| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.BoundedBase |
| import OAI.Probability.LatticeSurface.Geometry.CapReflection |
| import OAI.Probability.LatticeSurface.Geometry.CapSplit |
| import OAI.Probability.LatticeSurface.Geometry.CapUnpin |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| open scoped ENNReal |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
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| def weldHeight (pins : Set V) (w v : V) (h : Free pins → ℝ) (x : V) : ℝ := |
| if x = w then restore pins (fun _ => 0) h v else restore pins (fun _ => 0) h x |
|
|
| def weldBody (G : SimpleGraph V) (pins : Set V) (w v : V) : Set (Free pins → ℝ) := |
| {h | ∀ x y, G.Adj x y → |weldHeight pins w v h x - weldHeight pins w v h y| ≤ 1} |
|
|
| lemma continuous_weldHeight [Fintype V] (pins : Set V) (w v x : V) : |
| Continuous (fun h : Free pins → ℝ => weldHeight pins w v h x) := by |
| by_cases hx : x = w <;> simp only [weldHeight, hx, ite_true, ite_false] |
| · exact continuous_restore pins _ v |
| · exact continuous_restore pins _ x |
|
|
| lemma closed_weldBody [Fintype V] (G : SimpleGraph V) (pins : Set V) (w v : V) : |
| IsClosed (weldBody G pins w v) := by |
| simp only [weldBody, Set.ofPred_forall] |
| exact isClosed_iInter fun x => isClosed_iInter fun y => isClosed_iInter fun _ => |
| isClosed_le ((continuous_weldHeight pins w v x).sub (continuous_weldHeight pins w v y)).abs |
| continuous_const |
|
|
| lemma restore_boundary_congr (pins : Set V) {b c : V → ℝ} |
| (hbc : ∀ x ∈ pins, b x = c x) (h : Free pins → ℝ) (x : V) : |
| restore pins b h x = restore pins c h x := by |
| by_cases hx : x ∈ pins <;> simp [restore, hx, hbc x] |
|
|
| lemma body_boundary_congr (G : SimpleGraph V) (pins : Set V) {b c : V → ℝ} |
| (hbc : ∀ x ∈ pins, b x = c x) : |
| conditionalBody G pins b = conditionalBody G pins c := by |
| ext h |
| simp only [conditionalBody, Set.mem_ofPred_eq, restore_boundary_congr pins hbc] |
|
|
| lemma weld_pinned (G : SimpleGraph V) (pins : Set V) (w v : V) |
| (hw : w ∈ pins) (hv : v ∈ pins) : |
| weldBody G pins w v = conditionalBody G pins (fun _ => 0) := by |
| have hh : ∀ h x, weldHeight pins w v h x = restore pins (fun _ => 0) h x := by |
| intro h x |
| by_cases hx : x = w |
| · subst x; simp [weldHeight, restore, hw, hv] |
| · simp [weldHeight, hx] |
| ext h |
| simp only [weldBody, conditionalBody, Set.mem_ofPred_eq, hh] |
|
|
| def weldBoundary (w v : V) (a : ℝ) (x : V) : ℝ := if x = w ∨ x = v then a else 0 |
|
|
| lemma weld_coordinate (pins : Set V) (w v : V) (hw : w ∈ pins) (hv : v ∉ pins) |
| (h : Free (insert v pins) → ℝ) (a : ℝ) (x : V) : |
| weldHeight pins w v ((splitCoordinate pins v hv).symm (h,a)) x = |
| restore (insert v pins) (weldBoundary w v a) h x := by |
| have hvw : v ≠ w := fun he => hv (he ▸ hw) |
| rw [weldHeight, coordinate_restore, coordinate_restore] |
| by_cases hxw : x = w |
| · subst x |
| simp [restore, hw, weldBoundary] |
| · by_cases hxv : x = v |
| · subst x |
| simp [hxw, restore, weldBoundary] |
| · by_cases hxp : x ∈ pins |
| · simp [hxw, hxv, restore, hxp, weldBoundary] |
| · simp [hxw, hxv, restore, hxp] |
|
|
| lemma weld_volume_coordinate [Fintype V] (G : SimpleGraph V) (pins : Set V) |
| (w v : V) (hw : w ∈ pins) (hv : v ∉ pins) : |
| volume (weldBody G pins w v) = ∫⁻ a : ℝ, |
| volume (conditionalBody G (insert v pins) (weldBoundary w v a)) := by |
| rw [volume_coordinate_slices pins v hv _ (closed_weldBody G pins w v).measurableSet] |
| apply lintegral_congr |
| intro a |
| congr 1 |
| ext h |
| simp only [weldBody, conditionalBody, Set.mem_ofPred_eq, weld_coordinate pins w v hw hv] |
|
|
| lemma weld_boundary_extreme (pins : Set V) (w v : V) (a : ℝ) : |
| (∀ y ∈ pins, weldBoundary w v a y ≤ weldBoundary w v a w) ∨ |
| (∀ y ∈ pins, weldBoundary w v a w ≤ weldBoundary w v a y) := by |
| rcases le_total 0 a with ha | ha |
| · left; intro y hy; simp only [weldBoundary, true_or, ite_true]; split_ifs <;> linarith |
| · right; intro y hy; simp only [weldBoundary, true_or, ite_true]; split_ifs <;> linarith |
|
|
| lemma weld_released_body (G : SimpleGraph V) (pins : Set V) (w v : V) (a : ℝ) : |
| conditionalBody G ((insert v pins) \ {w}) (weldBoundary w v a) = |
| conditionalBody G ((insert v pins) \ {w}) (Function.update (fun _ => 0) v a) := by |
| apply body_boundary_congr |
| intro x hx |
| have hn : x ≠ w := hx.2 |
| by_cases hv : x = v |
| · subst x; simp [weldBoundary] |
| · simp [weldBoundary, hn, hv] |
|
|
| lemma volume_body_pin_congr [Fintype V] (G : SimpleGraph V) {p q : Set V} |
| (hpq : p = q) (b : V → ℝ) : |
| volume (conditionalBody G p b) = volume (conditionalBody G q b) := by |
| subst q |
| rfl |
|
|
| theorem volume_weld_le [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] |
| (pins : Set V) (w v : V) (hw : w ∈ pins) (hvw : v ≠ w) : |
| volume (weldBody G pins w v) ≤ |
| 2 ^ G.degree w * volume (conditionalBody G (pins \ {w}) (fun _ => 0)) := by |
| classical |
| by_cases hv : v ∈ pins |
| · rw [weld_pinned G pins w v hw hv] |
| exact volume_release_max_pin G pins w hw (fun _ => 0) (by simp) |
| · have hvs : v ∉ pins \ {w} := fun h => hv h.1 |
| rw [weld_volume_coordinate G pins w v hw hv, |
| volume_body_coordinate G (pins \ {w}) (fun _ => 0) v hvs, |
| ← lintegral_const_mul' _ _ (by finiteness)] |
| apply lintegral_mono |
| intro a |
| have hh := volume_release_extreme_pin G (insert v pins) w (Or.inr hw) |
| (weldBoundary w v a) (weld_boundary_extreme (insert v pins) w v a) |
| rw [weld_released_body] at hh |
| have he : (insert v pins) \ {w} = insert v (pins \ {w}) := by |
| ext x |
| simp only [mem_sdiff, mem_insert_iff, mem_singleton_iff] |
| constructor |
| · rintro ⟨h,hn⟩; exact h.elim Or.inl (fun hp => Or.inr ⟨hp,hn⟩) |
| · rintro (h | h) |
| · subst x; exact ⟨Or.inl rfl,hvw⟩ |
| · exact ⟨Or.inr h.1,h.2⟩ |
| rw [volume_body_pin_congr G he _] at hh |
| exact hh |
|
|
| end ExtremalEdges |
|
|
| end |
|
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| end OAI |
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