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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
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