| import OAI.Probability.LatticePassage.LocalPassage |
| import Mathlib.Algebra.Order.BigOperators.Ring.Finset |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open Finset |
|
|
| lemma trailCost_eq_sum (ω : Environment) {x y : Vertex} (p : lattice.Walk x y) |
| (hp : p.IsTrail) : pathCost ω p = ∑ e ∈ (pathEdges p).toFinset, ω e := by |
| classical |
| rw [pathCost_eq_edgeList_sum] |
| exact (List.sum_toFinset _ (pathEdges_nodup hp)).symm |
|
|
| theorem trailCost_one_edge_change (ω ω' : Environment) (e : Edge) |
| (h : ∀ f, f ≠ e → ω' f = ω f) {x y : Vertex} |
| (p : lattice.Walk x y) (hp : p.IsTrail) : |
| pathCost ω' p - pathCost ω p = |
| if e ∈ pathEdges p then ω' e - ω e else 0 := by |
| classical |
| rw [trailCost_eq_sum ω' p hp, trailCost_eq_sum ω p hp, ← sum_sub_distrib] |
| by_cases he : e ∈ pathEdges p |
| · rw [ite_eq_left he] |
| apply sum_eq_single_of_mem e (List.mem_toFinset.mpr he) |
| intro f _ hfe |
| rw [h f hfe, sub_self] |
| · rw [ite_eq_right he] |
| apply sum_eq_zero |
| intro f hf |
| have hfe : f ≠ e := by |
| intro hfe |
| exact he (hfe ▸ List.mem_toFinset.mp hf) |
| rw [h f hfe, sub_self] |
|
|
| theorem localPassageTime_one_edge_upper (ω ω' : Environment) (e : Edge) |
| (h : ∀ f, f ≠ e → ω' f = ω f) {x y : Vertex} {R : ℕ} |
| (p : lattice.Walk x y) (hp : p.IsPath) (hl : p.length < R) |
| (hm : pathCost ω p = localPassageTime ω x y R) : |
| localPassageTime ω' x y R - localPassageTime ω x y R ≤ |
| if e ∈ pathEdges p then ω' e - ω e else 0 := by |
| classical |
| have hc : localPassageTime ω' x y R ≤ pathCost ω' p := |
| csInf_le (local_costs_finite ω' x y R).bddBelow ⟨p,hp,hl,rfl⟩ |
| rw [← trailCost_one_edge_change ω ω' e h p hp.isTrail, ← hm] |
| linarith |
|
|
| theorem localPassageTime_one_edge_bound (ω ω' : Environment) (e : Edge) |
| (h : ∀ f, f ≠ e → ω' f = ω f) (x y : Vertex) (R : ℕ) |
| (hR : latticeDistance x y < R) : |
| localPassageTime ω' x y R - localPassageTime ω x y R ≤ max (ω' e - ω e) 0 := by |
| obtain ⟨p,hp,hl,hm⟩ := local_geodesic_exists ω x y R hR |
| apply (localPassageTime_one_edge_upper ω ω' e h p hp hl hm).trans |
| split_ifs |
| · exact le_max_left _ _ |
| · exact le_max_right _ _ |
|
|
| theorem localPassageTime_one_edge_lipschitz (ω ω' : Environment) (e : Edge) |
| (h : ∀ f, f ≠ e → ω' f = ω f) (x y : Vertex) (R : ℕ) |
| (hR : latticeDistance x y < R) : |
| |localPassageTime ω' x y R - localPassageTime ω x y R| ≤ |ω' e - ω e| := by |
| have h₁ := localPassageTime_one_edge_bound ω ω' e h x y R hR |
| have h₂ := localPassageTime_one_edge_bound ω' ω e (fun f hf => (h f hf).symm) x y R hR |
| have ha : max (ω' e - ω e) 0 ≤ |ω' e - ω e| := max_le (le_abs_self _) (abs_nonneg _) |
| have hb : max (ω e - ω' e) 0 ≤ |ω' e - ω e| := by |
| rw [abs_sub_comm] |
| exact max_le (le_abs_self _) (abs_nonneg _) |
| exact abs_le.mpr ⟨by linarith, h₁.trans ha⟩ |
|
|
| theorem local_pathCost_change_squared (ω ω' : Environment) {x y : Vertex} {R : ℕ} |
| (p : lattice.Walk x y) (hp : p.IsPath) (hl : p.length < R) : |
| (pathCost ω' p - pathCost ω p)^2 ≤ |
| (R : ℝ) * ∑ e ∈ localEdges x R, (ω' e - ω e)^2 := by |
| classical |
| let s := (pathEdges p).toFinset |
| have hc : s.card = p.length := by |
| rw [List.toFinset_card_of_nodup (pathEdges_nodup hp.isTrail), pathEdges_length] |
| have hsub : s ⊆ localEdges x R := by |
| intro e he |
| exact mem_biUnion.mpr ⟨⟨y,p⟩,mem_localCandidates p hp hl,he⟩ |
| have hsum : (∑ e ∈ s, (ω' e-ω e)^2) ≤ ∑ e ∈ localEdges x R, (ω' e-ω e)^2 := |
| sum_le_sum_of_subset_of_nonneg hsub (fun _ _ _ => sq_nonneg _) |
| have hcs := sum_mul_sq_le_sq_mul_sq s (fun _ => (1 : ℝ)) (fun e => ω' e-ω e) |
| simp only [one_mul, one_pow, sum_const, nsmul_eq_mul, mul_one, hc] at hcs |
| rw [trailCost_eq_sum ω' p hp.isTrail, trailCost_eq_sum ω p hp.isTrail, ← sum_sub_distrib] |
| apply hcs.trans |
| gcongr |
|
|
| private lemma localPassageTime_change_squared_of_le (ω ω' : Environment) |
| (x y : Vertex) (R : ℕ) (hR : latticeDistance x y < R) |
| (hle : localPassageTime ω x y R ≤ localPassageTime ω' x y R) : |
| (localPassageTime ω' x y R - localPassageTime ω x y R)^2 ≤ |
| (R : ℝ) * ∑ e ∈ localEdges x R, (ω' e - ω e)^2 := by |
| obtain ⟨p,hp,hl,hm⟩ := local_geodesic_exists ω x y R hR |
| have hc : localPassageTime ω' x y R ≤ pathCost ω' p := |
| csInf_le (local_costs_finite ω' x y R).bddBelow ⟨p,hp,hl,rfl⟩ |
| have hs := local_pathCost_change_squared ω ω' p hp hl |
| rw [hm] at hs |
| nlinarith |
|
|
| theorem localPassageTime_change_squared (ω ω' : Environment) |
| (x y : Vertex) (R : ℕ) (hR : latticeDistance x y < R) : |
| (localPassageTime ω' x y R - localPassageTime ω x y R)^2 ≤ |
| (R : ℝ) * ∑ e ∈ localEdges x R, (ω' e - ω e)^2 := by |
| rcases le_total (localPassageTime ω x y R) (localPassageTime ω' x y R) with h | h |
| · exact localPassageTime_change_squared_of_le ω ω' x y R hR h |
| · simpa only [sub_sq_comm] using localPassageTime_change_squared_of_le ω' ω x y R hR h |
|
|
| end CompanionFPP |
|
|
| end OAI |
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|
