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720fa25 | 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 | import Mathlib
namespace OAI
namespace SnakyPrototype
variable {α : Type*} [DecidableEq α]
abbrev Cell := ℤ × ℤ
def snaky : Finset Cell := {(0, 0), (1, 0), (2, 0), (3, 0), (3, 1), (4, 1)}
def orient (r : Fin 8) (p : Cell) : Cell :=
match r.val with
| 0 => (p.1, p.2)
| 1 => (p.2, p.1)
| 2 => (p.1, -p.2)
| 3 => (-p.2, p.1)
| 4 => (-p.1, p.2)
| 5 => (p.2, -p.1)
| 6 => (-p.1, -p.2)
| _ => (-p.2, -p.1)
def placement (r : Fin 8) (t p : Cell) : Cell := orient r p + t
def HasSnaky (M : Finset Cell) : Prop :=
∃ (r : Fin 8) (t : Cell), snaky.image (placement r t) ⊆ M
end SnakyPrototype
namespace SnakyPrototype.OrdinaryStrategy
variable {α : Type*} [DecidableEq α] [Infinite α]
abbrev Policy (α : Type*) := ℕ → Finset α → Finset α → α
def playState (σ : Policy α) (N : ℕ) (β : ℕ → α) (M₀ B₀ : Finset α) :
ℕ → Finset α × Finset α
| 0 => (M₀, B₀)
| k + 1 =>
let state := playState σ N β M₀ B₀ k
(insert (σ (N - k) state.1 state.2) state.1, insert (β k) state.2)
def makerAt (σ : Policy α) (N : ℕ) (β : ℕ → α) (M₀ B₀ : Finset α) (k : ℕ) : α :=
let state := playState σ N β M₀ B₀ k
σ (N - k) state.1 state.2
def LegalRepliesBeforeFinal (σ : Policy α) (N : ℕ) (β : ℕ → α)
(M₀ B₀ : Finset α) : Prop :=
∀ k, k + 1 < N →
β k ∉ insert (makerAt σ N β M₀ B₀ k) (playState σ N β M₀ B₀ k).1 ∧
β k ∉ (playState σ N β M₀ B₀ k).2
end SnakyPrototype.OrdinaryStrategy
end OAI
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