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 | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
open MeasureTheory Topology Metric Filter Set ENNReal NNReal
open scoped Topology ENNReal NNReal BoundedContinuousFunction
section auxiliary
namespace MeasureTheory
variable {Ω : Type*} [TopologicalSpace Ω] [Mea... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 95 | 105 | theorem measure_of_cont_bdd_of_tendsto_indicator [OpensMeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E)
(fs : ℕ → Ω →ᵇ ℝ≥0) (fs_bdd : ∀ n ω, fs n ω ≤ c)
(fs_lim : Tendsto (fun n ω ↦ fs n ω) atTop (𝓝 (indicator E fun _ ↦ (1 : ℝ≥0)))) :
Tendsto (fun ... |
have fs_lim' :
∀ ω, Tendsto (fun n : ℕ ↦ (fs n ω : ℝ≥0)) atTop (𝓝 (indicator E (fun _ ↦ (1 : ℝ≥0)) ω)) := by
rw [tendsto_pi_nhds] at fs_lim
exact fun ω ↦ fs_lim ω
apply measure_of_cont_bdd_of_tendsto_filter_indicator μ E_mble fs
(eventually_of_forall fun n ↦ eventually_of_forall (fs_bdd n)) (event... | 1,795 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
open MeasureTheory Topology Metric Filter Set ENNReal NNReal
open scoped Topology ENNReal NNReal BoundedContinuousFunction
section auxiliary
namespace MeasureTheory
variable {Ω : Type*} [TopologicalSpace Ω] [Mea... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 110 | 119 | theorem tendsto_lintegral_thickenedIndicator_of_isClosed {Ω : Type*} [MeasurableSpace Ω]
[PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] {F : Set Ω}
(F_closed : IsClosed F) {δs : ℕ → ℝ} (δs_pos : ∀ n, 0 < δs n)
(δs_lim : Tendsto δs atTop (𝓝 0)) :
Tendsto (fun n ↦ lin... |
apply measure_of_cont_bdd_of_tendsto_indicator μ F_closed.measurableSet
(fun n ↦ thickenedIndicator (δs_pos n) F) fun n ω ↦ thickenedIndicator_le_one (δs_pos n) F ω
have key := thickenedIndicator_tendsto_indicator_closure δs_pos δs_lim F
rwa [F_closed.closure_eq] at key
| 1,795 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 168 | 171 | theorem apply_mono (μ : FiniteMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ := by |
change ((μ : Measure Ω) s₁).toNNReal ≤ ((μ : Measure Ω) s₂).toNNReal
have key : (μ : Measure Ω) s₁ ≤ (μ : Measure Ω) s₂ := (μ : Measure Ω).mono h
apply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ s₁) (measure_ne_top _ s₂)).mpr key
| 1,796 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 200 | 204 | theorem mass_zero_iff (μ : FiniteMeasure Ω) : μ.mass = 0 ↔ μ = 0 := by |
refine ⟨fun μ_mass => ?_, fun hμ => by simp only [hμ, zero_mass]⟩
apply toMeasure_injective
apply Measure.measure_univ_eq_zero.mp
rwa [← ennreal_mass, ENNReal.coe_eq_zero]
| 1,796 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 207 | 209 | theorem mass_nonzero_iff (μ : FiniteMeasure Ω) : μ.mass ≠ 0 ↔ μ ≠ 0 := by |
rw [not_iff_not]
exact FiniteMeasure.mass_zero_iff μ
| 1,796 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 213 | 217 | theorem eq_of_forall_toMeasure_apply_eq (μ ν : FiniteMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by |
apply Subtype.ext
ext1 s s_mble
exact h s s_mble
| 1,796 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 220 | 223 | theorem eq_of_forall_apply_eq (μ ν : FiniteMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → μ s = ν s) : μ = ν := by |
ext1 s s_mble
simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (h s s_mble)
| 1,796 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 163 | 164 | theorem coeFn_univ_ne_zero (ν : ProbabilityMeasure Ω) : ν univ ≠ 0 := by |
simp only [coeFn_univ, Ne, one_ne_zero, not_false_iff]
| 1,797 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 193 | 196 | theorem ennreal_coeFn_eq_coeFn_toMeasure (ν : ProbabilityMeasure Ω) (s : Set Ω) :
(ν s : ℝ≥0∞) = (ν : Measure Ω) s := by |
rw [← coeFn_comp_toFiniteMeasure_eq_coeFn, FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure,
toMeasure_comp_toFiniteMeasure_eq_toMeasure]
| 1,797 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 199 | 201 | theorem apply_mono (μ : ProbabilityMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ := by |
rw [← coeFn_comp_toFiniteMeasure_eq_coeFn]
exact MeasureTheory.FiniteMeasure.apply_mono _ h
| 1,797 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 207 | 212 | theorem nonempty (μ : ProbabilityMeasure Ω) : Nonempty Ω := by |
by_contra maybe_empty
have zero : (μ : Measure Ω) univ = 0 := by
rw [univ_eq_empty_iff.mpr (not_nonempty_iff.mp maybe_empty), measure_empty]
rw [measure_univ] at zero
exact zero_ne_one zero.symm
| 1,797 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 216 | 220 | theorem eq_of_forall_toMeasure_apply_eq (μ ν : ProbabilityMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by |
apply toMeasure_injective
ext1 s s_mble
exact h s s_mble
| 1,797 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 44 | 52 | theorem withDensity_apply_le (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a in s, f a ∂μ ≤ μ.withDensity f s := by |
let t := toMeasurable (μ.withDensity f) s
calc
∫⁻ a in s, f a ∂μ ≤ ∫⁻ a in t, f a ∂μ :=
lintegral_mono_set (subset_toMeasurable (withDensity μ f) s)
_ = μ.withDensity f t :=
(withDensity_apply f (measurableSet_toMeasurable (withDensity μ f) s)).symm
_ = μ.withDensity f s := measure_toMeasurable s
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 68 | 75 | theorem withDensity_apply' [SFinite μ] (f : α → ℝ≥0∞) (s : Set α) :
μ.withDensity f s = ∫⁻ a in s, f a ∂μ := by |
apply le_antisymm ?_ (withDensity_apply_le f s)
let t := toMeasurable μ s
calc
μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
_ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
_ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sF... | 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 83 | 87 | theorem withDensity_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) :
μ.withDensity f = μ.withDensity g := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, withDensity_apply _ hs]
exact lintegral_congr_ae (ae_restrict_of_ae h)
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 97 | 102 | theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs,
← lintegral_add_left hf]
simp only [Pi.add_apply]
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 105 | 107 | theorem withDensity_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by |
simpa only [add_comm] using withDensity_add_left hg f
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 110 | 113 | theorem withDensity_add_measure {m : MeasurableSpace α} (μ ν : Measure α) (f : α → ℝ≥0∞) :
(μ + ν).withDensity f = μ.withDensity f + ν.withDensity f := by |
ext1 s hs
simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply]
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 116 | 119 | theorem withDensity_sum {ι : Type*} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) :
(sum μ).withDensity f = sum fun n => (μ n).withDensity f := by |
ext1 s hs
simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum μ hs, lintegral_sum_measure]
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 122 | 127 | theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
μ.withDensity (r • f) = r • μ.withDensity f := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf]
simp only [Pi.smul_apply, smul_eq_mul]
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 130 | 135 | theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
μ.withDensity (r • f) = r • μ.withDensity f := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul' r f hr]
simp only [Pi.smul_apply, smul_eq_mul]
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 138 | 142 | theorem withDensity_smul_measure (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
(r • μ).withDensity f = r • μ.withDensity f := by |
ext s hs
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, set_lintegral_smul_measure]
| 1,798 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 144 | 147 | theorem isFiniteMeasure_withDensity {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
IsFiniteMeasure (μ.withDensity f) :=
{ measure_univ_lt_top := by |
rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top] }
| 1,798 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 88 | 96 | theorem _root_.Measurable.lmarginal (hf : Measurable f) : Measurable (∫⋯∫⁻_s, f ∂μ) := by |
refine Measurable.lintegral_prod_right ?_
refine hf.comp ?_
rw [measurable_pi_iff]; intro i
by_cases hi : i ∈ s
· simp [hi, updateFinset]
exact measurable_pi_iff.1 measurable_snd _
· simp [hi, updateFinset]
exact measurable_pi_iff.1 measurable_fst _
| 1,799 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 105 | 108 | theorem lmarginal_congr {x y : ∀ i, π i} (f : (∀ i, π i) → ℝ≥0∞)
(h : ∀ i ∉ s, x i = y i) :
(∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y := by |
dsimp [lmarginal, updateFinset_def]; rcongr; exact h _ ‹_›
| 1,799 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 110 | 116 | theorem lmarginal_update_of_mem {i : δ} (hi : i ∈ s)
(f : (∀ i, π i) → ℝ≥0∞) (x : ∀ i, π i) (y : π i) :
(∫⋯∫⁻_s, f ∂μ) (Function.update x i y) = (∫⋯∫⁻_s, f ∂μ) x := by |
apply lmarginal_congr
intro j hj
have : j ≠ i := by rintro rfl; exact hj hi
apply update_noteq this
| 1,799 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 118 | 135 | theorem lmarginal_union (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f)
(hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_s, ∫⋯∫⁻_t, f ∂μ ∂μ := by |
ext1 x
let e := MeasurableEquiv.piFinsetUnion π hst
calc (∫⋯∫⁻_s ∪ t, f ∂μ) x
= ∫⁻ (y : (i : ↥(s ∪ t)) → π i), f (updateFinset x (s ∪ t) y)
∂.pi fun i' : ↥(s ∪ t) ↦ μ i' := rfl
_ = ∫⁻ (y : ((i : s) → π i) × ((j : t) → π j)), f (updateFinset x (s ∪ t) _)
∂(Measure.pi fun i : s ↦ μ i)... | 1,799 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 137 | 139 | theorem lmarginal_union' (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {s t : Finset δ}
(hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ := by |
rw [Finset.union_comm, lmarginal_union μ f hf hst.symm]
| 1,799 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 144 | 153 | theorem lmarginal_singleton (f : (∀ i, π i) → ℝ≥0∞) (i : δ) :
∫⋯∫⁻_{i}, f ∂μ = fun x => ∫⁻ xᵢ, f (Function.update x i xᵢ) ∂μ i := by |
let α : Type _ := ({i} : Finset δ)
let e := (MeasurableEquiv.piUnique fun j : α ↦ π j).symm
ext1 x
calc (∫⋯∫⁻_{i}, f ∂μ) x
= ∫⁻ (y : π (default : α)), f (updateFinset x {i} (e y)) ∂μ (default : α) := by
simp_rw [lmarginal, measurePreserving_piUnique (fun j : ({i} : Finset δ) ↦ μ j) |>.symm _
... | 1,799 |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 34 | 46 | theorem exists_null_pairwise_disjoint_diff [Countable ι] {s : ι → Set α}
(hd : Pairwise (AEDisjoint μ on s)) : ∃ t : ι → Set α, (∀ i, MeasurableSet (t i)) ∧
(∀ i, μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \ t i) := by |
refine ⟨fun i => toMeasurable μ (s i ∩ ⋃ j ∈ ({i}ᶜ : Set ι), s j), fun i =>
measurableSet_toMeasurable _ _, fun i => ?_, ?_⟩
· simp only [measure_toMeasurable, inter_iUnion]
exact (measure_biUnion_null_iff <| to_countable _).2 fun j hj => hd (Ne.symm hj)
· simp only [Pairwise, disjoint_left, onFun, mem_d... | 1,800 |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 94 | 96 | theorem iUnion_left_iff [Countable ι] {s : ι → Set α} :
AEDisjoint μ (⋃ i, s i) t ↔ ∀ i, AEDisjoint μ (s i) t := by |
simp only [AEDisjoint, iUnion_inter, measure_iUnion_null_iff]
| 1,800 |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 100 | 102 | theorem iUnion_right_iff [Countable ι] {t : ι → Set α} :
AEDisjoint μ s (⋃ i, t i) ↔ ∀ i, AEDisjoint μ s (t i) := by |
simp only [AEDisjoint, inter_iUnion, measure_iUnion_null_iff]
| 1,800 |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 106 | 107 | theorem union_left_iff : AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u := by |
simp [union_eq_iUnion, and_comm]
| 1,800 |
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 111 | 112 | theorem union_right_iff : AEDisjoint μ s (t ∪ u) ↔ AEDisjoint μ s t ∧ AEDisjoint μ s u := by |
simp [union_eq_iUnion, and_comm]
| 1,800 |
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.doubling from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filter Metric MeasureTheory TopologicalSpace ENNReal NN... | Mathlib/MeasureTheory/Measure/Doubling.lean | 69 | 99 | theorem exists_eventually_forall_measure_closedBall_le_mul (K : ℝ) :
∃ C : ℝ≥0, ∀ᶠ ε in 𝓝[>] 0, ∀ x, ∀ t ≤ K, μ (closedBall x (t * ε)) ≤ C * μ (closedBall x ε) := by |
let C := doublingConstant μ
have hμ :
∀ n : ℕ, ∀ᶠ ε in 𝓝[>] 0, ∀ x,
μ (closedBall x ((2 : ℝ) ^ n * ε)) ≤ ↑(C ^ n) * μ (closedBall x ε) := by
intro n
induction' n with n ih
· simp
replace ih := eventually_nhdsWithin_pos_mul_left (two_pos : 0 < (2 : ℝ)) ih
refine (ih.and (exists_measur... | 1,801 |
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.doubling from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filter Metric MeasureTheory TopologicalSpace ENNReal NN... | Mathlib/MeasureTheory/Measure/Doubling.lean | 113 | 129 | theorem eventually_measure_mul_le_scalingConstantOf_mul (K : ℝ) :
∃ R : ℝ,
0 < R ∧
∀ x t r, t ∈ Ioc 0 K → r ≤ R →
μ (closedBall x (t * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) := by |
have h := Classical.choose_spec (exists_eventually_forall_measure_closedBall_le_mul μ K)
rcases mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.1 h with ⟨R, Rpos, hR⟩
refine ⟨R, Rpos, fun x t r ht hr => ?_⟩
rcases lt_trichotomy r 0 with (rneg | rfl | rpos)
· have : t * r < 0 := mul_neg_of_pos_of_neg ht.1 rneg
s... | 1,801 |
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.doubling from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filter Metric MeasureTheory TopologicalSpace ENNReal NN... | Mathlib/MeasureTheory/Measure/Doubling.lean | 132 | 136 | theorem eventually_measure_le_scaling_constant_mul (K : ℝ) :
∀ᶠ r in 𝓝[>] 0, ∀ x, μ (closedBall x (K * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) := by |
filter_upwards [Classical.choose_spec
(exists_eventually_forall_measure_closedBall_le_mul μ K)] with r hr x
exact (hr x K le_rfl).trans (mul_le_mul_right' (ENNReal.coe_le_coe.2 (le_max_left _ _)) _)
| 1,801 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 94 | 98 | theorem idealOfSet_closed [T2Space R] (s : Set X) :
IsClosed (idealOfSet R s : Set C(X, R)) := by |
simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall]
exact isClosed_iInter fun x => isClosed_iInter fun _ =>
isClosed_eq (continuous_eval_const x) continuous_const
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 103 | 105 | theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 := by |
convert Iff.rfl
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 108 | 109 | theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by |
simp_rw [mem_idealOfSet]; push_neg; rfl
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 118 | 120 | theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0 := by |
rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_setOf]
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 123 | 125 | theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 := by |
simp_rw [setOfIdeal, Set.mem_compl_iff, Set.mem_setOf]; push_neg; rfl
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 128 | 132 | theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) := by |
simp only [setOfIdeal, Set.setOf_forall, isOpen_compl_iff]
exact
isClosed_iInter fun f =>
isClosed_iInter fun _ => isClosed_eq (map_continuous f) continuous_const
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 154 | 156 | theorem mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
f ∈ idealOfSet R ({x}ᶜ : Set X) ↔ f x = 0 := by |
simp only [mem_idealOfSet, compl_compl, Set.mem_singleton_iff, forall_eq]
| 1,802 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 161 | 168 | theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idealOfSet R) := by |
refine fun I s => ⟨fun h f hf => ?_, fun h x hx => ?_⟩
· by_contra h'
rcases not_mem_idealOfSet.mp h' with ⟨x, hx, hfx⟩
exact hfx (not_mem_setOfIdeal.mp (mt (@h x) hx) hf)
· obtain ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx
by_contra hx'
exact not_mem_idealOfSet.mpr ⟨x, hx', hfx⟩ (h hf)
| 1,802 |
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.polynomial.algebra_map from "leanprover-community/mathlib"@"e064a7bf82ad94c3c17b5128bbd860d1ec34874e"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
univer... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 123 | 127 | theorem algHom_eval₂_algebraMap {R A B : Type*} [CommSemiring R] [Semiring A] [Semiring B]
[Algebra R A] [Algebra R B] (p : R[X]) (f : A →ₐ[R] B) (a : A) :
f (eval₂ (algebraMap R A) a p) = eval₂ (algebraMap R B) (f a) p := by |
simp only [eval₂_eq_sum, sum_def]
simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast, AlgHom.commutes]
| 1,803 |
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.polynomial.algebra_map from "leanprover-community/mathlib"@"e064a7bf82ad94c3c17b5128bbd860d1ec34874e"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
univer... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 131 | 136 | theorem eval₂_algebraMap_X {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (p : R[X])
(f : R[X] →ₐ[R] A) : eval₂ (algebraMap R A) (f X) p = f p := by |
conv_rhs => rw [← Polynomial.sum_C_mul_X_pow_eq p]
simp only [eval₂_eq_sum, sum_def]
simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast]
simp [Polynomial.C_eq_algebraMap]
| 1,803 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.RingTheory.Derivation.Basic
noncomputable section
namespace Polynomial
section CommSemiring
variable {R A : Type*} [CommSemiring R]
@[simps]
def derivative' : D... | Mathlib/Algebra/Polynomial/Derivation.lean | 43 | 46 | theorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :
C a • D f = a • D f := by |
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul]
| 1,804 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.RingTheory.Derivation.Basic
noncomputable section
namespace Polynomial
section CommSemiring
variable {R A : Type*} [CommSemiring R]
@[simps]
def derivative' : D... | Mathlib/Algebra/Polynomial/Derivation.lean | 67 | 67 | theorem mkDerivation_X (a : A) : mkDerivation R a X = a := by | simp [mkDerivation_apply]
| 1,804 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section Semiring
variable [CommSemiring R] [... | Mathlib/RingTheory/Polynomial/Tower.lean | 37 | 38 | theorem aeval_map_algebraMap (x : B) (p : R[X]) : aeval x (map (algebraMap R A) p) = aeval x p := by |
rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]
| 1,805 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section CommSemiring
variable [CommSemiring ... | Mathlib/RingTheory/Polynomial/Tower.lean | 54 | 56 | theorem aeval_algebraMap_apply (x : A) (p : R[X]) :
aeval (algebraMap A B x) p = algebraMap A B (aeval x p) := by |
rw [aeval_def, aeval_def, hom_eval₂, ← IsScalarTower.algebraMap_eq]
| 1,805 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section CommSemiring
variable [CommSemiring ... | Mathlib/RingTheory/Polynomial/Tower.lean | 60 | 63 | theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X]) :
aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,
iff_false_intro (one_ne_zero' B), or_false_iff]
| 1,805 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section CommSemiring
variable [CommSemiring ... | Mathlib/RingTheory/Polynomial/Tower.lean | 68 | 70 | theorem aeval_algebraMap_eq_zero_iff_of_injective {x : A} {p : R[X]}
(h : Function.Injective (algebraMap A B)) : aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, ← (algebraMap A B).map_zero, h.eq_iff]
| 1,805 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 56 | 61 | theorem toFunBilinear_apply_eq_sum (a : A) (p : R[X]) :
toFunBilinear R A a p = p.sum fun n r => monomial n (a * algebraMap R A r) := by |
simp only [toFunBilinear_apply_apply, aeval_def, eval₂_eq_sum, Polynomial.sum, Finset.smul_sum]
congr with i : 1
rw [← Algebra.smul_def, ← C_mul', mul_smul_comm, C_mul_X_pow_eq_monomial, ← Algebra.commutes,
← Algebra.smul_def, smul_monomial]
| 1,806 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 80 | 82 | theorem toFunLinear_mul_tmul_mul_aux_1 (p : R[X]) (k : ℕ) (h : Decidable ¬p.coeff k = 0) (a : A) :
ite (¬coeff p k = 0) (a * (algebraMap R A) (coeff p k)) 0 =
a * (algebraMap R A) (coeff p k) := by | classical split_ifs <;> simp [*]
| 1,806 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 85 | 91 | theorem toFunLinear_mul_tmul_mul_aux_2 (k : ℕ) (a₁ a₂ : A) (p₁ p₂ : R[X]) :
a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) =
(Finset.antidiagonal k).sum fun x =>
a₁ * (algebraMap R A) (coeff p₁ x.1) * (a₂ * (algebraMap R A) (coeff p₂ x.2)) := by |
simp_rw [mul_assoc, Algebra.commutes, ← Finset.mul_sum, mul_assoc, ← Finset.mul_sum]
congr
simp_rw [Algebra.commutes (coeff p₂ _), coeff_mul, map_sum, RingHom.map_mul]
| 1,806 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 94 | 106 | theorem toFunLinear_mul_tmul_mul (a₁ a₂ : A) (p₁ p₂ : R[X]) :
(toFunLinear R A) ((a₁ * a₂) ⊗ₜ[R] (p₁ * p₂)) =
(toFunLinear R A) (a₁ ⊗ₜ[R] p₁) * (toFunLinear R A) (a₂ ⊗ₜ[R] p₂) := by |
classical
simp only [toFunLinear_tmul_apply, toFunBilinear_apply_eq_sum]
ext k
simp_rw [coeff_sum, coeff_monomial, sum_def, Finset.sum_ite_eq', mem_support_iff, Ne]
conv_rhs => rw [coeff_mul]
simp_rw [finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', mem_support_iff, Ne, mul_ite,
mul_ze... | 1,806 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 109 | 111 | theorem toFunLinear_one_tmul_one :
toFunLinear R A (1 ⊗ₜ[R] 1) = 1 := by |
rw [toFunLinear_tmul_apply, toFunBilinear_apply_apply, Polynomial.aeval_one, one_smul]
| 1,806 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 138 | 139 | theorem invFun_add {p q} : invFun R A (p + q) = invFun R A p + invFun R A q := by |
simp only [invFun, eval₂_add]
| 1,806 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Finsupp.Fin
import Mathlib.Logic.Equiv.Fin
#align_import data.mv_polynomial.equiv from "leanprover-community/mathlib"@"2f5b500... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 143 | 147 | theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) := by |
ext
simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map]
rfl
| 1,807 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.Polynomial.Eval
#align_import data.mv_polynomial.polynomial from "leanprover-community/mathlib"@"0b89934139d3be96f9dab477f10c20f9f93da580"
namespace MvPolynomial
variable {R S σ : Type*}
| Mathlib/Algebra/MvPolynomial/Polynomial.lean | 19 | 28 | theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring S]
{x : S} (f : R →+* Polynomial S) (g : σ → Polynomial S) (p : MvPolynomial σ R) :
Polynomial.eval x (eval₂ f g p) =
eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p := by |
apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp]
| 1,808 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.Polynomial.Eval
#align_import data.mv_polynomial.polynomial from "leanprover-community/mathlib"@"0b89934139d3be96f9dab477f10c20f9f93da580"
namespace MvPolynomial
variable {R S σ : Type*}
theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring ... | Mathlib/Algebra/MvPolynomial/Polynomial.lean | 30 | 40 | theorem eval_polynomial_eval_finSuccEquiv {n : ℕ} {x : Fin n → R}
[CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R) :
(eval x) (Polynomial.eval q (finSuccEquiv R n f)) = eval (Fin.cases (eval x q) x) f := by |
simp only [finSuccEquiv_apply, coe_eval₂Hom, polynomial_eval_eval₂, eval_eval₂]
conv in RingHom.comp _ _ =>
refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_
simp
simp only [eval₂_id]
congr
funext i
refine Fin.cases (by simp) (by simp) i
| 1,808 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 44 | 45 | theorem integralNormalization_zero : integralNormalization (0 : R[X]) = 0 := by |
simp [integralNormalization]
| 1,809 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 48 | 53 | theorem integralNormalization_coeff {f : R[X]} {i : ℕ} :
(integralNormalization f).coeff i =
if f.degree = i then 1 else coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) := by |
have : f.coeff i = 0 → f.degree ≠ i := fun hc hd => coeff_ne_zero_of_eq_degree hd hc
simp (config := { contextual := true }) [integralNormalization, coeff_monomial, this,
mem_support_iff]
| 1,809 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 56 | 59 | theorem integralNormalization_support {f : R[X]} :
(integralNormalization f).support ⊆ f.support := by |
intro
simp (config := { contextual := true }) [integralNormalization, coeff_monomial, mem_support_iff]
| 1,809 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 62 | 63 | theorem integralNormalization_coeff_degree {f : R[X]} {i : ℕ} (hi : f.degree = i) :
(integralNormalization f).coeff i = 1 := by | rw [integralNormalization_coeff, if_pos hi]
| 1,809 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 71 | 73 | theorem integralNormalization_coeff_ne_degree {f : R[X]} {i : ℕ} (hi : f.degree ≠ i) :
coeff (integralNormalization f) i = coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) := by |
rw [integralNormalization_coeff, if_neg hi]
| 1,809 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 67 | 74 | theorem Subalgebra.isAlgebraic_iff (S : Subalgebra R A) :
S.IsAlgebraic ↔ @Algebra.IsAlgebraic R S _ _ S.algebra := by |
delta Subalgebra.IsAlgebraic
rw [Subtype.forall', Algebra.isAlgebraic_def]
refine forall_congr' fun x => exists_congr fun p => and_congr Iff.rfl ?_
have h : Function.Injective S.val := Subtype.val_injective
conv_rhs => rw [← h.eq_iff, AlgHom.map_zero]
rw [← aeval_algHom_apply, S.val_apply]
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 78 | 80 | theorem Algebra.isAlgebraic_iff : Algebra.IsAlgebraic R A ↔ (⊤ : Subalgebra R A).IsAlgebraic := by |
delta Subalgebra.IsAlgebraic
simp only [Algebra.isAlgebraic_def, Algebra.mem_top, forall_prop_of_true, iff_self_iff]
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 83 | 85 | theorem isAlgebraic_iff_not_injective {x : A} :
IsAlgebraic R x ↔ ¬Function.Injective (Polynomial.aeval x : R[X] →ₐ[R] A) := by |
simp only [IsAlgebraic, injective_iff_map_eq_zero, not_forall, and_comm, exists_prop]
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 113 | 115 | theorem isAlgebraic_one [Nontrivial R] : IsAlgebraic R (1 : A) := by |
rw [← _root_.map_one (algebraMap R A)]
exact isAlgebraic_algebraMap 1
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 118 | 120 | theorem isAlgebraic_nat [Nontrivial R] (n : ℕ) : IsAlgebraic R (n : A) := by |
rw [← map_natCast (_ : R →+* A) n]
exact isAlgebraic_algebraMap (Nat.cast n)
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 123 | 125 | theorem isAlgebraic_int [Nontrivial R] (n : ℤ) : IsAlgebraic R (n : A) := by |
rw [← _root_.map_intCast (algebraMap R A)]
exact isAlgebraic_algebraMap (Int.cast n)
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 128 | 131 | theorem isAlgebraic_rat (R : Type u) {A : Type v} [DivisionRing A] [Field R] [Algebra R A] (n : ℚ) :
IsAlgebraic R (n : A) := by |
rw [← map_ratCast (algebraMap R A)]
exact isAlgebraic_algebraMap (Rat.cast n)
| 1,810 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 330 | 338 | theorem algHom_bijective [Algebra.IsAlgebraic K L] (f : L →ₐ[K] L) :
Function.Bijective f := by |
refine ⟨f.injective, fun b ↦ ?_⟩
obtain ⟨p, hp, he⟩ := Algebra.IsAlgebraic.isAlgebraic (R := K) b
let f' : p.rootSet L → p.rootSet L := (rootSet_maps_to' (fun x ↦ x) f).restrict f _ _
have : f'.Surjective := Finite.injective_iff_surjective.1
fun _ _ h ↦ Subtype.eq <| f.injective <| Subtype.ext_iff.1 h
ob... | 1,810 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.RingTheory.Algebraic
#align_import field_theory.ax_grothendieck from "leanprover-community/mathlib"@"4e529b03dd62b7b7d13806c3fb974d9d4848910e"
noncomputable section
open MvPolynomial Finset Function
| Mathlib/FieldTheory/AxGrothendieck.lean | 33 | 66 | theorem ax_grothendieck_of_locally_finite {ι K R : Type*} [Field K] [Finite K] [CommRing R]
[Finite ι] [Algebra K R] [Algebra.IsAlgebraic K R] (ps : ι → MvPolynomial ι R)
(hinj : Injective fun v i => MvPolynomial.eval v (ps i)) :
Surjective fun v i => MvPolynomial.eval v (ps i) := by |
classical
intro v
cases nonempty_fintype ι
/- `s` is the set of all coefficients of the polynomial, as well as all of
the coordinates of `v`, the point I am trying to find the preimage of. -/
let s : Finset R :=
(Finset.biUnion (univ : Finset ι) fun i => (ps i).support.image fun x => coef... | 1,811 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 32 | 34 | theorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, x ≠ a / b := by |
simp only [Irrational, Rat.forall, cast_mk, not_exists, Set.mem_range, cast_intCast, cast_div,
eq_comm]
| 1,812 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 38 | 40 | theorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by |
rintro ⟨a, rfl⟩
exact tr (isAlgebraic_algebraMap a)
| 1,812 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 50 | 65 | theorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = m)
(hv : ¬∃ y : ℤ, x = y) (hnpos : 0 < n) : Irrational x := by |
rintro ⟨⟨N, D, P, C⟩, rfl⟩
rw [← cast_pow] at hxr
have c1 : ((D : ℤ) : ℝ) ≠ 0 := by
rw [Int.cast_ne_zero, Int.natCast_ne_zero]
exact P
have c2 : ((D : ℤ) : ℝ) ^ n ≠ 0 := pow_ne_zero _ c1
rw [mk'_eq_divInt, cast_pow, cast_mk, div_pow, div_eq_iff_mul_eq c2, ← Int.cast_pow,
← Int.cast_pow, ← Int.cas... | 1,812 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 70 | 85 | theorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)
[hp : Fact p.Prime] (hxr : x ^ n = m)
(hv : (multiplicity (p : ℤ) m).get (finite_int_iff.2 ⟨hp.1.ne_one, hm⟩) % n ≠ 0) :
Irrational x := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr
simp [hxr, multiplicity.one_right (mt isUnit_iff_dvd_one.1
(mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one)), Nat.zero_mod] at hv
refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos
rintro ⟨y,... | 1,812 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 103 | 104 | theorem irrational_sqrt_two : Irrational (√2) := by |
simpa using Nat.prime_two.irrational_sqrt
| 1,812 |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 93 | 132 | theorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by |
let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋
have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _
have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)
conv in |_| ≤ _ => rw [mul_comm, le_div_iff hn, ← abs_of_pos hn, ← abs_mul]
let D := Icc (0 : ℤ) n
by_cases H : ∃ m ∈ D, f m = ... | 1,813 |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 139 | 144 | theorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ k : ℕ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - round (↑k * ξ)| ≤ 1 / (n + 1) := by |
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk := toNat_of_nonneg hk₀.le
rw [← hk] at hk₀ hk₁ h
exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩
| 1,813 |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 152 | 163 | theorem exists_rat_abs_sub_le_and_den_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ q : ℚ, |ξ - q| ≤ 1 / ((n + 1) * q.den) ∧ q.den ≤ n := by |
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀
have hden : ((j / k : ℚ).den : ℤ) ≤ k := by
convert le_of_dvd hk₀ (Rat.den_dvd j k)
exact Rat.intCast_div_eq_divInt _ _
refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩
rw [← div_div... | 1,813 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
| Mathlib/NumberTheory/Pell.lean | 83 | 85 | theorem is_pell_solution_iff_mem_unitary {d : ℤ} {a : ℤ√d} :
a.re ^ 2 - d * a.im ^ 2 = 1 ↔ a ∈ unitary (ℤ√d) := by |
rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
| 1,814 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 133 | 133 | theorem prop_x (a : Solution₁ d) : a.x ^ 2 = 1 + d * a.y ^ 2 := by | rw [← a.prop]; ring
| 1,814 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 137 | 137 | theorem prop_y (a : Solution₁ d) : d * a.y ^ 2 = a.x ^ 2 - 1 := by | rw [← a.prop]; ring
| 1,814 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 178 | 180 | theorem x_mul (a b : Solution₁ d) : (a * b).x = a.x * b.x + d * (a.y * b.y) := by |
rw [← mul_assoc]
rfl
| 1,814 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 209 | 214 | theorem eq_zero_of_d_neg (h₀ : d < 0) (a : Solution₁ d) : a.x = 0 ∨ a.y = 0 := by |
have h := a.prop
contrapose! h
have h1 := sq_pos_of_ne_zero h.1
have h2 := sq_pos_of_ne_zero h.2
nlinarith
| 1,814 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 218 | 222 | theorem x_ne_zero (h₀ : 0 ≤ d) (a : Solution₁ d) : a.x ≠ 0 := by |
intro hx
have h : 0 ≤ d * a.y ^ 2 := mul_nonneg h₀ (sq_nonneg _)
rw [a.prop_y, hx, sq, zero_mul, zero_sub] at h
exact not_le.mpr (neg_one_lt_zero : (-1 : ℤ) < 0) h
| 1,814 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 367 | 434 | theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by |
let ξ : ℝ := √d
have hξ : Irrational ξ := by
refine irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) ?_ two_pos
rintro ⟨x, hx⟩
refine hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) ?_⟩
rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
obtain ⟨M, hM₁⟩ := exists... | 1,814 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 45 | 51 | theorem dense_irrational : Dense { x : ℝ | Irrational x } := by |
refine Real.isTopologicalBasis_Ioo_rat.dense_iff.2 ?_
simp only [gt_iff_lt, Rat.cast_lt, not_lt, ge_iff_le, Rat.cast_le, mem_iUnion, mem_singleton_iff,
exists_prop, forall_exists_index, and_imp]
rintro _ a b hlt rfl _
rw [inter_comm]
exact exists_irrational_btwn (Rat.cast_lt.2 hlt)
| 1,815 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 78 | 89 | theorem eventually_forall_le_dist_cast_div (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n) := by |
have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with... | 1,815 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 56 | 62 | theorem not_countably_generated_cocompact : ¬IsCountablyGenerated (cocompact ℚ) := by |
intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩)
| 1,816 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 65 | 69 | theorem not_countably_generated_nhds_infty_opc : ¬IsCountablyGenerated (𝓝 (∞ : ℚ∞)) := by |
intro
have : IsCountablyGenerated (comap (OnePoint.some : ℚ → ℚ∞) (𝓝 ∞)) := by infer_instance
rw [OnePoint.comap_coe_nhds_infty, coclosedCompact_eq_cocompact] at this
exact not_countably_generated_cocompact this
| 1,816 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 72 | 74 | theorem not_firstCountableTopology_opc : ¬FirstCountableTopology ℚ∞ := by |
intro
exact not_countably_generated_nhds_infty_opc inferInstance
| 1,816 |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 77 | 79 | theorem not_secondCountableTopology_opc : ¬SecondCountableTopology ℚ∞ := by |
intro
exact not_firstCountableTopology_opc inferInstance
| 1,816 |
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 | 44 | 47 | theorem inv_gold : φ⁻¹ = -ψ := by |
have : 1 + √5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
| 1,817 |
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 | 51 | 53 | theorem inv_goldConj : ψ⁻¹ = -φ := by |
rw [inv_eq_iff_eq_inv, ← neg_inv, ← neg_eq_iff_eq_neg]
exact inv_gold.symm
| 1,817 |
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 | 57 | 60 | theorem gold_mul_goldConj : φ * ψ = -1 := by |
field_simp
rw [← sq_sub_sq]
norm_num
| 1,817 |
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