| import OAI.Probability.DilutedSpin.CommonTreeSample |
| import OAI.Probability.DilutedSpin.UpperCountState |
|
|
| namespace OAI |
|
|
| section |
| namespace DilutedSpinGlass.PrescribedTree |
| open KernelTower |
| variable {ι Ω Λ R : Type} [Fintype ι] [DecidableEq ι] |
| [Fintype Ω] [Fintype Λ] [Fintype R] {n p : ℕ} |
|
|
| /-- Duplicate every block at every site before selecting its actual allocation. |
| The unused duplicates disappear by prior-law projection, not by independence |
| of tilted physical sites. -/ |
| def cavityJoin : (l : ℕ) → RootPath ι l → |
| (ι → CavityState Ω Λ p l) → CavityState (ι → Ω) Λ p l |
| | 0,_,z => z |
| | l+1,c,z => (cavityJoin l c.2 (fun i => (z i).1),(z c.1).2) |
|
|
| omit [Fintype R] in |
| lemma cavityJoin_projects (T : ι → KernelTower Ω n) (U : R → KernelTower Λ n) |
| (l : ℕ) (r : RootPath (Fin p → R) l) (c : RootPath ι l) : |
| KernelTower.Projects (cavityJoin l c) n |
| (KernelTower.pi n (fun i => cavityTower (T i) U l r)) |
| (cavityTower (KernelTower.pi n T) U l r) := by |
| induction l with |
| | zero => exact KernelTower.Projects.refl n _ |
| | succ l ih => |
| have h := KernelTower.pi_prod_projects n (fun i => cavityTower (T i) U l r.2) |
| (fun _ : ι => piTower n (fun j => U (r.1 j))) c.1 |
| have g := KernelTower.Projects.prod n (ih r.2 c.2) |
| (KernelTower.Projects.refl n (piTower n (fun j => U (r.1 j)))) |
| exact KernelTower.Projects.comp n h g |
|
|
| omit [Fintype ι] [DecidableEq ι] [Fintype Ω] [Fintype Λ] in |
| lemma pathMap_cavityJoin_zero (y : FinitePath (ι → Ω) n) (c : RootPath ι 0) : |
| pathMap (cavityJoin (Ω := Ω) (Λ := Λ) (p := p) 0 c) n y=y := |
| pathMap_id n y |
|
|
| omit [Fintype ι] [DecidableEq ι] [Fintype Ω] [Fintype Λ] in |
| lemma pathFst_cavityJoin (l : ℕ) (c : RootPath ι (l+1)) |
| (y : FinitePath (ι → CavityState Ω Λ p (l+1)) n) : |
| pathFst n (pathMap (cavityJoin (Ω := Ω) (Λ := Λ) (p := p) (l+1) c) n y)= |
| pathMap (cavityJoin l c.2) n (pathMap (fun a i => (a i).1) n y) := by |
| induction n with |
| | zero => rfl |
| | succ n ih => exact Prod.ext rfl (ih y.2) |
|
|
| omit [Fintype ι] [DecidableEq ι] [Fintype Ω] [Fintype Λ] in |
| lemma pathSnd_cavityJoin (l : ℕ) (c : RootPath ι (l+1)) |
| (y : FinitePath (ι → CavityState Ω Λ p (l+1)) n) : |
| pathSnd n (pathMap (cavityJoin (Ω := Ω) (Λ := Λ) (p := p) (l+1) c) n y)= |
| pathSnd n (FinitePath.proj n y c.1) := by |
| induction n with |
| | zero => rfl |
| | succ n ih => exact Prod.ext rfl (ih y.2) |
|
|
| omit [Fintype ι] [DecidableEq ι] [Fintype Ω] [Fintype Λ] in |
| lemma cavityProject_join (l : ℕ) (c : RootPath ι l) |
| (a : ι → CavityState Ω Λ p l) (i : ι) : |
| cavityProject l (cavityJoin l c a) i=cavityProject l (a i) := by |
| induction l with |
| | zero => rfl |
| | succ l ih => exact ih c.2 (fun i => (a i).1) |
|
|
| omit [Fintype ι] [DecidableEq ι] [Fintype Ω] [Fintype Λ] in |
| lemma cavityProject_pathJoin (l : ℕ) (c : RootPath ι l) |
| (y : FinitePath (ι → CavityState Ω Λ p l) n) (i : ι) : |
| pathMap (fun a : CavityState (ι → Ω) Λ p l => cavityProject l a i) n (pathMap (cavityJoin l c) n y)= |
| pathMap (cavityProject l) n (FinitePath.proj n y i) := by |
| induction n with |
| | zero => rfl |
| | succ n ih => exact Prod.ext (cavityProject_join l c y.1 i) (ih y.2) |
|
|
| end DilutedSpinGlass.PrescribedTree |
|
|
| end |
|
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| end OAI |
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|
