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import OAI.Probability.CriticalPercolation.Entropy.BinaryLaw
namespace OAI
section
section
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph BinaryInterpolation
variable {V E : Type*}
def sides : (n : ℕ) → History E n → Fin n → Bool
| 0, _ => Fin.elim0
| n+1, h => Fin.snoc (sides n h.1) h.2.1
def marks : (n : ℕ) → History E n → Fin n → Option E
| 0, _ => Fin.elim0
| n+1, h => Fin.snoc (marks n h.1) h.2.2
def fromWords : (n : ℕ) → (Fin n → Bool) → (Fin n → Option E) → History E n
| 0, _, _ => PUnit.unit
| n+1, b, m => (fromWords n (Fin.init b) (Fin.init m),b (Fin.last n),m (Fin.last n))
lemma fromWords_sides_marks (n : ℕ) (h : History E n) :
fromWords n (sides n h) (marks n h) = h := by
induction n with
| zero => cases h; rfl
| succ n ih =>
simp only [fromWords,sides,marks,Fin.init_snoc,Fin.snoc_last,ih]
rfl
lemma sides_fromWords (n : ℕ) (b : Fin n → Bool) (m : Fin n → Option E) :
sides n (fromWords n b m) = b := by
induction n with
| zero => exact Subsingleton.elim _ _
| succ n ih => simp only [fromWords,sides,ih,Fin.snoc_init_self]
lemma marks_fromWords (n : ℕ) (b : Fin n → Bool) (m : Fin n → Option E) :
marks n (fromWords n b m) = m := by
induction n with
| zero => exact Subsingleton.elim _ _
| succ n ih => simp only [fromWords,marks,ih,Fin.snoc_init_self]
def wordsEquiv (n : ℕ) : History E n ≃ (Fin n → Bool) × (Fin n → Option E) where
toFun h := (sides n h, marks n h)
invFun h := fromWords n h.1 h.2
left_inv := fromWords_sides_marks n
right_inv := fun h => Prod.ext (sides_fromWords n h.1 h.2) (marks_fromWords n h.1 h.2)
lemma interval_fromWords (d n : ℕ) (b : Fin n → Bool) (m m' : Fin n → Option E) :
interval d n (fromWords n b m) = interval d n (fromWords n b m') := by
induction n with
| zero => rfl
| succ n ih => simp only [fromWords,interval,ih (Fin.init b) (Fin.init m) (Fin.init m')]
lemma nextIndex_fromWords (d n : ℕ) (b : Fin n → Bool) (m m' : Fin n → Option E) :
nextIndex d n (fromWords n b m) = nextIndex d n (fromWords n b m') := by
simp only [nextIndex,interval_fromWords d n b m m']
lemma anchors_fromWords (d : ℕ) (x : ℕ → V) (n : ℕ) (b : Fin n → Bool)
(m m' : Fin n → Option E) :
anchors d x n (fromWords n b m) = anchors d x n (fromWords n b m') := by
induction n with
| zero => rfl
| succ n ih =>
simp only [fromWords,anchors,ih (Fin.init b) (Fin.init m) (Fin.init m'),
nextIndex_fromWords d n (Fin.init b) (Fin.init m) (Fin.init m')]
lemma anchorSet_fromWords (d : ℕ) (x : ℕ → V) (n : ℕ) (b : Fin n → Bool)
(m m' : Fin n → Option E) :
anchorSet d x n (fromWords n b m) = anchorSet d x n (fromWords n b m') := by
simp only [anchorSet,anchors_fromWords d x n b m m']
def historyPrefix : (n : ℕ) → History E n → (k : ℕ) → History E k
| 0, _, 0 => PUnit.unit
| 0, _, k+1 => fromWords (k+1) (fun _ => true) (fun _ => none)
| n+1, h, k => if he : k = n+1 then he.symm ▸ h else historyPrefix n h.1 k
lemma prefix_self (n : ℕ) (h : History E n) : historyPrefix n h n = h := by
cases n with
| zero => cases h; rfl
| succ n => simp only [historyPrefix,dite_true]
lemma prefix_prev (n : ℕ) (h : History E (n+1)) (k : ℕ) (hk : k ≤ n) :
historyPrefix (n+1) h k = historyPrefix n h.1 k := by
rw [historyPrefix,dite_eq_right (by omega : k ≠ n+1)]
lemma prefix_last (n : ℕ) (h : History E (n+1)) : historyPrefix (n+1) h n = h.1 := by
rw [prefix_prev _ _ _ le_rfl,prefix_self]
lemma prefix_fromWords (n : ℕ) (b : Fin n → Bool) (m m' : Fin n → Option E)
(k : ℕ) (hk : k ≤ n) :
sides k (historyPrefix n (fromWords n b m) k) = sides k (historyPrefix n (fromWords n b m') k) := by
induction n with
| zero =>
have he : k=0 := by omega
subst k
exact Subsingleton.elim _ _
| succ n ih =>
by_cases he : k=n+1
· subst k
simp only [prefix_self,sides_fromWords]
· have hk' : k ≤ n := by omega
rw [prefix_prev _ _ _ hk',prefix_prev _ _ _ hk']
exact ih (Fin.init b) (Fin.init m) (Fin.init m') hk'
lemma anchors_sides (d : ℕ) (x : ℕ → V) (n : ℕ) (h h' : History E n)
(hs : sides n h = sides n h') : anchors d x n h = anchors d x n h' := by
conv_lhs => rw [←fromWords_sides_marks n h]
conv_rhs => rw [←fromWords_sides_marks n h']
rw [hs]
exact anchors_fromWords d x n _ _ _
lemma interval_sides (d n : ℕ) (h h' : History E n) (hs : sides n h = sides n h') :
interval d n h = interval d n h' := by
conv_lhs => rw [←fromWords_sides_marks n h]
conv_rhs => rw [←fromWords_sides_marks n h']
rw [hs]
exact interval_fromWords d n _ _ _
lemma nextIndex_sides (d n : ℕ) (h h' : History E n) (hs : sides n h = sides n h') :
nextIndex d n h = nextIndex d n h' := by
simp only [nextIndex,interval_sides d n h h' hs]
section Density
variable [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
variable (G : BondGraph V E) (order : List (LazyDFS.Dart V E)) (d : ℕ) (x : ℕ → V)
variable (p : ℝ) (w : E → Prop → ℝ) (r₀ : (E → Prop) → ℝ)
lemma density_prefix_support (n : ℕ) (h : History E n) (ω : E → Prop)
(hr : density G order d x p w r₀ n h ω ≠ 0) (k : ℕ) (hk : k ≤ n) :
density G order d x p w r₀ k (historyPrefix n h k) ω ≠ 0 := by
induction n with
| zero =>
have he : k=0 := by omega
subst k
simpa only [prefix_self] using hr
| succ n ih =>
by_cases he : k=n+1
· subst k
simpa only [prefix_self] using hr
· have hk' : k ≤ n := by omega
rw [prefix_prev _ _ _ hk']
exact ih h.1 (density_step_support G order d x p w r₀ n h ω hr).1 hk'
end Density
end CriticalPercolation.BinaryLaw
end
end
end OAI
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