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import OAI.Probability.Ising.Marginals
namespace OAI
open scoped BigOperators Classical
namespace BufferedIsing
/-- The actual contracted marginal retains arbitrary fields and all effects
of the common pins. It is distinct from the zero-field comparison weight. -/
noncomputable def contractedPinnedWeight {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (I J F : Finset (Fin n)) (τ : Spins n)
(σ : KeptVertex F → Bool) : ℝ :=
if Respects (comparisonSetoid G I J F ∅) σ then weight G h (pinExtension F τ σ) else 0
lemma respects_terminals_iff {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (σ : KeptVertex F → Bool) :
Respects (comparisonSetoid G I J F ∅) σ ↔
(∀ x y, x.val ∈ I → y.val ∈ I → σ x = σ y) ∧
(∀ x y, x.val ∈ J → y.val ∈ J → σ x = σ y) := by
constructor
· intro H
exact ⟨fun x y hx hy => H x y (Relation.EqvGen.rel _ _ (Or.inl ⟨hx, hy⟩)),
fun x y hx hy => H x y (Relation.EqvGen.rel _ _ (Or.inr (Or.inl ⟨hx, hy⟩)))⟩
· rintro ⟨hI, hJ⟩ x y hxy
induction hxy with
| rel x y h =>
rcases h with h | h | ⟨e, he, _⟩
· exact hI x y h.1 h.2
· exact hJ x y h.1 h.2
· exact (Finset.notMem_empty e he).elim
| refl => rfl
| symm _ _ _ ih => exact ih.symm
| trans _ _ _ _ _ ih ih' => exact ih.trans ih'
lemma contracted_prescription_iff {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (hIF : Disjoint I F) (hJF : Disjoint J F)
(i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J)
(τ : Spins n) (σ : KeptVertex F → Bool) (s t : Bool) :
(σ i = s ∧ σ j = t ∧ Respects (comparisonSetoid G I J F ∅) σ) ↔
AgreesOn I (pinExtension F τ σ) (fun _ => s) ∧
AgreesOn J (pinExtension F τ σ) (fun _ => t) := by
constructor
· rintro ⟨his, hjt, hresp⟩
obtain ⟨hI, hJ⟩ := (respects_terminals_iff G I J F σ).mp hresp
constructor
· intro v hv
have hvF : v ∉ F := fun hvF => Finset.disjoint_left.mp hIF hv hvF
exact (pinExtension_apply F τ σ ⟨v, hvF⟩).trans ((hI _ i hv hi).trans his)
· intro v hv
have hvF : v ∉ F := fun hvF => Finset.disjoint_left.mp hJF hv hvF
exact (pinExtension_apply F τ σ ⟨v, hvF⟩).trans ((hJ _ j hv hj).trans hjt)
· rintro ⟨hI, hJ⟩
have hI' : ∀ v : KeptVertex F, v.val ∈ I → σ v = s := by
intro v hv
simpa only [pinExtension_apply] using hI v.val hv
have hJ' : ∀ v : KeptVertex F, v.val ∈ J → σ v = t := by
intro v hv
simpa only [pinExtension_apply] using hJ v.val hv
refine ⟨hI' i hi, hJ' j hj, (respects_terminals_iff G I J F σ).mpr ?_⟩
exact ⟨fun x y hx hy => (hI' x hx).trans (hI' y hy).symm,
fun x y hx hy => (hJ' x hx).trans (hJ' y hy).symm⟩
/-- Identifying each terminal set separately is *exactly* restriction to its
constant configurations; no new model comparison enters this identity. -/
lemma pairTable_contractedPinned {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (I J F : Finset (Fin n))
(hIF : Disjoint I F) (hJF : Disjoint J F)
(i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J)
(τ : Spins n) (s t : Bool) :
pairTable (contractedPinnedWeight G h I J F τ) i j s t =
jointWeight G h I J F τ (fun _ => s) (fun _ => t) := by
classical
have hdel := constrainedSum_delete_variables G h F τ
(fun ρ => AgreesOn I ρ (fun _ => s) ∧ AgreesOn J ρ (fun _ => t))
have hswap : jointWeight G h I J F τ (fun _ => s) (fun _ => t) =
constrainedSum G h (fun ρ => AgreesOn F ρ τ ∧
(AgreesOn I ρ (fun _ => s) ∧ AgreesOn J ρ (fun _ => t))) := by
unfold jointWeight constrainedSum
apply Finset.sum_congr rfl
intro ρ _
congr 1
exact propext and_rotate.symm
rw [hswap, hdel]
unfold pairTable
apply Finset.sum_congr rfl
intro σ _
unfold contractedPinnedWeight
have H := contracted_prescription_iff G I J F hIF hJF i j hi hj τ σ s t
by_cases hc : σ i = s ∧ σ j = t
· rw [ite_eq_left hc]
by_cases hr : Respects (comparisonSetoid G I J F ∅) σ
· rw [ite_eq_left hr, ite_eq_left (H.mp ⟨hc.1, hc.2, hr⟩)]
· rw [ite_eq_right hr, ite_eq_right (fun hp => hr (H.mpr hp).2.2)]
· rw [ite_eq_right hc, ite_eq_right (fun hp => hc ⟨(H.mpr hp).1, (H.mpr hp).2.1⟩)]
lemma extremeLogOdds_eq_effective {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (I J F : Finset (Fin n))
(hIF : Disjoint I F) (hJF : Disjoint J F)
(i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J) (τ : Spins n) :
extremeLogOdds G h I J F τ =
4 * tableCoupling (pairTable (contractedPinnedWeight G h I J F τ) i j) := by
simp only [tableCoupling, pairTable_contractedPinned G h I J F hIF hJF i j hi hj]
unfold extremeLogOdds
change _ = 4 * ((Real.log (jointWeight G h I J F τ ⊤ ⊤) +
Real.log (jointWeight G h I J F τ ⊥ ⊥) - Real.log (jointWeight G h I J F τ ⊤ ⊥) -
Real.log (jointWeight G h I J F τ ⊥ ⊤)) / 4)
ring
lemma sum_pin_incidence {n m : ℕ} (l r : Fin m → Fin n) (F : Finset (Fin n)) (f : Fin m → ℝ) (σ : Fin n → ℝ) :
(∑ v : Fin n, if v ∉ F then (∑ e, if l e = v ∧ r e ∈ F then f e else 0) * σ v else 0) =
∑ e, if l e ∉ F ∧ r e ∈ F then f e * σ (l e) else 0 := by
classical
calc
_ = ∑ v : Fin n, ∑ e, if v ∉ F ∧ l e = v ∧ r e ∈ F then f e * σ v else 0 := by
apply Finset.sum_congr rfl
intro v _
by_cases hv : v ∈ F
· simp [hv]
· simp only [hv, not_false_eq_true, ite_true, true_and, Finset.sum_mul]
apply Finset.sum_congr rfl
intro e _
split_ifs <;> simp
_ = _ := by
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro e _
have H : ∀ v : Fin n, (v ∉ F ∧ l e = v ∧ r e ∈ F) ↔
(l e = v ∧ l e ∉ F ∧ r e ∈ F) := by
intro v
constructor
· rintro ⟨hv, he, hr⟩
exact ⟨he, he.symm ▸ hv, hr⟩
· rintro ⟨he, hl, hr⟩
exact ⟨he ▸ hl, he, hr⟩
simp_rw [H]
by_cases he : l e ∉ F ∧ r e ∈ F
· simp [he]
· simp [he]
/-- The finite-field contribution induced by the actual prescribed spins.
Neither these induced fields nor the original fields survive in `qZero`. -/
noncomputable def pinField {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (F : Finset (Fin n)) (τ : Spins n) (v : KeptVertex F) : ℝ :=
h v + (∑ e, if G.left e = v.val ∧ G.right e ∈ F then
G.coupling e * spin (τ (G.right e)) else 0) +
∑ e, if G.right e = v.val ∧ G.left e ∈ F then
G.coupling e * spin (τ (G.left e)) else 0
/-- Field and bond terms depending only on fixed sites. They cancel from
all normalized laws, but recording them makes the elimination exact. -/
noncomputable def pinConstant {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (F : Finset (Fin n)) (τ : Spins n) : ℝ :=
(∑ e, if G.left e ∈ F ∧ G.right e ∈ F then
G.coupling e * spin (τ (G.left e)) * spin (τ (G.right e)) else 0) +
∑ v ∈ F, h v * spin (τ v)
lemma energy_pin_expansion {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (F : Finset (Fin n)) (τ ρ : Spins n) (hρ : AgreesOn F ρ τ) :
energy G h ρ = pinConstant G h F τ +
(∑ e, if G.left e ∉ F ∧ G.right e ∉ F then
G.coupling e * spin (ρ (G.left e)) * spin (ρ (G.right e)) else 0) +
∑ v : Fin n, if v ∉ F then
(h v + (∑ e, if G.left e = v ∧ G.right e ∈ F then
G.coupling e * spin (τ (G.right e)) else 0) +
∑ e, if G.right e = v ∧ G.left e ∈ F then
G.coupling e * spin (τ (G.left e)) else 0) * spin (ρ v) else 0 := by
classical
have hsplit : (∑ v : Fin n, if v ∉ F then
(h v + (∑ e, if G.left e = v ∧ G.right e ∈ F then
G.coupling e * spin (τ (G.right e)) else 0) +
∑ e, if G.right e = v ∧ G.left e ∈ F then
G.coupling e * spin (τ (G.left e)) else 0) * spin (ρ v) else 0) =
(∑ v : Fin n, if v ∉ F then h v * spin (ρ v) else 0) +
(∑ e, if G.left e ∉ F ∧ G.right e ∈ F then
(G.coupling e * spin (τ (G.right e))) * spin (ρ (G.left e)) else 0) +
(∑ e, if G.right e ∉ F ∧ G.left e ∈ F then
(G.coupling e * spin (τ (G.left e))) * spin (ρ (G.right e)) else 0) := by
rw [← sum_pin_incidence G.left G.right F
(fun e => G.coupling e * spin (τ (G.right e))) (fun v => spin (ρ v)),
← sum_pin_incidence G.right G.left F
(fun e => G.coupling e * spin (τ (G.left e))) (fun v => spin (ρ v))]
rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro v _
by_cases hv : v ∈ F <;> simp [hv, add_mul]
rw [hsplit]
unfold energy pinConstant
have he : (∑ e, G.coupling e * spin (ρ (G.left e)) * spin (ρ (G.right e))) =
(∑ e, if G.left e ∈ F ∧ G.right e ∈ F then
G.coupling e * spin (τ (G.left e)) * spin (τ (G.right e)) else 0) +
(∑ e, if G.left e ∉ F ∧ G.right e ∉ F then
G.coupling e * spin (ρ (G.left e)) * spin (ρ (G.right e)) else 0) +
(∑ e, if G.left e ∉ F ∧ G.right e ∈ F then
(G.coupling e * spin (τ (G.right e))) * spin (ρ (G.left e)) else 0) +
(∑ e, if G.right e ∉ F ∧ G.left e ∈ F then
(G.coupling e * spin (τ (G.left e))) * spin (ρ (G.right e)) else 0) := by
rw [← Finset.sum_add_distrib, ← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro e _
by_cases hl : G.left e ∈ F <;> by_cases hr : G.right e ∈ F
· simp [hl, hr, hρ _ hl, hρ _ hr]
· simp [hl, hr, hρ _ hl]
· simp [hl, hr, hρ _ hr, mul_comm, mul_left_comm]
· simp [hl, hr]
have hv : (∑ v, h v * spin (ρ v)) =
(∑ v ∈ F, h v * spin (τ v)) +
(∑ v : Fin n, if v ∉ F then h v * spin (ρ v) else 0) := by
rw [← Fintype.sum_extend_by_zero F (fun v => h v * spin (τ v)),
← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro v _
by_cases hv : v ∈ F <;> simp [hv, hρ v]
rw [he, hv]
ring
lemma sum_kept {n : ℕ} (F : Finset (Fin n)) (f : Fin n → ℝ) :
(∑ v : KeptVertex F, f v.val) = ∑ v : Fin n, if v ∉ F then f v else 0 := by
classical
rw [← Finset.sum_filter]
exact (Finset.sum_subtype _ (fun v => by simp only [Finset.mem_filter,
Finset.mem_univ, true_and]) f).symm
/-- Exact deletion of fixed sites: their incident bonds become finite real
fields, while only bonds with two free endpoints remain as interactions. -/
lemma energy_pinExtension {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (F : Finset (Fin n)) (τ : Spins n) (σ : KeptVertex F → Bool) :
energy G h (pinExtension F τ σ) = pinConstant G h F τ +
deletedEnergy G F σ + ∑ v : KeptVertex F, pinField G h F τ v * spin (σ v) := by
classical
rw [energy_pin_expansion G h F τ _ (pinExtension_agrees F τ σ)]
congr 1
· congr 1
unfold deletedEnergy keptEdges
rw [Finset.sum_filter]
apply Finset.sum_congr rfl
intro e _
split_ifs with he
· simp only [pinExtension, extendKept, dite_eq_right he.1, dite_eq_right he.2, dite_eq_left he.1, dite_eq_left he.2]
· rfl
· have hs : (∑ v : KeptVertex F, pinField G h F τ v * spin (σ v)) =
∑ v : Fin n, if v ∉ F then
(h v + (∑ e, if G.left e = v ∧ G.right e ∈ F then
G.coupling e * spin (τ (G.right e)) else 0) +
∑ e, if G.right e = v ∧ G.left e ∈ F then
G.coupling e * spin (τ (G.left e)) else 0) *
spin (pinExtension F τ σ v) else 0 := by
rw [← sum_kept]
simp only [pinField, pinExtension_apply]
exact hs.symm
lemma weight_pinExtension {n m : ℕ} (G : FiniteGraph n m)
(h : Fin n → ℝ) (F : Finset (Fin n)) (τ : Spins n) (σ : KeptVertex F → Bool) :
weight G h (pinExtension F τ σ) = Real.exp (pinConstant G h F τ) *
Real.exp (deletedEnergy G F σ + ∑ v : KeptVertex F, pinField G h F τ v * spin (σ v)) := by
unfold weight
rw [energy_pinExtension]
rw [← Real.exp_add]
congr 1
ring
end BufferedIsing
end OAI
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