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e82fac3 | 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
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
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