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f205175 | 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.BinaryWords
import OAI.Probability.CriticalPercolation.Entropy.BinarySwitches
namespace OAI
section
section
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph BinaryInterpolation InterpolationForest
variable {V E : Type*}
lemma anchors_prefix_getElem (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n)
(k : ℕ) (hk : k ≤ n) (i : ℕ) (hi : i < k+2) :
(anchors d x k (historyPrefix n h k))[i]? = (anchors d x n h)[i]? := by
induction n with
| zero => have he : k=0 := by omega
subst k
rw [prefix_self]
| succ n ih =>
by_cases he : k=n+1
· subst k; rw [prefix_self]
· have hk' : k ≤ n := by omega
rw [prefix_prev n h k hk',ih h.1 hk']
change _ = (anchors d x n h.1 ++ [_])[i]?
rw [List.getElem?_append_left (by rw [anchors_length]; omega)]
lemma node_prefix (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n)
(k : ℕ) (hk : k ≤ n) (i : ℕ) (hi : i < k+2) :
node d x k (historyPrefix n h k) i = node d x n h i := by
simp only [node,anchors_prefix_getElem d x n h k hk i hi]
lemma nodeMark_prefix (n : ℕ) (h : History E n) (k : ℕ) (hk : k ≤ n)
(i : ℕ) (hi : i < k+2) : nodeMark k (historyPrefix n h k) i = nodeMark n h i := by
induction n with
| zero => have he : k=0 := by omega
subst k
rw [prefix_self]
| succ n ih =>
by_cases he : k=n+1
· subst k; rw [prefix_self]
· have hk' : k ≤ n := by omega
rw [prefix_prev n h k hk',ih h.1 hk',nodeMark_old n h i (by omega)]
lemma marks_nodeMark (n : ℕ) (h : History E n) (i : Fin n) :
marks n h i = nodeMark n h (i.val+2) := by
induction n with
| zero => exact Fin.elim0 i
| succ n ih =>
refine Fin.lastCases ?_ (fun j => ?_) i
· simp only [marks,Fin.snoc_last,Fin.val_last,nodeMark_current]
· rw [marks,Fin.snoc_castSucc,ih h.1 j,nodeMark_old n h _ (by simp)]
rfl
lemma targets_prefix (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n)
(k : ℕ) (hk : k ≤ n) :
targets d x n h (k+2) = anchorSet d x k (historyPrefix n h k) := by
ext v
constructor
· rintro ⟨j,hj,hjv⟩
refine ⟨(node d x k (historyPrefix n h k) j).2,?_⟩
have hm := node_mem d x k (historyPrefix n h k) j hj
rw [node_prefix d x n h k hk j hj] at hm ⊢
have heq : node d x n h j = (v, (node d x n h j).2) := Prod.ext hjv rfl
rw [← heq]
exact hm
· rintro ⟨b,hb⟩
obtain ⟨j,hj⟩ := List.mem_iff_getElem?.mp hb
have hjlt : j < k+2 := by
have ht := List.getElem?_eq_some_iff.mp hj
simpa only [anchors_length] using ht.1
have hnode : node d x k (historyPrefix n h k) j = (v,b) := by
simp only [node,hj,Option.getD_some]
exact ⟨j,hjlt,congrArg Prod.fst ((node_prefix d x n h k hk j hjlt).symm.trans hnode)⟩
lemma current_prefix_anchor (d : ℕ) (x : ℕ → V) (n : ℕ) (h : History E n)
(k : ℕ) (hk : k < n) :
(node d x n h (k+2)).1 = x (nextIndex d k (historyPrefix n h k)) := by
have he := node_prefix d x n h (k+1) hk (k+2) (by omega)
rw [←he,node_current]
have hp : (historyPrefix n h (k+1)).1 = historyPrefix n h k := by
clear he
induction n with
| zero => omega
| succ n ih =>
by_cases hn : k=n
· subst k; rw [prefix_self,prefix_last]
· have hk' : k<n := by omega
rw [prefix_prev n h (k+1) (by omega),prefix_prev n h k hk'.le]
exact ih h.1 hk'
rw [hp]
end CriticalPercolation.BinaryLaw
end
end
end OAI
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