| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace PolynomialVacuumOpenPath |
| open Classical |
| variable {α : Type*} |
|
|
| def follow (f : α → Option α) (a : α) : ℕ → Option α |
| | 0 => some a |
| | n+1 => (follow f a n).bind f |
|
|
| @[simp] theorem follow_zero (f : α → Option α) (a : α) : follow f a 0=some a := rfl |
| @[simp] theorem follow_succ (f : α → Option α) (a : α) (n : ℕ) : |
| follow f a (n+1)=(follow f a n).bind f := rfl |
|
|
| |
| theorem follow_injective (f : α → Option α) |
| (hf : ∀ u v w,f u=some w → f v=some w → u=v) (a : α) |
| (ha : ∀ u,f u≠some a) {i j : ℕ} {b : α} |
| (hi : follow f a i=some b) (hj : follow f a j=some b) : i=j := by |
| induction i generalizing j b with |
| | zero => |
| have hb : a=b := Option.some.inj hi |
| subst b |
| cases j with |
| | zero => rfl |
| | succ j => |
| rw [follow_succ] at hj |
| cases hh : follow f a j with |
| | none => simp [hh] at hj |
| | some u => exact (ha u (by simpa [hh] using hj)).elim |
| | succ i ih => |
| cases j with |
| | zero => |
| have hb : a=b := Option.some.inj hj |
| subst b |
| rw [follow_succ] at hi |
| cases hh : follow f a i with |
| | none => simp [hh] at hi |
| | some u => exact (ha u (by simpa [hh] using hi)).elim |
| | succ j => |
| rw [follow_succ] at hi hj |
| cases hu : follow f a i with |
| | none => simp [hu] at hi |
| | some u => |
| cases hv : follow f a j with |
| | none => simp [hv] at hj |
| | some v => |
| have he : u=v := hf u v b (by simpa [hu] using hi) (by simpa [hv] using hj) |
| subst v |
| exact congrArg Nat.succ (ih hu hv) |
|
|
| theorem eventually_exits [Fintype α] (f : α → Option α) |
| (hf : ∀ u v w,f u=some w → f v=some w → u=v) (a : α) |
| (ha : ∀ u,f u≠some a) : ∃ n,follow f a n=none := by |
| by_contra hn |
| push Not at hn |
| have hex (n : ℕ) : ∃ b,follow f a n=some b := Option.ne_none_iff_exists'.mp (hn n) |
| let g : Fin (Fintype.card α+1) → α:=fun n => (hex n.val).choose |
| have hg : Function.Injective g := by |
| intro i j he |
| apply Fin.ext |
| apply follow_injective f hf a ha ((hex i.val).choose_spec) |
| change (hex i.val).choose=(hex j.val).choose at he |
| rw [he] |
| exact (hex j.val).choose_spec |
| have hc:=Fintype.card_le_of_injective g hg |
| simp only [Fintype.card_fin] at hc |
| omega |
|
|
| section Exit |
| variable [Fintype α] (f : α → Option α) |
| (hf : ∀ u v w,f u=some w → f v=some w → u=v) (a : α) |
| (ha : ∀ u,f u≠some a) |
| noncomputable def exitTime : ℕ := Nat.find (eventually_exits f hf a ha) |
|
|
| theorem exitTime_spec : follow f a (exitTime f hf a ha)=none := |
| Nat.find_spec (eventually_exits f hf a ha) |
|
|
| theorem exitTime_pos : 0<exitTime f hf a ha := by |
| have h:=exitTime_spec f hf a ha |
| by_contra hn |
| have hz : exitTime f hf a ha=0 := by omega |
| simp [hz] at h |
|
|
| theorem beforeExit_some (j : Fin (exitTime f hf a ha)) : ∃ b,follow f a j.val=some b := by |
| apply Option.ne_none_iff_exists'.mp |
| exact Nat.find_min (eventually_exits f hf a ha) j.isLt |
|
|
| noncomputable def pathAt (j : Fin (exitTime f hf a ha)) : α := |
| (beforeExit_some f hf a ha j).choose |
|
|
| theorem pathAt_spec (j : Fin (exitTime f hf a ha)) : |
| follow f a j.val=some (pathAt f hf a ha j) := |
| (beforeExit_some f hf a ha j).choose_spec |
|
|
| theorem pathAt_injective : Function.Injective (pathAt f hf a ha) := by |
| intro i j hij |
| apply Fin.ext |
| apply follow_injective f hf a ha (pathAt_spec f hf a ha i) |
| rw [hij] |
| exact pathAt_spec f hf a ha j |
|
|
| theorem pathAt_start : pathAt f hf a ha ⟨0,exitTime_pos f hf a ha⟩=a := by |
| exact (Option.some.inj (pathAt_spec f hf a ha ⟨0,exitTime_pos f hf a ha⟩)).symm |
|
|
| theorem pathAt_next (i j : Fin (exitTime f hf a ha)) (hij : j.val=i.val+1) : |
| f (pathAt f hf a ha i)=some (pathAt f hf a ha j) := by |
| have h:=pathAt_spec f hf a ha j |
| rw [hij,follow_succ,pathAt_spec] at h |
| exact h |
|
|
| theorem pathAt_exit (i : Fin (exitTime f hf a ha)) (hi : i.val+1=exitTime f hf a ha) : |
| f (pathAt f hf a ha i)=none := by |
| have h:=exitTime_spec f hf a ha |
| rw [← hi,follow_succ,pathAt_spec] at h |
| exact h |
| end Exit |
|
|
| end PolynomialVacuumOpenPath |
|
|
| end OAI |
|
|
