| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.MatrixReplicaBridge |
| import OAI.Probability.SKBarriers.Replicas.TriplePerturbation |
| import OAI.Probability.SKBarriers.Replicas.PairFactor |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def overlapCode {n : ℕ} (x y : Config n) : Fin (n+1) := |
| ⟨(Finset.univ.filter (fun i => x i=y i)).card, |
| Nat.lt_succ_of_le ((Finset.card_filter_le _ _).trans_eq (by simp))⟩ |
|
|
| def codeOverlap (n : ℕ) (c : Fin (n+1)) : ℝ := (2*(c.val:ℝ)-(n:ℝ))/(n:ℝ) |
|
|
| theorem overlap_eq_codeOverlap {n : ℕ} (x y : Config n) : |
| overlap x y=codeOverlap n (overlapCode x y) := by |
| have he (i : Fin n) : spin (x i)*spin (y i)=2*(if x i=y i then (1:ℝ) else 0)-1 := by |
| cases hx : x i <;> cases hy : y i <;> norm_num [spin] |
| simp only [overlap,he,Finset.sum_sub_distrib,← Finset.mul_sum, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] |
| congr 1 |
| simp [overlapCode,Finset.sum_boole] |
|
|
| def pairReplicaCode {n : ℕ} (s : ReplicaConfig n 2) : Fin (n+1) := overlapCode (s 0) (s 1) |
|
|
| def tripleReplicaCode {n : ℕ} (s : ReplicaConfig n 3) : Fin 3 → Fin (n+1) := |
| ![overlapCode (s 0) (s 1),overlapCode (s 0) (s 2),overlapCode (s 1) (s 2)] |
|
|
| theorem replicaGram_pair {n : ℕ} (hn : 0 < n) (s : ReplicaConfig n 2) : |
| replicaGram s=pairMatrix 1 (codeOverlap n (pairReplicaCode s)) := by |
| funext i j |
| fin_cases i <;> fin_cases j <;> |
| simp [replicaGram,pairMatrix,pairReplicaCode,← overlap_eq_codeOverlap,overlap_comm,overlap_self hn] |
|
|
| theorem replicaGram_triple {n : ℕ} (hn : 0 < n) (s : ReplicaConfig n 3) : |
| replicaGram s=tripleGram |
| ((codeOverlap n (tripleReplicaCode s 0)+codeOverlap n (tripleReplicaCode s 1))/2) |
| ((codeOverlap n (tripleReplicaCode s 0)-codeOverlap n (tripleReplicaCode s 1))/2) |
| (codeOverlap n (tripleReplicaCode s 2)) := by |
| funext i j |
| fin_cases i <;> fin_cases j <;> |
| simp [replicaGram,tripleGram,tripleReplicaCode,← overlap_eq_codeOverlap,overlap_comm,overlap_self hn] <;> ring_nf <;> exact overlap_comm _ _ |
|
|
| theorem pairReplicaCode_gram {n : ℕ} (hn : 0 < n) {s t : ReplicaConfig n 2} |
| (h : pairReplicaCode s=pairReplicaCode t) : replicaGram s=replicaGram t := by |
| rw [replicaGram_pair hn,replicaGram_pair hn,h] |
|
|
| theorem tripleReplicaCode_gram {n : ℕ} (hn : 0 < n) {s t : ReplicaConfig n 3} |
| (h : tripleReplicaCode s=tripleReplicaCode t) : replicaGram s=replicaGram t := by |
| rw [replicaGram_triple hn,replicaGram_triple hn,h] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
