openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /CircleCouplingForcing.lean
| import OAI.Analysis.IrrationalRotation.Couplings.CircleCouplingSupport | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped ENNReal | |
| namespace Invariant233.CircleCoupling | |
| open CircleDigitCoupling CircleParameterData OneCoordinateCoupling | |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) | |
| include hp | |
| lemma ae_endpoint_unit (j : ℕ) (i : ℕ) (side : Bool) : | |
| ∀ᵐ ω ∂law p j, endpoint side (ω i) ∈ Ico (0 : ℝ) 1 := by | |
| let := coordinateLaw_isProbabilityMeasure hp j | |
| apply (measurePreserving_eval_infinitePi (coordinateLaw p j) i).quasiMeasurePreserving.ae | |
| (p := fun z => endpoint side z ∈ Ico (0 : ℝ) 1) | |
| by_cases hi : i = 0 | |
| · rw [coordinateLaw, ite_eq_left hi, initialLaw] | |
| apply (ae_map_iff (show Measurable (fun x : ℝ => ((none : Label), (x, x))) from by fun_prop).aemeasurable | |
| (show MeasurableSet {z : CoordinateSample | endpoint side z ∈ Ico (0 : ℝ) 1} from by | |
| apply measurableSet_Ico.preimage | |
| cases side | |
| · exact measurable_fst.comp measurable_snd | |
| · exact measurable_snd.comp measurable_snd)).2 | |
| cases side <;> exact CouplingMixture.intervalLaw_ae_mem 0 1 | |
| · have hs := p.blockIndex_pos hp.m_pos (show 1 ≤ i by omega) | |
| have hm : (100 : ℝ) ≤ p.m (p.blockIndex i) := by exact_mod_cast | |
| (show 100 ≤ p.m (p.blockIndex i) by have := hp.m_ge_200 _ hs; omega) | |
| rw [coordinateLaw, ite_eq_right hi] | |
| exact rawLaw_ae_endpoint_unit hm (hp.coupling_a_bounds _ hs).1 (hp.coupling_a_bounds _ hs).2 | |
| (hp.coupling_b_bounds _ hs).1 (hp.coupling_b_bounds _ hs).2 side | |
| lemma ae_rounding_bound (j : ℕ) (hr : ∀ i, 0 < MixedRadix.radices p.q i) | |
| (i : ℕ) (side : Bool) : | |
| ∀ᵐ ω ∂law p j, |p.coordinate i (point p hr side ω) - endpoint side (ω i)| ≤ | |
| 1 / (MixedRadix.radices p.q i : ℝ) := by | |
| let : NeZero (MixedRadix.radices p.q i) := ⟨Nat.ne_of_gt (hr i)⟩ | |
| filter_upwards [ae_endpoint_unit hp j i side, ae_nonterminal_digits hp j hr side] with ω hunit hnt | |
| have hraw : endpoint side (ω i) ∈ bin _ (digits p hr side ω i) := | |
| (UniformQuantization.quantize_eq_iff _ hunit).mp rfl | |
| exact (abs_sub_lt_bin_width _ (coordinate_point_mem_bin hp hr side ω hnt i) hraw).le | |
| lemma ae_top_raw_support (j : ℕ) {s i : ℕ} (hs : 1 ≤ s) (hi : i ∈ p.block s) (which : Bool) : | |
| ∀ᵐ ω ∂law p j, label i ω = topLabel which → | |
| 1 - width (p.a s) (p.b s) which / 2 ≤ (ω i).2.2 := by | |
| let := coordinateLaw_isProbabilityMeasure hp j | |
| apply (measurePreserving_eval_infinitePi (coordinateLaw p j) i).quasiMeasurePreserving.ae | |
| (p := fun z => z.1 = topLabel which → 1 - width (p.a s) (p.b s) which / 2 ≤ z.2.2) | |
| have hi0 : i ≠ 0 := by have := (p.mem_block_iff s i).mp hi; omega | |
| rw [coordinateLaw, ite_eq_right hi0, p.blockIndex_eq_of_mem hp.m_pos hi] | |
| have hm : (100 : ℝ) ≤ p.m s := by exact_mod_cast (show 100 ≤ p.m s by have := hp.m_ge_200 s hs; omega) | |
| exact rawLaw_ae_top_support hm (hp.coupling_a_bounds s hs).1 (hp.coupling_a_bounds s hs).2 | |
| (hp.coupling_b_bounds s hs).1 (hp.coupling_b_bounds s hs).2 which | |
| lemma ae_bottom_raw_support (j : ℕ) {s i : ℕ} (hs : 1 ≤ s) (hi : i ∈ p.block s) (which : Bool) : | |
| ∀ᵐ ω ∂law p j, label i ω = bottomLabel which → | |
| (ω i).2.1 < width (p.a s) (p.b s) which / 2 := by | |
| let := coordinateLaw_isProbabilityMeasure hp j | |
| apply (measurePreserving_eval_infinitePi (coordinateLaw p j) i).quasiMeasurePreserving.ae | |
| (p := fun z => z.1 = bottomLabel which → z.2.1 < width (p.a s) (p.b s) which / 2) | |
| have hi0 : i ≠ 0 := by have := (p.mem_block_iff s i).mp hi; omega | |
| rw [coordinateLaw, ite_eq_right hi0, p.blockIndex_eq_of_mem hp.m_pos hi] | |
| have hm : (100 : ℝ) ≤ p.m s := by exact_mod_cast (show 100 ≤ p.m s by have := hp.m_ge_200 s hs; omega) | |
| exact rawLaw_ae_bottom_support hm (hp.coupling_a_bounds s hs).1 (hp.coupling_a_bounds s hs).2 | |
| (hp.coupling_b_bounds s hs).1 (hp.coupling_b_bounds s hs).2 which | |
| lemma ae_top_forces (j : ℕ) (hr : ∀ i, 0 < MixedRadix.radices p.q i) | |
| {s i : ℕ} (hs : 1 ≤ s) (hi : i ∈ p.block s) (which : Bool) : | |
| ∀ᵐ ω ∂law p j, ∀ u : ℝ, label i ω = topLabel which → | |
| u ≤ 1 - width (p.a s) (p.b s) which → u ≤ p.coordinate i (point p hr true ω) := by | |
| have hw : 1 / (p.m s : ℝ) ≤ width (p.a s) (p.b s) which := by | |
| cases which | |
| · exact (hp.coupling_a_bounds s hs).1 | |
| · exact (hp.coupling_b_bounds s hs).1 | |
| filter_upwards [ae_top_raw_support hp j hs hi which, ae_rounding_bound hp j hr i true] with ω hsupp hround | |
| intro u hl hu | |
| exact top_forces_crossing (hsupp hl) hround (hp.coupling_bin_small hs hi hw) hu | |
| lemma ae_bottom_forces (j : ℕ) (hr : ∀ i, 0 < MixedRadix.radices p.q i) | |
| {s i : ℕ} (hs : 1 ≤ s) (hi : i ∈ p.block s) (which : Bool) : | |
| ∀ᵐ ω ∂law p j, ∀ u : ℝ, label i ω = bottomLabel which → | |
| width (p.a s) (p.b s) which ≤ u → p.coordinate i (point p hr false ω) < u := by | |
| have hw : 1 / (p.m s : ℝ) ≤ width (p.a s) (p.b s) which := by | |
| cases which | |
| · exact (hp.coupling_a_bounds s hs).1 | |
| · exact (hp.coupling_b_bounds s hs).1 | |
| filter_upwards [ae_bottom_raw_support hp j hs hi which, ae_rounding_bound hp j hr i false] with ω hsupp hround | |
| intro u hl hu | |
| exact bottom_forces_crossing (hsupp hl) hround (hp.coupling_bin_small hs hi hw) hu | |
| end Invariant233.CircleCoupling | |
| end | |
| end OAI | |