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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
namespace OAI
section
section
namespace CriticalPercolation.BinaryInterpolation
open scoped Classical
def midpoint (l r : ℕ) : ℕ := (l+r)/2
def bisect (l r : ℕ) (side : Bool) : ℕ × ℕ :=
if r ≤ l+1 then (l,r) else if side then (midpoint l r,r) else (l,midpoint l r)
lemma midpoint_strict {l r : ℕ} (h : l+1 < r) :
l < midpoint l r ∧ midpoint l r < r := by
dsimp [midpoint]
omega
lemma bisect_bounds {l r : ℕ} (h : l < r) (b : Bool) :
l ≤ (bisect l r b).1 ∧ (bisect l r b).1 < (bisect l r b).2 ∧
(bisect l r b).2 ≤ r := by
unfold bisect
split_ifs with ht hb
· simp [h]
· have hm := midpoint_strict (by omega : l+1 < r)
simp only
omega
· have hm := midpoint_strict (by omega : l+1 < r)
simp only
omega
lemma bisect_length {l r : ℕ} (h : l+1 < r) (b : Bool) :
(bisect l r b).2 - (bisect l r b).1 ≤ (r-l+1)/2 := by
unfold bisect
rw [ite_eq_right (by omega : ¬ r ≤ l+1)]
cases b <;> simp only [Bool.false_eq_true, ↓reduceIte,midpoint] <;> omega
lemma bisect_clog {l r : ℕ} (h : l+1 < r) (b : Bool) :
Nat.clog 2 ((bisect l r b).2 - (bisect l r b).1) + 1 ≤ Nat.clog 2 (r-l) := by
have hh := Nat.clog_mono_right 2 (bisect_length h b)
rw [Nat.clog_of_one_lt (by decide : 1 < 2) (by omega : 1 < r-l)]
have he : r-l+2-1 = r-l+1 := by omega
rw [he]
exact Nat.add_le_add_right hh 1
def terminal (l r : ℕ) : List Bool → ℕ × ℕ
| [] => (l,r)
| b::bs => terminal (bisect l r b).1 (bisect l r b).2 bs
lemma terminal_bounds {l r : ℕ} (h : l < r) (bs : List Bool) :
l ≤ (terminal l r bs).1 ∧ (terminal l r bs).1 < (terminal l r bs).2 ∧
(terminal l r bs).2 ≤ r := by
induction bs generalizing l r with
| nil => simpa [terminal] using h
| cons b bs ih =>
have hb := bisect_bounds h b
have hi := ih hb.2.1
change l ≤ (terminal (bisect l r b).1 (bisect l r b).2 bs).1 ∧ _
exact ⟨hb.1.trans hi.1,hi.2.1,hi.2.2.trans hb.2.2⟩
lemma terminal_eq_of_adjacent {l r : ℕ} (h : r = l+1) (bs : List Bool) :
terminal l r bs = (l,r) := by
induction bs with
| nil => rfl
| cons b bs ih => simpa only [terminal,bisect,ite_eq_left (by omega : r ≤ l+1)] using ih
theorem terminal_adjacent {l r : ℕ} (h : l < r) (bs : List Bool)
(hlen : Nat.clog 2 (r-l) ≤ bs.length) :
(terminal l r bs).2 = (terminal l r bs).1 + 1 := by
induction bs generalizing l r with
| nil =>
simp only [List.length_nil] at hlen
have hh := (Nat.clog_le_iff_le_pow (by decide : 1 < 2)).mp hlen
simp only [pow_zero] at hh
simp only [terminal]
omega
| cons b bs ih =>
by_cases ht : r ≤ l+1
· rw [terminal_eq_of_adjacent (by omega : r = l+1)]
dsimp only
omega
· have hb := bisect_bounds h b
have hc := bisect_clog (by omega : l+1 < r) b
simp only [List.length_cons] at hlen
exact ih hb.2.1 (by omega)
def anchors (l r : ℕ) : List Bool → Finset ℕ
| [] => {l,r}
| b::bs => {l,r,midpoint l r} ∪ anchors (bisect l r b).1 (bisect l r b).2 bs
lemma terminal_mem_anchors (l r : ℕ) (bs : List Bool) :
(terminal l r bs).1 ∈ anchors l r bs ∧ (terminal l r bs).2 ∈ anchors l r bs := by
induction bs generalizing l r with
| nil => simp [terminal,anchors]
| cons b bs ih =>
exact ⟨Finset.mem_union_right _ (ih _ _).1, Finset.mem_union_right _ (ih _ _).2⟩
lemma midpoint_internal {d l r : ℕ} (_hl : l ≤ r) (hr : r ≤ d) (h : l+1 < r) :
midpoint l r < d ∧ 0 < midpoint l r ∧ midpoint l r-l < d ∧ r-midpoint l r < d := by
have hm := midpoint_strict h
omega
abbrev Descriptor (J : ℕ) := (n : Fin (J+1)) × (Fin n.val → Fin 8)
lemma descriptor_card (J : ℕ) : Fintype.card (Descriptor J) = ∑ j ∈ Finset.range (J+1), 8^j := by
simp only [Descriptor,Fintype.card_sigma,Fintype.card_fun,Fintype.card_fin]
exact Fin.sum_univ_eq_sum_range (fun j => 8^j) (J+1)
lemma descriptor_card_le (J : ℕ) : Fintype.card (Descriptor J) ≤ 8^(J+1) := by
rw [descriptor_card]
have hh : ∀ n : ℕ, (∑ j ∈ Finset.range n, 8^j) < 8^n := by
intro n
induction n with
| zero => simp
| succ n ih => rw [Finset.sum_range_succ,pow_succ]; omega
exact (hh (J+1)).le
end CriticalPercolation.BinaryInterpolation
end
end
end OAI
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