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namespace OAI
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
def IsRetainedSuffix (A D : Finset ℕ) : Prop :=
D ⊆ A ∧ ∀ ⦃i j : ℕ⦄, i ∈ D → j ∈ A → i ≤ j → j ∈ D
theorem retainedSuffix_empty_of_last {A D : Finset ℕ} {j : ℕ}
(hD : IsRetainedSuffix A D) (hA : ∀ i ∈ A,i ≤ j) (hj : j ∈ A) (hcompat : j ∉ D) :
D=∅ := by
apply Finset.eq_empty_iff_forall_notMem.mpr
intro i hi
exact hcompat (hD.2 hi hj (hA i (hD.1 hi)))
theorem retainedSuffix_preserves_before_block {A D : Finset ℕ} {s B H : ℕ}
(hD : IsRetainedSuffix A D) (hcard : D.card ≤ H) (hHB : H < B)
(hblock : Finset.Ico s (s+B) ⊆ A) {i : ℕ} (hi : i ∈ A) (his : i ≤ s) :
i ∈ A\D := by
refine Finset.mem_sdiff.mpr ⟨hi,?_⟩
intro hid
have hsub : Finset.Ico s (s+B) ⊆ D := by
intro k hk
exact hD.2 hid (hblock hk) (his.trans (Finset.mem_Ico.mp hk).1)
have h := Finset.card_le_card hsub
rw [Nat.card_Ico] at h
omega
theorem protected_block_markers (B H p : ℕ) (hB : 0 < B) (hHB : H < B)
(A D : ℕ → Finset ℕ)
(hzero : A 0=∅)
(hstep : ∀ j < p,A (j+1)=insert j (A j\D j))
(hrange : ∀ j ≤ p,∀ i ∈ A j,i < j)
(hsuffix : ∀ j < p,IsRetainedSuffix (A j) (D j))
(hcard : ∀ j < p,(D j).card ≤ H)
(hcompatible : ∀ j < p,j%B≠0 → j-1 ∉ D j) :
∀ j ≤ p,
(∀ i, (j-1)/B*B ≤ i → i < j → i ∈ A j) ∧
(∀ k,k*B < j → k*B ∈ A j) := by
intro j hj
induction j with
| zero => simp
| succ j ih =>
have hjp : j < p := by omega
obtain ⟨hc,hm⟩ := ih (by omega)
rw [hstep j hjp]
simp only [Nat.add_sub_cancel]
have hdecomp : j%B+j/B*B=j := by simpa only [Nat.mul_comm B] using Nat.mod_add_div j B
have hprevle := Nat.div_mul_le_self (j-1) B
by_cases hr : j%B=0
· have hjmul : j/B*B=j := by omega
have hsurvive (i : ℕ) (hi : i ∈ A j) (hij : i ≤ j-B) : i ∈ A j\D j := by
by_cases hj0 : j=0
· subst j
rw [hzero] at hi
exact False.elim (Finset.notMem_empty i hi)
· have hBj : B ≤ j := Nat.le_of_dvd (Nat.pos_of_ne_zero hj0) (Nat.dvd_of_mod_eq_zero hr)
have hq : j/B=(j-1)/B+1 := by
simpa only [Nat.sub_add_cancel (by omega : 1 ≤ j)] using
(Nat.succ_div_of_dvd (a:=j-1) (b:=B) (by simpa only [Nat.sub_add_cancel (by omega : 1 ≤ j)] using Nat.dvd_of_mod_eq_zero hr))
have ht : (j-1)/B*B=j-B := by
have hx := congrArg (fun z : ℕ => z*B) hq
simp only [Nat.add_mul,one_mul] at hx
omega
have hb : Finset.Ico (j-B) (j-B+B) ⊆ A j := by
intro k hk
have hkj := Finset.mem_Ico.mp hk
exact hc k (by omega) (by omega)
exact retainedSuffix_preserves_before_block (hsuffix j hjp) (hcard j hjp) hHB hb hi hij
constructor
· intro i hil hij
have : i=j := by omega
subst i
exact Finset.mem_insert_self _ _
· intro k hk
by_cases hkj : k*B=j
· rw [hkj]; exact Finset.mem_insert_self _ _
· apply Finset.mem_insert_of_mem
apply hsurvive _ (hm k (by omega))
have hkj : k < j/B := (Nat.mul_lt_mul_right hB).mp (by omega)
have hx := Nat.mul_le_mul_right B (show k+1 ≤ j/B by omega)
simp only [Nat.add_mul,one_mul] at hx
omega
· have hj0 : 0 < j := by
by_contra h
have hjz : j=0 := by omega
subst j
exact hr (by simp)
have hq : j/B=(j-1)/B := by
simpa only [Nat.sub_add_cancel hj0] using
(Nat.succ_div_of_not_dvd (a:=j-1) (b:=B) (by
simpa only [Nat.sub_add_cancel hj0] using
(fun h => hr (Nat.mod_eq_zero_of_dvd h))))
have hqmul := congrArg (fun z : ℕ => z*B) hq
have hjlast : j-1 ∈ A j := hc (j-1) (by omega) (by omega)
have hD0 := retainedSuffix_empty_of_last (hsuffix j hjp)
(fun i hi => by have := hrange j (by omega) i hi; omega) hjlast
(hcompatible j hjp hr)
rw [hD0,Finset.sdiff_empty]
constructor
· intro i hil hij
by_cases he : i=j
· rw [he]; exact Finset.mem_insert_self _ _
· exact Finset.mem_insert_of_mem (hc i (by omega) (by omega))
· intro k hk
by_cases he : k*B=j
· rw [he]; exact Finset.mem_insert_self _ _
· exact Finset.mem_insert_of_mem (hm k (by omega))
end SK.Analytic
end
end
end OAI
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