File size: 3,954 Bytes
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 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 | import OAI.GameTheory.SnakyTwentyOne.Model
import OAI.GameTheory.SnakyTwentyOne.Semantics.Game
namespace OAI
namespace SnakyPrototype
variable {α : Type*} [DecidableEq α]
def inverseOrientation (r : Fin 8) : Fin 8 :=
if r = 3 then 5 else if r = 5 then 3 else r
theorem orient_inverse_left (r : Fin 8) (p : Cell) :
orient (inverseOrientation r) (orient r p) = p := by
fin_cases r <;> rcases p with ⟨x, y⟩ <;> simp [orient, inverseOrientation]
theorem orient_inverse_right (r : Fin 8) (p : Cell) :
orient r (orient (inverseOrientation r) p) = p := by
fin_cases r <;> rcases p with ⟨x, y⟩ <;> simp [orient, inverseOrientation]
def orientationEquiv (r : Fin 8) : Cell ≃ Cell where
toFun := orient r
invFun := orient (inverseOrientation r)
left_inv := orient_inverse_left r
right_inv := orient_inverse_right r
def translationEquiv (t : Cell) : Cell ≃ Cell where
toFun := fun p => p + t
invFun := fun p => p - t
left_inv := fun p => add_sub_cancel_right p t
right_inv := fun p => sub_add_cancel p t
def placementEquiv (r : Fin 8) (t : Cell) : Cell ≃ Cell :=
(orientationEquiv r).trans (translationEquiv t)
@[simp] theorem placementEquiv_apply (r : Fin 8) (t p : Cell) :
placementEquiv r t p = placement r t p := rfl
@[simp] theorem coe_placementEquiv (r : Fin 8) (t : Cell) :
(placementEquiv r t : Cell → Cell) = placement r t := rfl
def composeOrientation (r s : Fin 8) : Fin 8 :=
![![0,1,2,3,4,5,6,7], ![1,0,3,2,5,4,7,6],
![2,5,0,7,6,1,4,3], ![3,4,1,6,7,0,5,2],
![4,3,6,1,0,7,2,5], ![5,2,7,0,1,6,3,4],
![6,7,4,5,2,3,0,1], ![7,6,5,4,3,2,1,0]] r s
theorem orient_comp (r s : Fin 8) (p : Cell) :
orient r (orient s p) = orient (composeOrientation r s) p := by
fin_cases r <;> fin_cases s <;> rcases p with ⟨x, y⟩ <;>
simp [orient, composeOrientation]
theorem orient_add (r : Fin 8) (p q : Cell) :
orient r (p + q) = orient r p + orient r q := by
fin_cases r <;> rcases p with ⟨x, y⟩ <;> rcases q with ⟨u, v⟩ <;>
simp [orient, add_comm]
theorem placement_comp (r s : Fin 8) (t u p : Cell) :
placement r t (placement s u p) =
placement (composeOrientation r s) (orient r u + t) p := by
simp only [placement, orient_add, orient_comp, add_assoc]
theorem HasSnaky.mono {M N : Finset Cell} (h : HasSnaky M) (hMN : M ⊆ N) :
HasSnaky N := by
obtain ⟨r, t, hrt⟩ := h
exact ⟨r, t, hrt.trans hMN⟩
theorem HasSnaky.placed (r : Fin 8) (t : Cell) (M : Finset Cell)
(h : HasSnaky M) : HasSnaky (M.image (placement r t)) := by
obtain ⟨s, u, hsu⟩ := h
refine ⟨composeOrientation r s, orient r u + t, ?_⟩
intro z hz
obtain ⟨p, hp, hpz⟩ := Finset.mem_image.mp hz
refine Finset.mem_image.mpr ⟨placement s u p,
hsu (Finset.mem_image_of_mem (placement s u) hp), ?_⟩
exact (placement_comp r s t u p).trans hpz
theorem placed_template {n : ℕ} {A H : Finset Cell}
(h : Template HasSnaky n A H) (r : Fin 8) (t : Cell) :
Template HasSnaky n (A.image (placement r t)) (H.image (placement r t)) := by
have hw : ∀ X, HasSnaky X → HasSnaky (X.image (placementEquiv r t)) := by
intro X hx
simpa only [coe_placementEquiv] using HasSnaky.placed r t X hx
simpa only [coe_placementEquiv] using h.transport (placementEquiv r t) hw
def basePoint (i : Fin 6) : Cell := ![(0,0),(1,0),(2,0),(3,0),(3,1),(4,1)] i
def baseEnvelope (i : Fin 6) : Finset Cell :=
snaky.image (placement 0 (-basePoint i))
def baseRequired (i : Fin 6) : Finset Cell := (baseEnvelope i).erase (0, 0)
theorem basePoint_mem (i : Fin 6) : basePoint i ∈ snaky := by
fin_cases i <;> decide
theorem base_works (i : Fin 6) :
Template HasSnaky 1 (baseRequired i) (baseEnvelope i) := by
apply immediate_template HasSnaky (baseEnvelope i) (0, 0)
· exact Finset.mem_image.mpr ⟨basePoint i, basePoint_mem i, by simp [placement, orient]; rfl⟩
· intro M hM
exact ⟨0, -basePoint i, hM⟩
end SnakyPrototype
end OAI
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