ⇑κ)) a) s = ((κ ∘ₖ const α μ) a) s",
" (κ ∘ₖ const α μ) a = μ.bind ... |
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import order.succ_pred.linear_locally_finite from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Order
variable {ι : Type*}... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 72 | 74 | theorem le_succFn (i : ι) : i ≤ succFn i := by |
rw [le_isGLB_iff (succFn_spec i), mem_lowerBounds]
exact fun x hx ↦ le_of_lt hx
| [
" i ≤ succFn i",
" ∀ x ∈ Set.Ioi i, i ≤ x"
] | [] |
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
#align_import cat... | Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean | 51 | 54 | theorem pullbackFst_eq_pullback_snd (F : WalkingParallelPair ⥤ C) :
pullbackFst F = pullback.snd := by |
convert (eq_whisker pullback.condition Limits.prod.fst :
(_ : constructEqualizer F ⟶ F.obj WalkingParallelPair.zero) = _) <;> simp
| [
" pullbackFst F = pullback.snd",
" pullbackFst F =\n (pullback.fst ≫ prod.lift (𝟙 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.left)) ≫ prod.fst",
" pullback.snd =\n (pullback.snd ≫ prod.lift (𝟙 (F.obj WalkingParallelPair.zero)) (F.map WalkingParallelPairHom.right)) ≫ prod.fst"
] | [] |
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u w v' u' w'
open ... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 113 | 116 | theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (forget A)) := by |
simp only [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top]
rfl
| [
" ∀ {Y Z : C} {f_1 : Y ⟶ U},\n (fun V i => ∃ t, (f.app { unop := V }) t = (G.map i.op) s) Y f_1 →\n ∀ (g : Z ⟶ Y), (fun V i => ∃ t, (f.app { unop := V }) t = (G.map i.op) s) Z (g ≫ f_1)",
" ∃ t, (f.app { unop := W }) t = (G.map (j ≫ i).op) s",
" (f.app { unop := W }) ((F.map j.op) t) = (G.map (j ≫ i).op... | [
" ∀ {Y Z : C} {f_1 : Y ⟶ U},\n (fun V i => ∃ t, (f.app { unop := V }) t = (G.map i.op) s) Y f_1 →\n ∀ (g : Z ⟶ Y), (fun V i => ∃ t, (f.app { unop := V }) t = (G.map i.op) s) Z (g ≫ f_1)",
" ∃ t, (f.app { unop := W }) t = (G.map (j ≫ i).op) s",
" (f.app { unop := W }) ((F.map j.op) t) = (G.map (j ≫ i).op... |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.determinant from "lea... | Mathlib/LinearAlgebra/Determinant.lean | 96 | 99 | theorem det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) :
det (M * N * M') = det N := by |
rw [← det_comm' hM'M hMM', ← Matrix.mul_assoc, hM'M, Matrix.one_mul]
| [
" (M * N).det = (N * M).det",
" (M * N * M').det = N.det"
] | [
" (M * N).det = (N * M).det"
] |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
namespace SimpleGraph
universe u v
variable {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'}
namespace Subgraph
protected structure Preconnected (H : G.Subgraph) : Prop where
protected coe : H.coe.Preconnected
instance {H : G.Subgraph}... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 64 | 69 | theorem singletonSubgraph_connected {v : V} : (G.singletonSubgraph v).Connected := by |
refine ⟨⟨?_⟩⟩
rintro ⟨a, ha⟩ ⟨b, hb⟩
simp only [singletonSubgraph_verts, Set.mem_singleton_iff] at ha hb
subst_vars
rfl
| [
" H.Connected ↔ H.Preconnected ∧ H.verts.Nonempty",
" H.Preconnected",
" H.verts.Nonempty",
" (G.singletonSubgraph v).Connected",
" (G.singletonSubgraph v).coe.Preconnected",
" (G.singletonSubgraph v).coe.Reachable ⟨a, ha⟩ ⟨b, hb⟩",
" (G.singletonSubgraph v).coe.Reachable ⟨a, ha✝⟩ ⟨b, hb✝⟩",
" (G.sing... | [
" H.Connected ↔ H.Preconnected ∧ H.verts.Nonempty",
" H.Preconnected",
" H.verts.Nonempty"
] |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 48 | 49 | theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iic]
| [
" ⇑e ⁻¹' Iic b = Iic (e.symm b)",
" x ∈ ⇑e ⁻¹' Iic b ↔ x ∈ Iic (e.symm b)",
" ⇑e ⁻¹' Ici b = Ici (e.symm b)",
" x ∈ ⇑e ⁻¹' Ici b ↔ x ∈ Ici (e.symm b)",
" ⇑e ⁻¹' Iio b = Iio (e.symm b)",
" x ∈ ⇑e ⁻¹' Iio b ↔ x ∈ Iio (e.symm b)",
" ⇑e ⁻¹' Ioi b = Ioi (e.symm b)",
" x ∈ ⇑e ⁻¹' Ioi b ↔ x ∈ Ioi (e.symm b)"... | [
" ⇑e ⁻¹' Iic b = Iic (e.symm b)",
" x ∈ ⇑e ⁻¹' Iic b ↔ x ∈ Iic (e.symm b)",
" ⇑e ⁻¹' Ici b = Ici (e.symm b)",
" x ∈ ⇑e ⁻¹' Ici b ↔ x ∈ Ici (e.symm b)",
" ⇑e ⁻¹' Iio b = Iio (e.symm b)",
" x ∈ ⇑e ⁻¹' Iio b ↔ x ∈ Iio (e.symm b)",
" ⇑e ⁻¹' Ioi b = Ioi (e.symm b)",
" x ∈ ⇑e ⁻¹' Ioi b ↔ x ∈ Ioi (e.symm b)"... |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 59 | 60 | theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 - e2) = t1 - t2 := by | simp [left_distrib, *, sub_eq_add_neg]
| [
" k * (e1 * e2) = t1 * t2",
" k * (e1 / e2) = t1",
" e * n = e'",
" n * (e1 + e2) = t1 + t2",
" n * (e1 - e2) = t1 - t2"
] | [
" k * (e1 * e2) = t1 * t2",
" k * (e1 / e2) = t1",
" e * n = e'",
" n * (e1 + e2) = t1 + t2"
] |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
section
open Euclidean... | Mathlib/RingTheory/EuclideanDomain.lean | 42 | 47 | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by |
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, mul_div_cancel_right₀ _ pq0]
exact r0
| [
" p / GCDMonoid.gcd p q ≠ 0",
" GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0",
" r ≠ 0"
] | [] |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
namespace CategoryTheory
namespace Bicategory
open Category
open scoped Bicategory
open Mathlib.Tactic.BicategoryCoherence (bicategoricalComp bicategoricalIsoComp)
universe w v u
variable {B : Type u} [Bicategory... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 232 | 247 | theorem adjointifyCounit_left_triangle (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :
leftZigzagIso η (adjointifyCounit η ε) = λ_ f ≪≫ (ρ_ f).symm := by |
apply Iso.ext
dsimp [adjointifyCounit, bicategoricalIsoComp]
calc
_ = 𝟙 _ ⊗≫ (η.hom ▷ (f ≫ 𝟙 b) ≫ (f ≫ g) ◁ f ◁ ε.inv) ⊗≫
f ◁ g ◁ η.inv ▷ f ⊗≫ f ◁ ε.hom := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫ f ◁ ε.inv ⊗≫ (η.hom ▷ (f ≫ g) ≫ (f ≫ g) ◁ η.inv) ▷ f ⊗≫ f ◁ ε.hom := by
rw... | [
" rightZigzag η ε ⊗≫ rightZigzag η ε = rightZigzag η ε",
" (g ◁ η ⊗≫ ε ▷ g) ⊗≫ g ◁ η ⊗≫ ε ▷ g = g ◁ η ⊗≫ ε ▷ g",
" (g ◁ η ⊗≫ ε ▷ g) ⊗≫ g ◁ η ⊗≫ ε ▷ g = g ◁ η ⊗≫ (ε ▷ g ▷ 𝟙 a ≫ (𝟙 b ≫ g) ◁ η) ⊗≫ ε ▷ g",
" g ◁ η ≫ (α_ g f g).inv ≫ ε ▷ g ≫ (λ_ g).hom ≫ (ρ_ g).inv ≫ g ◁ η ≫ (α_ g f g).inv ≫ ε ▷ g =\n g ◁ η ≫... | [
" rightZigzag η ε ⊗≫ rightZigzag η ε = rightZigzag η ε",
" (g ◁ η ⊗≫ ε ▷ g) ⊗≫ g ◁ η ⊗≫ ε ▷ g = g ◁ η ⊗≫ ε ▷ g",
" (g ◁ η ⊗≫ ε ▷ g) ⊗≫ g ◁ η ⊗≫ ε ▷ g = g ◁ η ⊗≫ (ε ▷ g ▷ 𝟙 a ≫ (𝟙 b ≫ g) ◁ η) ⊗≫ ε ▷ g",
" g ◁ η ≫ (α_ g f g).inv ≫ ε ▷ g ≫ (λ_ g).hom ≫ (ρ_ g).inv ≫ g ◁ η ≫ (α_ g f g).inv ≫ ε ▷ g =\n g ◁ η ≫... |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 81 | 85 | theorem add_one_le_of_lt {n m : WithBot ℕ} (h : n < m) : n + 1 ≤ m := by |
cases n
· exact bot_le
cases m
exacts [(not_lt_bot h).elim, WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 h)]
| [
" n + m = 0 ↔ n = 0 ∧ m = 0",
" some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
" none + none = 0 ↔ none = 0 ∧ none = 0",
" none + some val✝ = 0 ↔ none = 0 ∧ some val✝ = 0",
" some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0",
" (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ some val✝¹ = 0 ∧ some va... | [
" n + m = 0 ↔ n = 0 ∧ m = 0",
" some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
" none + none = 0 ↔ none = 0 ∧ none = 0",
" none + some val✝ = 0 ↔ none = 0 ∧ some val✝ = 0",
" some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0",
" (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ some val✝¹ = 0 ∧ some va... |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 73 | 76 | theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
| [
" IsCoprime m n ↔ m.gcd n = 1",
" IsCoprime m n → m.gcd n = 1",
" m.gcd n = 1",
" 1 = m * a + n * b",
" m.gcd n = 1 → IsCoprime m n",
" m.gcdA n * m + m.gcdB n * n = 1 → ∃ a b, a * m + b * n = 1",
" ∃ a b, a * m + b * n = 1",
" IsCoprime ↑m ↑n ↔ m.Coprime n",
" IsCoprime ↑a ↑b",
" IsCoprime ↑↑a ↑↑... | [
" IsCoprime m n ↔ m.gcd n = 1",
" IsCoprime m n → m.gcd n = 1",
" m.gcd n = 1",
" 1 = m * a + n * b",
" m.gcd n = 1 → IsCoprime m n",
" m.gcdA n * m + m.gcdB n * n = 1 → ∃ a b, a * m + b * n = 1",
" ∃ a b, a * m + b * n = 1",
" IsCoprime ↑m ↑n ↔ m.Coprime n",
" IsCoprime ↑a ↑b",
" IsCoprime ↑↑a ↑↑... |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 126 | 136 | theorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' := by |
dsimp [destruct]
induction' f0 : s.1 0 with a' <;> intro h
· injection h with h'
rw [← h']
cases' s with f al
apply Subtype.eq
dsimp [think, tail]
rw [← f0]
exact (Stream'.eta f).symm
· contradiction
| [
" Stream'.cons none (↑c) (n + 1) = some a",
" Stream'.cons none (↑c) (0 + 1) = some a",
" Stream'.cons none (↑c) (n + 1 + 1) = some a",
" s.destruct = Sum.inl a → s = pure a",
" (match ↑s 0 with\n | none => Sum.inr s.tail\n | some a => Sum.inl a) =\n Sum.inl a →\n s = pure a",
" (match n... | [
" Stream'.cons none (↑c) (n + 1) = some a",
" Stream'.cons none (↑c) (0 + 1) = some a",
" Stream'.cons none (↑c) (n + 1 + 1) = some a",
" s.destruct = Sum.inl a → s = pure a",
" (match ↑s 0 with\n | none => Sum.inr s.tail\n | some a => Sum.inl a) =\n Sum.inl a →\n s = pure a",
" (match n... |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
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
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 95 | 100 | theorem length_mul_right_ne (w : W) : ℓ (t * w) ≠ ℓ w := by |
suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| [
" cs.IsReflection (cs.simple i)",
" cs.simple i = 1 * cs.simple i * 1⁻¹",
" t ^ 2 = 1",
" (w * cs.simple i * w⁻¹) ^ 2 = 1",
" t * t = 1",
" w * cs.simple i * w⁻¹ * (w * cs.simple i * w⁻¹) = 1",
" t⁻¹ = t",
" (w * cs.simple i * w⁻¹)⁻¹ = w * cs.simple i * w⁻¹",
" cs.IsReflection t⁻¹",
" Odd (cs.leng... | [
" cs.IsReflection (cs.simple i)",
" cs.simple i = 1 * cs.simple i * 1⁻¹",
" t ^ 2 = 1",
" (w * cs.simple i * w⁻¹) ^ 2 = 1",
" t * t = 1",
" w * cs.simple i * w⁻¹ * (w * cs.simple i * w⁻¹) = 1",
" t⁻¹ = t",
" (w * cs.simple i * w⁻¹)⁻¹ = w * cs.simple i * w⁻¹",
" cs.IsReflection t⁻¹",
" Odd (cs.leng... |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 46 | 47 | theorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0 := by |
rwa [zero_opow', Ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero]
| [
" 0 ^ a = 1 - a",
" 0 ^ a = 0"
] | [
" 0 ^ a = 1 - a"
] |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Rat.BigOperators
#align_import combinatorics.simple_graph.regularity.energy from "leanprover-community/mathlib"@"bf7ef0... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 61 | 63 | theorem coe_energy {𝕜 : Type*} [LinearOrderedField 𝕜] : (P.energy G : 𝕜) =
(∑ uv ∈ P.parts.offDiag, (G.edgeDensity uv.1 uv.2 : 𝕜) ^ 2) / (P.parts.card : 𝕜) ^ 2 := by |
rw [energy]; norm_cast
| [
" 0 ≤ P.energy G",
" ↑P.parts.offDiag.card ≤ 1 * ↑P.parts.card ^ 2",
" ↑(P.parts.card * P.parts.card - P.parts.card) ≤ ↑P.parts.card ^ 2",
" P.parts.card * P.parts.card - P.parts.card ≤ P.parts.card ^ 2",
" P.parts.card * P.parts.card - P.parts.card ≤ P.parts.card * P.parts.card",
" ↑(P.energy G) = (∑ uv ... | [
" 0 ≤ P.energy G",
" ↑P.parts.offDiag.card ≤ 1 * ↑P.parts.card ^ 2",
" ↑(P.parts.card * P.parts.card - P.parts.card) ≤ ↑P.parts.card ^ 2",
" P.parts.card * P.parts.card - P.parts.card ≤ P.parts.card ^ 2",
" P.parts.card * P.parts.card - P.parts.card ≤ P.parts.card * P.parts.card"
] |
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 215 | 216 | theorem symmetric_symmetrizeRel (V : Set (α × α)) : SymmetricRel (symmetrizeRel V) := by |
simp [SymmetricRel, symmetrizeRel, preimage_inter, inter_comm, ← preimage_comp]
| [
" idRel ⊆ s ↔ ∀ (a : α), (a, a) ∈ s",
" (a, b) ∈ Prod.swap '' idRel ↔ (a, b) ∈ idRel",
" (a, b) ∈ idRel ○ r ↔ (a, b) ∈ r",
" r ○ s ○ t = r ○ (s ○ t)",
" (a, b) ∈ r ○ s ○ t ↔ (a, b) ∈ r ○ (s ○ t)",
" (∃ z, (∃ z_1, (a, z_1) ∈ r ∧ (z_1, z) ∈ s) ∧ (z, b) ∈ t) ↔ ∃ z, (a, z) ∈ r ∧ ∃ z_1, (z, z_1) ∈ s ∧ (z_1, b)... | [
" idRel ⊆ s ↔ ∀ (a : α), (a, a) ∈ s",
" (a, b) ∈ Prod.swap '' idRel ↔ (a, b) ∈ idRel",
" (a, b) ∈ idRel ○ r ↔ (a, b) ∈ r",
" r ○ s ○ t = r ○ (s ○ t)",
" (a, b) ∈ r ○ s ○ t ↔ (a, b) ∈ r ○ (s ○ t)",
" (∃ z, (∃ z_1, (a, z_1) ∈ r ∧ (z_1, z) ∈ s) ∧ (z, b) ∈ t) ↔ ∃ z, (a, z) ∈ r ∧ ∃ z_1, (z, z_1) ∈ s ∧ (z_1, b)... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 213 | 214 | theorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
(f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by | simp only [← mul_apply, f.inv_mul, coe_one, id]
| [
" f (1 + x) = 1 + f x",
" (f.comp g.toOrderHom).toFun (x + 1) = (f.comp g.toOrderHom).toFun x + 1",
" ↑f⁻¹ (↑f x) = x"
] | [
" f (1 + x) = 1 + f x",
" (f.comp g.toOrderHom).toFun (x + 1) = (f.comp g.toOrderHom).toFun x + 1"
] |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.isometric_smul from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set
open ENNReal Pointwise
universe u v w
vari... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 121 | 123 | theorem edist_div_right [DivInvMonoid M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
(a b c : M) : edist (a / c) (b / c) = edist a b := by |
simp only [div_eq_mul_inv, edist_mul_right]
| [
" edist ((fun x => c • x) x) ((fun x => c • x) y) = edist x y",
" edist (a / c) (b / c) = edist a b"
] | [
" edist ((fun x => c • x) x) ((fun x => c • x) y) = edist x y"
] |
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous
import Mathlib.Algebra.Polynomial.Roots
#align_i... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 124 | 127 | theorem totalDegree_zero_iff_isHomogeneous {p : MvPolynomial σ R} :
p.totalDegree = 0 ↔ IsHomogeneous p 0 := by |
rw [← weightedTotalDegree_one,
← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsHomogeneous]
| [
" (weightedDegree 1) d = degree d",
" weightedTotalDegree 1 φ = φ.totalDegree",
" (weightedDegree 1) c = n",
" coeff c a ≠ 0 ∨ coeff c b ≠ 0",
" coeff c a + coeff c b = 0",
" coeff c a ≠ 0",
" False",
" r • coeff c a = 0",
" r • 0 = 0",
" homogeneousSubmodule σ R n = Finsupp.supported R R {d | deg... | [
" (weightedDegree 1) d = degree d",
" weightedTotalDegree 1 φ = φ.totalDegree",
" (weightedDegree 1) c = n",
" coeff c a ≠ 0 ∨ coeff c b ≠ 0",
" coeff c a + coeff c b = 0",
" coeff c a ≠ 0",
" False",
" r • coeff c a = 0",
" r • 0 = 0",
" homogeneousSubmodule σ R n = Finsupp.supported R R {d | deg... |
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import category_theory.preadditive.schur from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 114 | 125 | theorem finrank_endomorphism_eq_one {X : C} (isIso_iff_nonzero : ∀ f : X ⟶ X, IsIso f ↔ f ≠ 0)
[I : FiniteDimensional 𝕜 (X ⟶ X)] : finrank 𝕜 (X ⟶ X) = 1 := by |
have id_nonzero := (isIso_iff_nonzero (𝟙 X)).mp (by infer_instance)
refine finrank_eq_one (𝟙 X) id_nonzero ?_
intro f
have : Nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero
have : FiniteDimensional 𝕜 (End X) := I
obtain ⟨c, nu⟩ := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional 𝕜 (End.of f)... | [
" f ≠ 0",
" False",
" 𝟙 X = 0",
" f * f⁻¹ = 1",
" (if h : f = 0 then 0 else inv f) ≫ f = 1",
" inv f ≫ f = 1",
" f = 0",
" finrank 𝕜 (X ⟶ X) = 1",
" IsIso (𝟙 X)",
" ∀ (w : X ⟶ X), ∃ c, c • 𝟙 X = w",
" ∃ c, c • 𝟙 X = f",
" c • 𝟙 X = f"
] | [
" f ≠ 0",
" False",
" 𝟙 X = 0",
" f * f⁻¹ = 1",
" (if h : f = 0 then 0 else inv f) ≫ f = 1",
" inv f ≫ f = 1",
" f = 0"
] |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 142 | 143 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [← card_Ioc, Fintype.card_ofFinset]
| [
" map valEmbedding (Icc a b) = Icc ↑a ↑b",
" map valEmbedding (Ico a b) = Ico ↑a ↑b",
" map valEmbedding (Ioc a b) = Ioc ↑a ↑b",
" map valEmbedding (Ioo a b) = Ioo ↑a ↑b",
" (Icc a b).card = ↑b + 1 - ↑a",
" (Ico a b).card = ↑b - ↑a",
" (Ioc a b).card = ↑b - ↑a",
" (Ioo a b).card = ↑b - ↑a - 1",
" (u... | [
" map valEmbedding (Icc a b) = Icc ↑a ↑b",
" map valEmbedding (Ico a b) = Ico ↑a ↑b",
" map valEmbedding (Ioc a b) = Ioc ↑a ↑b",
" map valEmbedding (Ioo a b) = Ioo ↑a ↑b",
" (Icc a b).card = ↑b + 1 - ↑a",
" (Ico a b).card = ↑b - ↑a",
" (Ioc a b).card = ↑b - ↑a",
" (Ioo a b).card = ↑b - ↑a - 1",
" (u... |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ι κ α β R M : Type*}
| Mathlib/Algebra/Module/BigOperators.lean | 50 | 51 | theorem Finset.cast_card [CommSemiring R] (s : Finset α) : (s.card : R) = ∑ a ∈ s, 1 := by |
rw [Finset.sum_const, Nat.smul_one_eq_cast]
| [
" ↑s.card = ∑ a ∈ s, 1"
] | [] |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.Antichain
import Mathlib.Order.Interval.Finset.Nat
#align_import data.finset.slice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Finset Nat
variable {α : Type*} {ι : Sort*} {κ : ι → Sort*}
namespace Set
... | Mathlib/Data/Finset/Slice.lean | 64 | 66 | theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r := by |
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_swap
| [
" Sized r ∅",
" Sized r {s} ↔ s.card = r",
" Sized r (⋃ i, f i) ↔ ∀ (i : ι), Sized r (f i)",
" (∀ ⦃x : Finset α⦄ (x_1 : ι), x ∈ f x_1 → x.card = r) ↔ ∀ (i : ι) ⦃x : Finset α⦄, x ∈ f i → x.card = r"
] | [
" Sized r ∅",
" Sized r {s} ↔ s.card = r"
] |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 106 | 112 | theorem summable_inner (f g : lp G 2) : Summable fun i => ⟪f i, g i⟫ := by |
-- Apply the Direct Comparison Test, comparing with ∑' i, ‖f i‖ * ‖g i‖ (summable by Hölder)
refine .of_norm_bounded (fun i => ‖f i‖ * ‖g i‖) (lp.summable_mul ?_ f g) ?_
· rw [Real.isConjExponent_iff]; norm_num
intro i
-- Then apply Cauchy-Schwarz pointwise
exact norm_inner_le_norm (𝕜 := 𝕜) _ _
| [
" Summable fun i => ⟪↑f i, ↑g i⟫_𝕜",
" (ENNReal.toReal 2).IsConjExponent (ENNReal.toReal 2)",
" 1 < ENNReal.toReal 2 ∧ (ENNReal.toReal 2)⁻¹ + (ENNReal.toReal 2)⁻¹ = 1",
" ∀ (i : ι), ‖⟪↑f i, ↑g i⟫_𝕜‖ ≤ (fun i => ‖↑f i‖ * ‖↑g i‖) i",
" ‖⟪↑f i, ↑g i⟫_𝕜‖ ≤ (fun i => ‖↑f i‖ * ‖↑g i‖) i"
] | [] |
import Mathlib.CategoryTheory.FinCategory.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.P... | Mathlib/CategoryTheory/Filtered/Basic.lean | 372 | 388 | theorem of_cocone_nonempty (h : ∀ {J : Type w} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
Nonempty (Cocone F)) : IsFiltered C := by |
have : Nonempty C := by
obtain ⟨c⟩ := h (Functor.empty _)
exact ⟨c.pt⟩
have : IsFilteredOrEmpty C := by
refine ⟨?_, ?_⟩
· intros X Y
obtain ⟨c⟩ := h (ULiftHom.down ⋙ ULift.downFunctor ⋙ pair X Y)
exact ⟨c.pt, c.ι.app ⟨⟨WalkingPair.left⟩⟩, c.ι.app ⟨⟨WalkingPair.right⟩⟩, trivial⟩
· in... | [
" f ≫ 𝟙 Y = g ≫ 𝟙 Y",
" (f ≫ 𝟙 Y).down = (g ≫ 𝟙 Y).down",
" IsFiltered α",
" X.as = { as := PUnit.unit }.as",
" Y.as = { as := PUnit.unit }.as",
" f ≫ { down := { down := ⋯ } } = g ≫ { down := { down := ⋯ } }",
" (f ≫ { down := { down := ⋯ } }).down = (g ≫ { down := { down := ⋯ } }).down",
" IsFil... | [
" f ≫ 𝟙 Y = g ≫ 𝟙 Y",
" (f ≫ 𝟙 Y).down = (g ≫ 𝟙 Y).down",
" IsFiltered α",
" X.as = { as := PUnit.unit }.as",
" Y.as = { as := PUnit.unit }.as",
" f ≫ { down := { down := ⋯ } } = g ≫ { down := { down := ⋯ } }",
" (f ≫ { down := { down := ⋯ } }).down = (g ≫ { down := { down := ⋯ } }).down"
] |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 95 | 99 | theorem const_smul [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (c • f) μ := by |
rcases hf with ⟨f', h_f'_meas, hff'⟩
refine ⟨c • f', h_f'_meas.const_smul c, ?_⟩
exact EventuallyEq.fun_comp hff' fun x => c • x
| [
" AEStronglyMeasurable' m g μ",
" AEStronglyMeasurable' m (f + g) μ",
" AEStronglyMeasurable' m (-f) μ",
" (-f) x = (-f') x",
" -f x = -f' x",
" AEStronglyMeasurable' m (f - g) μ",
" (f - g) x = (f' - g') x",
" f x - g x = f' x - g' x",
" AEStronglyMeasurable' m (c • f) μ",
" c • f =ᶠ[ae μ] c • f'... | [
" AEStronglyMeasurable' m g μ",
" AEStronglyMeasurable' m (f + g) μ",
" AEStronglyMeasurable' m (-f) μ",
" (-f) x = (-f') x",
" -f x = -f' x",
" AEStronglyMeasurable' m (f - g) μ",
" (f - g) x = (f' - g') x",
" f x - g x = f' x - g' x"
] |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 99 | 105 | theorem zpow_log_le_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) : (b : R) ^ log b r ≤ r := by |
rcases le_total 1 r with hr1 | hr1
· rw [log_of_one_le_right _ hr1]
rw [zpow_natCast, ← Nat.cast_pow, ← Nat.le_floor_iff hr.le]
exact Nat.pow_log_le_self b (Nat.floor_pos.mpr hr1).ne'
· rw [log_of_right_le_one _ hr1, zpow_neg, zpow_natCast, ← Nat.cast_pow]
exact inv_le_of_inv_le hr (Nat.ceil_le.1 <| ... | [
" log b r = -↑(b.clog ⌈r⁻¹⌉₊)",
" log b 1 = -↑(b.clog ⌈1⁻¹⌉₊)",
" log b ↑n = ↑(b.log n)",
" log b ↑0 = ↑(b.log 0)",
" log b ↑(n✝ + 1) = ↑(b.log (n✝ + 1))",
" 1 ≤ ↑(n✝ + 1)",
" log b r = 0",
" ↑b ^ log b r ≤ r",
" ↑b ^ ↑(b.log ⌊r⌋₊) ≤ r",
" b ^ b.log ⌊r⌋₊ ≤ ⌊r⌋₊",
" (↑(b ^ b.clog ⌈r⁻¹⌉₊))⁻¹ ≤ r"
... | [
" log b r = -↑(b.clog ⌈r⁻¹⌉₊)",
" log b 1 = -↑(b.clog ⌈1⁻¹⌉₊)",
" log b ↑n = ↑(b.log n)",
" log b ↑0 = ↑(b.log 0)",
" log b ↑(n✝ + 1) = ↑(b.log (n✝ + 1))",
" 1 ≤ ↑(n✝ + 1)",
" log b r = 0"
] |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 103 | 106 | theorem support_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).support = (unop p).support := by |
induction' p using MulOpposite.rec' with p
cases p
exact Finsupp.support_mapRange_of_injective (map_zero _) _ op_injective
| [
" (opRingEquiv R) (op ((monomial n) r)) = (monomial n) (op r)",
" (opRingEquiv R) (op (C r * X ^ n)) = C (op r) * X ^ n",
" (opRingEquiv R) ((opRingEquiv R).symm ((monomial n) r)) = (opRingEquiv R) (op ((monomial n) r.unop))",
" (opRingEquiv R).symm (C r * X ^ n) = op (C r.unop * X ^ n)",
" ((opRingEquiv R)... | [
" (opRingEquiv R) (op ((monomial n) r)) = (monomial n) (op r)",
" (opRingEquiv R) (op (C r * X ^ n)) = C (op r) * X ^ n",
" (opRingEquiv R) ((opRingEquiv R).symm ((monomial n) r)) = (opRingEquiv R) (op ((monomial n) r.unop))",
" (opRingEquiv R).symm (C r * X ^ n) = op (C r.unop * X ^ n)",
" ((opRingEquiv R)... |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 33 | 38 | theorem forall_sum {γ : α ⊕ β → Sort _} (p : (∀ ab, γ ab) → Prop) :
(∀ fab, p fab) ↔ (∀ fa fb, p (Sum.rec fa fb)) := by |
refine ⟨fun h fa fb => h _, fun h fab => ?_⟩
have h1 : fab = Sum.rec (fun a => fab (Sum.inl a)) (fun b => fab (Sum.inr b)) := by
ext ab; cases ab <;> rfl
rw [h1]; exact h _ _
| [
" (∀ (fab : (ab : α ⊕ β) → γ ab), p fab) ↔\n ∀ (fa : (val : α) → γ (inl val)) (fb : (val : β) → γ (inr val)), p fun t => rec fa fb t",
" p fab",
" fab = fun t => rec (fun a => fab (inl a)) (fun b => fab (inr b)) t",
" fab ab = rec (fun a => fab (inl a)) (fun b => fab (inr b)) ab",
" fab (inl val✝) = rec ... | [] |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
open Set Function
open scoped MeasureTheory
namespace MeasurableSpace
variable {α : Type*}
def invariants [m : MeasurableSpace α] (f : α → α) : MeasurableSpace α :=
{ m ⊓ ⟨fun s ↦ f ⁻¹' s = s, by simp, by simp, fun f hf ↦ by simp [hf]⟩ with
MeasurableSet' :... | Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 50 | 54 | theorem le_invariants_iterate (f : α → α) (n : ℕ) :
invariants f ≤ invariants (f^[n]) := by |
induction n with
| zero => simp [invariants_le]
| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)
| [
" (fun s => f ⁻¹' s = s) ∅",
" ∀ (s : Set α), (fun s => f ⁻¹' s = s) s → (fun s => f ⁻¹' s = s) sᶜ",
" (fun s => f✝ ⁻¹' s = s) (⋃ i, f i)",
" f ∘ g ⁻¹' s = s",
" invariants f ≤ invariants f^[n]",
" invariants f ≤ invariants f^[0]",
" invariants f ≤ invariants f^[n + 1]"
] | [
" (fun s => f ⁻¹' s = s) ∅",
" ∀ (s : Set α), (fun s => f ⁻¹' s = s) s → (fun s => f ⁻¹' s = s) sᶜ",
" (fun s => f✝ ⁻¹' s = s) (⋃ i, f i)",
" f ∘ g ⁻¹' s = s"
] |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 81 | 84 | theorem lift_rank_eq_of_surjective {f : M →ₗ[R] M'} (h : Surjective f) :
lift.{v} (Module.rank R M) =
lift.{u} (Module.rank R M') + lift.{v} (Module.rank R (LinearMap.ker f)) := by |
rw [← lift_rank_range_add_rank_ker f, ← rank_range_of_surjective f h]
| [
" Nontrivial R",
" False",
" lift.{u, v} (Module.rank R ↥(LinearMap.range f)) + lift.{v, u} (Module.rank R ↥(LinearMap.ker f)) =\n lift.{v, u} (Module.rank R M)",
" Module.rank R ↥(LinearMap.range f) + Module.rank R ↥(LinearMap.ker f) = Module.rank R M",
" lift.{v, u} (Module.rank R M) = lift.{u, v} (Mod... | [
" Nontrivial R",
" False",
" lift.{u, v} (Module.rank R ↥(LinearMap.range f)) + lift.{v, u} (Module.rank R ↥(LinearMap.ker f)) =\n lift.{v, u} (Module.rank R M)",
" Module.rank R ↥(LinearMap.range f) + Module.rank R ↥(LinearMap.ker f) = Module.rank R M"
] |
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 | 490 | 490 | theorem div_one (a : G) : a / 1 = a := by | simp [div_eq_mul_inv]
| [
" a / 1 = a"
] | [] |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 60 | 61 | theorem factorization_def (n : ℕ) {p : ℕ} (pp : p.Prime) : n.factorization p = padicValNat p n := by |
simpa [factorization] using absurd pp
| [
" ∀ (a : ℕ), a ∈ n.primeFactors ↔ (fun p => if p.Prime then padicValNat p n else 0) a ≠ 0",
" ∀ (a : ℕ), a.Prime → (a ∣ n ∧ ¬n = 0 ↔ ¬a = 1 ∧ ¬n = 0 ∧ a ∣ n)",
" n.factorization p = padicValNat p n"
] | [
" ∀ (a : ℕ), a ∈ n.primeFactors ↔ (fun p => if p.Prime then padicValNat p n else 0) a ≠ 0",
" ∀ (a : ℕ), a.Prime → (a ∣ n ∧ ¬n = 0 ↔ ¬a = 1 ∧ ¬n = 0 ∧ a ∣ n)"
] |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 94 | 94 | theorem sInf_div : sInf (s / t) = sInf s / sSup t := by | simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]
| [
" sSup s⁻¹ = (sInf s)⁻¹",
" ⨆ a ∈ s, a⁻¹ = (sInf s)⁻¹",
" sInf s⁻¹ = (sSup s)⁻¹",
" ⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹",
" sSup (s / t) = sSup s / sInf t",
" sInf (s / t) = sInf s / sSup t"
] | [
" sSup s⁻¹ = (sInf s)⁻¹",
" ⨆ a ∈ s, a⁻¹ = (sInf s)⁻¹",
" sInf s⁻¹ = (sSup s)⁻¹",
" ⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹",
" sSup (s / t) = sSup s / sInf t"
] |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 81 | 81 | theorem logb_inv (x : ℝ) : logb b x⁻¹ = -logb b x := by | simp [logb, neg_div]
| [
" b.logb 0 = 0",
" b.logb 1 = 0",
" False",
" b.logb |x| = b.logb x",
" b.logb (-x) = b.logb x",
" b.logb (x * y) = b.logb x + b.logb y",
" b.logb (x / y) = b.logb x - b.logb y",
" b.logb x⁻¹ = -b.logb x"
] | [
" b.logb 0 = 0",
" b.logb 1 = 0",
" False",
" b.logb |x| = b.logb x",
" b.logb (-x) = b.logb x",
" b.logb (x * y) = b.logb x + b.logb y",
" b.logb (x / y) = b.logb x - b.logb y"
] |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 115 | 120 | theorem IsSymmetric.coe_reApplyInnerSelf_apply {T : E →L[𝕜] E} (hT : IsSymmetric (T : E →ₗ[𝕜] E))
(x : E) : (T.reApplyInnerSelf x : 𝕜) = ⟪T x, x⟫ := by |
rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ⟪T x, x⟫ = r
· simp [hr, T.reApplyInnerSelf_apply]
rw [← conj_eq_iff_real]
exact hT.conj_inner_sym x x
| [
" (starRingEnd 𝕜) ⟪T x, y⟫_𝕜 = ⟪T y, x⟫_𝕜",
" (T + S).IsSymmetric",
" ⟪(T + S) x, y⟫_𝕜 = ⟪x, (T + S) y⟫_𝕜",
" ⟪x, T y + S y⟫_𝕜 = ⟪x, (T + S) y⟫_𝕜",
" Continuous ⇑T",
" y = T x",
" ⟪y - T x, y - T x⟫_𝕜 = 0",
" ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫_𝕜 = ⟪u k - x, T (y - T x)⟫_𝕜",
" ⟪T (u k) - T... | [
" (starRingEnd 𝕜) ⟪T x, y⟫_𝕜 = ⟪T y, x⟫_𝕜",
" (T + S).IsSymmetric",
" ⟪(T + S) x, y⟫_𝕜 = ⟪x, (T + S) y⟫_𝕜",
" ⟪x, T y + S y⟫_𝕜 = ⟪x, (T + S) y⟫_𝕜",
" Continuous ⇑T",
" y = T x",
" ⟪y - T x, y - T x⟫_𝕜 = 0",
" ∀ (k : ℕ), ⟪T (u k) - T x, y - T x⟫_𝕜 = ⟪u k - x, T (y - T x)⟫_𝕜",
" ⟪T (u k) - T... |
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.MonotoneConvergence
#align_import analysis.box_integral.box.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Fu... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 157 | 168 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by |
tfae_have 1 ↔ 2
· exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
tfae_have 3 ↔ 4
· exact Icc_subset_Icc_iff I.lower_le_upper
tfae_have 4 → 2
· exact fun h x hx i ↦ Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i)
tfae_finish
| [
" [I ≤ J, ↑I ⊆ ↑J, Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper].TFAE",
" I ≤ J ↔ ↑I ⊆ ↑J",
" ↑I ⊆ ↑J → Icc I.lower I.upper ⊆ Icc J.lower J.upper",
" Icc I.lower I.upper ⊆ Icc J.lower J.upper",
" Icc I.lower I.upper ⊆ Icc J.lower J.upper ↔ J.lower ≤ I.lower ∧ I.upper ≤ J.... | [] |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 32 | 38 | theorem tan_add {x y : ℝ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨
(∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := by |
simpa only [← Complex.ofReal_inj, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_div,
Complex.ofReal_mul, Complex.ofReal_tan] using
@Complex.tan_add (x : ℂ) (y : ℂ) (by convert h <;> norm_cast)
| [
" (x + y).tan = (x.tan + y.tan) / (1 - x.tan * y.tan)",
" ((∀ (k : ℤ), ↑x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), ↑y ≠ (2 * ↑l + 1) * ↑π / 2) ∨\n (∃ k, ↑x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, ↑y = (2 * ↑l + 1) * ↑π / 2",
" ↑x = (2 * ↑a✝ + 1) * ↑π / 2 ↔ x = (2 * ↑a✝ + 1) * π / 2",
" ↑y = (2 * ↑a✝ + 1) * ↑π / 2 ↔ y... | [] |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 131 | 134 | theorem pullback_fst_eq :
CompHaus.pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
| [
" fst f g ≫ f = snd f g ≫ g",
" (fst f g ≫ f) ⟨val✝, h⟩ = (snd f g ≫ g) ⟨val✝, h⟩",
" (a z, b z) ∈ {xy | f xy.1 = g xy.2}",
" Continuous fun z => ⟨(a z, b z), ⋯⟩",
" Continuous fun x => (a x, b x)",
" (Continuous fun x => a x) ∧ Continuous fun x => b x",
" a = b",
" a z = b z",
" ↑(a z) = ↑(b z)",
... | [
" fst f g ≫ f = snd f g ≫ g",
" (fst f g ≫ f) ⟨val✝, h⟩ = (snd f g ≫ g) ⟨val✝, h⟩",
" (a z, b z) ∈ {xy | f xy.1 = g xy.2}",
" Continuous fun z => ⟨(a z, b z), ⋯⟩",
" Continuous fun x => (a x, b x)",
" (Continuous fun x => a x) ∧ Continuous fun x => b x",
" a = b",
" a z = b z",
" ↑(a z) = ↑(b z)",
... |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 72 | 77 | theorem tendsto_zero_smul_of_tendsto_zero_of_bounded {ι 𝕜 𝔸 : Type*} [NormedDivisionRing 𝕜]
[NormedAddCommGroup 𝔸] [Module 𝕜 𝔸] [BoundedSMul 𝕜 𝔸] {l : Filter ι} {ε : ι → 𝕜} {f : ι → 𝔸}
(hε : Tendsto ε l (𝓝 0)) (hf : Filter.IsBoundedUnder (· ≤ ·) l (norm ∘ f)) :
Tendsto (ε • f) l (𝓝 0) := by |
rw [← isLittleO_one_iff 𝕜] at hε ⊢
simpa using IsLittleO.smul_isBigO hε (hf.isBigO_const (one_ne_zero : (1 : 𝕜) ≠ 0))
| [
" Summable f",
" ∀ (i : ℕ), 0 ≤ ‖f i‖",
" Tendsto (fun n => ∑ i ∈ Finset.range n, ‖f i‖) atTop (𝓝 r)",
" Tendsto (fun x => ‖x ^ m‖) (𝓝[≠] 0) atTop",
" Tendsto (fun x => ‖x ^ (-m)‖) (𝓝[≠] 0) atTop",
" Tendsto (fun x => ‖x ^ (-↑m)‖) (𝓝[≠] 0) atTop",
" Tendsto (fun x => ‖x⁻¹‖ ^ m) (𝓝[≠] 0) atTop",
"... | [
" Summable f",
" ∀ (i : ℕ), 0 ≤ ‖f i‖",
" Tendsto (fun n => ∑ i ∈ Finset.range n, ‖f i‖) atTop (𝓝 r)",
" Tendsto (fun x => ‖x ^ m‖) (𝓝[≠] 0) atTop",
" Tendsto (fun x => ‖x ^ (-m)‖) (𝓝[≠] 0) atTop",
" Tendsto (fun x => ‖x ^ (-↑m)‖) (𝓝[≠] 0) atTop",
" Tendsto (fun x => ‖x⁻¹‖ ^ m) (𝓝[≠] 0) atTop"
] |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 146 | 147 | theorem type_cardinal : @type Cardinal (· < ·) _ = Ordinal.univ.{u, u + 1} := by |
rw [Ordinal.univ_id]; exact Quotient.sound ⟨alephIdx.relIso⟩
| [
" c.ord.IsLimit",
" ℵ₀ = 0",
" c.ord ≤ a",
" c ≤ a.card",
" ℵ₀ ≤ a.card",
" ℵ₀ ≤ (succ a).card",
" ℵ₀.ord.IsLimit",
" ω.IsLimit",
" a.alephIdx ≤ b.alephIdx ↔ a ≤ b",
" False",
" ∀ (α : Type u) (r : α → α → Prop) [inst : IsWellOrder α r], (∀ (c : Cardinal.{u}), c.alephIdx < type r) → False",
" ... | [
" c.ord.IsLimit",
" ℵ₀ = 0",
" c.ord ≤ a",
" c ≤ a.card",
" ℵ₀ ≤ a.card",
" ℵ₀ ≤ (succ a).card",
" ℵ₀.ord.IsLimit",
" ω.IsLimit",
" a.alephIdx ≤ b.alephIdx ↔ a ≤ b",
" False",
" ∀ (α : Type u) (r : α → α → Prop) [inst : IsWellOrder α r], (∀ (c : Cardinal.{u}), c.alephIdx < type r) → False",
" ... |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions.Fin... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 83 | 87 | theorem NatTrans.equifibered_of_discrete {ι : Type*} {F G : Discrete ι ⥤ C}
(α : F ⟶ G) : NatTrans.Equifibered α := by |
rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
simp only [Discrete.functor_map_id]
exact IsPullback.of_horiz_isIso ⟨by rw [Category.id_comp, Category.comp_id]⟩
| [
" NatTrans.Equifibered α",
" IsPullback (F.map { down := { down := ⋯ } }) (α.app { as := WalkingPair.right })\n (α.app { as := { as := WalkingPair.right }.as }) (F'.map { down := { down := ⋯ } })",
" IsPullback (F.map { down := { down := ⋯ } }) (α.app { as := WalkingPair.left })\n (α.app { as := { as := W... | [
" NatTrans.Equifibered α",
" IsPullback (F.map { down := { down := ⋯ } }) (α.app { as := WalkingPair.right })\n (α.app { as := { as := WalkingPair.right }.as }) (F'.map { down := { down := ⋯ } })",
" IsPullback (F.map { down := { down := ⋯ } }) (α.app { as := WalkingPair.left })\n (α.app { as := { as := W... |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 56 | 59 | theorem fold_insert [DecidableEq α] (h : a ∉ s) :
(insert a s).fold op b f = f a * s.fold op b f := by |
unfold fold
rw [insert_val, ndinsert_of_not_mem h, Multiset.map_cons, fold_cons_left]
| [
" fold op b f (cons a s h) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))",
" fold op b f (insert a s) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b ... | [
" fold op b f (cons a s h) = op (f a) (fold op b f s)",
" Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))"
] |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 150 | 150 | theorem fourier_one {x : AddCircle T} : fourier 1 x = toCircle x := by | rw [fourier_apply, one_zsmul]
| [
" (fourier n) ↑x = (2 * ↑π * Complex.I * ↑n * ↑x / ↑T).exp",
" (↑2 * ↑π / ↑T * (↑n * ↑x) * Complex.I).exp = (2 * ↑π * Complex.I * ↑n * ↑x / ↑T).exp",
" (2 * ↑π / ↑T * (↑n * ↑x) * Complex.I).exp = (2 * ↑π * Complex.I * ↑n * ↑x / ↑T).exp",
" 2 * ↑π / ↑T * (↑n * ↑x) * Complex.I = 2 * ↑π * Complex.I * ↑n * ↑x / ↑... | [
" (fourier n) ↑x = (2 * ↑π * Complex.I * ↑n * ↑x / ↑T).exp",
" (↑2 * ↑π / ↑T * (↑n * ↑x) * Complex.I).exp = (2 * ↑π * Complex.I * ↑n * ↑x / ↑T).exp",
" (2 * ↑π / ↑T * (↑n * ↑x) * Complex.I).exp = (2 * ↑π * Complex.I * ↑n * ↑x / ↑T).exp",
" 2 * ↑π / ↑T * (↑n * ↑x) * Complex.I = 2 * ↑π * Complex.I * ↑n * ↑x / ↑... |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 130 | 136 | theorem liftOn_condition_of_liftOn'_condition {P : Sort v} {f : K[X] → K[X] → P}
(H : ∀ {p q a} (hq : q ≠ 0) (_ha : a ≠ 0), f (a * p) (a * q) = f p q) ⦃p q p' q' : K[X]⦄
(hq : q ≠ 0) (hq' : q' ≠ 0) (h : q' * p = q * p') : f p q = f p' q' :=
calc
f p q = f (q' * p) (q' * q) := (H hq hq').symm
_ = f (q ... | rw [h, mul_comm q']
_ = f p' q' := H hq' hq
| [
" { toFractionRing := x } = { toFractionRing := y }",
" { toFractionRing := x } = { toFractionRing := { toFractionRing := x }.toFractionRing }",
" P",
" ∀ {a c : K[X]} {b d : ↥K[X]⁰},\n (Localization.r K[X]⁰) (a, b) (c, d) → (fun p q => f p ↑q) a b = (fun p q => f p ↑q) c d",
" (fun p q => f p ↑q) p q = ... | [
" { toFractionRing := x } = { toFractionRing := y }",
" { toFractionRing := x } = { toFractionRing := { toFractionRing := x }.toFractionRing }",
" P",
" ∀ {a c : K[X]} {b d : ↥K[X]⁰},\n (Localization.r K[X]⁰) (a, b) (c, d) → (fun p q => f p ↑q) a b = (fun p q => f p ↑q) c d",
" (fun p q => f p ↑q) p q = ... |
import Mathlib.Data.Vector.Basic
set_option autoImplicit true
namespace Vector
def snoc : Vector α n → α → Vector α (n+1) :=
fun xs x => append xs (x ::ᵥ Vector.nil)
section Simp
variable (xs : Vector α n)
@[simp]
theorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=
rfl
@[simp]
theorem snoc_nil... | Mathlib/Data/Vector/Snoc.lean | 48 | 52 | theorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) := by |
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, Vector.append, Append.append]
| [
" (x ::ᵥ xs).reverse = xs.reverse.snoc x",
" (x ::ᵥ ⟨val✝, property✝⟩).reverse = (reverse ⟨val✝, property✝⟩).snoc x",
" ⟨val✝.reverse ++ [x], ⋯⟩ = append ⟨val✝.reverse, ⋯⟩ ⟨[x], ⋯⟩",
" (xs.snoc x).reverse = x ::ᵥ xs.reverse",
" (snoc ⟨val✝, property✝⟩ x).reverse = x ::ᵥ reverse ⟨val✝, property✝⟩",
" ⟨(app... | [
" (x ::ᵥ xs).reverse = xs.reverse.snoc x",
" (x ::ᵥ ⟨val✝, property✝⟩).reverse = (reverse ⟨val✝, property✝⟩).snoc x",
" ⟨val✝.reverse ++ [x], ⋯⟩ = append ⟨val✝.reverse, ⋯⟩ ⟨[x], ⋯⟩"
] |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 295 | 297 | theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by |
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
| [
" (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x",
" IsRelPrime (∏ i ∈ insert b t, s i) x",
" IsRelPrime (s b * ∏ x ∈ t, s x) x",
" (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i)",
" IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x",
" x✝ ∈ ∅ → IsRelPrime (s x✝) x",
" ... | [
" (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x",
" IsRelPrime (∏ i ∈ insert b t, s i) x",
" IsRelPrime (s b * ∏ x ∈ t, s x) x",
" (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i)",
" IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x",
" x✝ ∈ ∅ → IsRelPrime (s x✝) x",
" ... |
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 65 | 66 | theorem isCaratheodory_compl : IsCaratheodory m s₁ → IsCaratheodory m s₁ᶜ := by |
simp [IsCaratheodory, diff_eq, add_comm]
| [
" m.IsCaratheodory ∅",
" m.IsCaratheodory s₁ → m.IsCaratheodory s₁ᶜ"
] | [
" m.IsCaratheodory ∅"
] |
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