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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Dynamics.ProtectedBlock
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace SK.Analytic
def BlockProtected (B j : ℕ) (A : Finset ℕ) : Prop :=
(∀ i,(j-1)/B*B ≤ i → i < j → i ∈ A) ∧ (∀ k, k*B < j → k*B ∈ A)
theorem protected_block_step (B H j : ℕ) (hB : 0 < B) (hHB : H < B)
(A D : Finset ℕ) (hA : ∀ i ∈ A, i < j) (hc : BlockProtected B j A)
(hs : IsRetainedSuffix A D) (hcard : D.card ≤ H)
(hcompat : j%B ≠ 0 → j-1∉D) :
BlockProtected B (j+1) (insert j (A\D)) ∧ (0 < j → 0 ∈ A\D) := by
classical
obtain ⟨hc,hm⟩ := hc
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) (hij : i ≤ j-B) : i ∈ A\D := by
have hj0 : 0 < j := by have := hA i hi; omega
have hBj : B ≤ j := Nat.le_of_dvd hj0 (Nat.dvd_of_mod_eq_zero hr)
have hq : j/B=(j-1)/B+1 := by
simpa only [Nat.sub_add_cancel hj0] using
(Nat.succ_div_of_dvd (a:=j-1) (b:=B) (by simpa only [Nat.sub_add_cancel hj0] 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 := by
intro k hk
have hkj := Finset.mem_Ico.mp hk
exact hc k (by omega) (by omega)
exact retainedSuffix_preserves_before_block hs hcard hHB hb hi hij
refine ⟨⟨?_,?_⟩,?_⟩
· intro i hil hij
simp only [Nat.add_sub_cancel] at hil
have : i=j := by omega
subst i
exact Finset.mem_insert_self _ _
· intro k hk
by_cases he : k*B=j
· rw [he]; exact Finset.mem_insert_self _ _
· apply Finset.mem_insert_of_mem
apply hsurvive _ (hm k (by omega))
have hlt : 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
· intro hj
apply hsurvive 0
· simpa using hm 0 (by simpa using hj)
· 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 := hc (j-1) (by omega) (by omega)
have hD0 := retainedSuffix_empty_of_last hs (fun i hi => by have := hA i hi; omega) hjlast (hcompat hr)
rw [hD0,Finset.sdiff_empty]
refine ⟨⟨?_,?_⟩,?_⟩
· intro i hil hij
simp only [Nat.add_sub_cancel] at hil
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))
· intro _
simpa using hm 0 (by simpa using hj0)
end SK.Analytic
end
end
end OAI
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