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5d5048e | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionContextCells
import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionPassCells
import OAI.Combinatorics.LowSpaceSubsetSum.Compression.BudgetedPreparedCompressionPass
namespace OAI
/-! # Prepaid four-context compression setup and its live data -/
namespace LowSpaceSubsetSum.CompressionPreparedCells
open CompressionPassCells (Fits Ready)
open PreparedCompressionPass (Arrays)
variable {n : ℕ}
theorem context_fits (left right : Array ℤ) (P : ℤ) (hP : 0 < P) (mode : FilteredStream.Mode)
(C : ℕ) (hL : left.size ≤ C) (hR : right.size ≤ C) (hC : C ≤ 2^n) :
Fits n C (CompressionContextCells.build left right P hP mode).1.1 := by
rw [congrArg Prod.fst (CompressionContextCells.build_value left right P hP mode)]
constructor
· change (CompressionContext.sort P right).1.size ≤ C
rwa [CompressionContext.sort_size]
· change Nat.clog 2 left.size ≤ n
exact Nat.clog_le_of_le_pow (hL.trans hC)
· change 2^(Nat.clog 2 left.size+1) ≤ 4*(C+1)
have h := Tournament.padded_nodes_le left.size
(Tournament.build (fun _ : Unit => (0 : ℤ)) (Nat.clog 2 left.size) (fun _ => none))
rw [Tournament.nodes_exact] at h
exact h.trans (Nat.mul_le_mul_left 4 (Nat.add_le_add_right hL 1))
def prepare (A : Arrays) (P t z : ℤ) (hP : 0 < P) :
(BudgetedCompressionPass.Data P t z × ℕ) × ℕ :=
let a := CompressionContextCells.build (A.first 0) (A.second 0) P hP .direct
let b := CompressionContextCells.build (A.first 1) (A.second 1) P hP (.complementary z)
let c := CompressionContextCells.build (A.first 2) (A.second 2) P hP .direct
let d := CompressionContextCells.build (A.first 3) (A.second 3) P hP (.complementary (t-z))
let out : BudgetedCompressionPass.Data P t z :=
⟨a.1.1,b.1.1,c.1.1,d.1.1,
(A.first 0).size*(A.second 0).size,(A.first 1).size*(A.second 1).size,
(A.first 2).size*(A.second 2).size,(A.first 3).size*(A.second 3).size⟩
((out,a.1.2+b.1.2+c.1.2+d.1.2+16),a.2+b.2+c.2+d.2+
CompressionPassCells.resident out+4*PreparedCompressionPass.size A+64)
theorem prepare_value (A : Arrays) (P t z : ℤ) (hP : 0 < P) :
(prepare A P t z hP).1 = PreparedCompressionPass.prepare A P t z hP := by
simp only [prepare,CompressionContextCells.build_value,PreparedCompressionPass.prepare]
theorem prepare_ready (A : Arrays) (P t z : ℤ) (hP : 0 < P)
(C : ℕ) (hL : ∀ i, (A.first i).size ≤ C) (hR : ∀ i, (A.second i).size ≤ C) (hC : C ≤ 2^n) :
Ready n C (prepare A P t z hP).1.1 := by
constructor
· exact context_fits _ _ _ _ _ C (hL 0) (hR 0) hC
· exact context_fits _ _ _ _ _ C (hL 1) (hR 1) hC
· exact context_fits _ _ _ _ _ C (hL 2) (hR 2) hC
· exact context_fits _ _ _ _ _ C (hL 3) (hR 3) hC
theorem arrays_size (A : Arrays) (C : ℕ) (hL : ∀ i, (A.first i).size ≤ C)
(hR : ∀ i, (A.second i).size ≤ C) : PreparedCompressionPass.size A ≤ 8*C+1 := by
have h := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin 4))) =>
Nat.add_le_add (hL i) (hR i))
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,smul_eq_mul] at h
unfold PreparedCompressionPass.size
omega
theorem prepare_cells (I : Instance n) (hn : 10000 ≤ n) (A : Arrays) (P t z : ℤ) (hP : 0 < P)
(C : ℕ) (hL : ∀ i, (A.first i).size ≤ C) (hR : ∀ i, (A.second i).size ≤ C) (hC : C ≤ 2^n) :
(prepare A P t z hP).2 ≤ inputScale I^7*(C+1) := by
let u := inputScale I
have hu : 10000 ≤ u := by unfold u inputScale; omega
have ha := CompressionContextCells.build_cells I hn (A.first 0) (A.second 0) P hP .direct C (hL 0) (hR 0) hC
have hb := CompressionContextCells.build_cells I hn (A.first 1) (A.second 1) P hP (.complementary z)
C (hL 1) (hR 1) hC
have hc := CompressionContextCells.build_cells I hn (A.first 2) (A.second 2) P hP .direct C (hL 2) (hR 2) hC
have hd := CompressionContextCells.build_cells I hn (A.first 3) (A.second 3) P hP (.complementary (t-z))
C (hL 3) (hR 3) hC
have hready := prepare_ready A P t z hP C hL hR hC
have hresident := CompressionPassCells.resident_bound C (prepare A P t z hP).1.1 hready
have hsize := arrays_size A C hL hR
have hlarge : 1024 ≤ u^6 := by
have hp : u ≤ u^6 := by
simpa only [pow_one] using Nat.pow_le_pow_right (show 1 ≤ u by omega) (by decide : 1 ≤ 6)
omega
have hm : 6*u^6*(C+1) ≤ u^7*(C+1) := by
apply Nat.mul_le_mul_right
calc
_ ≤ u*u^6 := Nat.mul_le_mul_right _ (by omega)
_ = _ := (pow_succ' _ _).symm
change _ ≤ u^7*(C+1)
change _+_+_+_+CompressionPassCells.resident (prepare A P t z hP).1.1+
4*PreparedCompressionPass.size A+64 ≤ _
calc
_ ≤ 6*u^6*(C+1) := by nlinarith only [ha,hb,hc,hd,hresident,hsize,hlarge]
_ ≤ _ := hm
def run (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P) (budget : ℕ) :=
if BudgetedPreparedCompressionPass.charge I A ≤ budget then
let prep := prepare A P t z hP
let out := CompressionPassCells.run I C B prep.1.1
(budget-BudgetedPreparedCompressionPass.charge I A)
((out.1.1,(prep.1.2+1)+out.1.2),prep.2+out.2+4*PreparedCompressionPass.size A+64)
else ((none,1),4*PreparedCompressionPass.size A+16)
theorem value (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P) (budget : ℕ) :
(run I C B A P t z hP budget).1 = BudgetedPreparedCompressionPass.run I C B A P t z hP budget := by
simp only [run,prepare_value,CompressionPassCells.value,BudgetedPreparedCompressionPass.run,
BudgetedSeparatorCall.run]
split <;> rfl
theorem cells (I : Instance n) (hn : 10000 ≤ n) (C B : ℕ) (A : Arrays)
(P t z : ℤ) (hP : 0 < P) (budget : ℕ) (hB : B ≤ C)
(hL : ∀ i, (A.first i).size ≤ C) (hR : ∀ i, (A.second i).size ≤ C) (hC : C ≤ 2^n) :
(run I C B A P t z hP budget).2 ≤ inputScale I^8*(C+1) := by
let u := inputScale I
have hu : 10000 ≤ u := by unfold u inputScale; omega
have hp := prepare_cells I hn A P t z hP C hL hR hC
have hs := arrays_size A C hL hR
have hr := CompressionPassCells.cells I C B (prepare A P t z hP).1.1
(budget-BudgetedPreparedCompressionPass.charge I A) hn hB (prepare_ready A P t z hP C hL hR hC)
have h67 : u^6 ≤ u^7 := Nat.pow_le_pow_right (by omega) (by decide)
have hlarge : 100 ≤ u^7 := by
have hp : u ≤ u^7 := by
simpa only [pow_one] using Nat.pow_le_pow_right (show 1 ≤ u by omega) (by decide : 1 ≤ 7)
omega
have hm : 3*u^7*(C+1) ≤ u^8*(C+1) := by
apply Nat.mul_le_mul_right
calc
_ ≤ u*u^7 := Nat.mul_le_mul_right _ (by omega)
_ = _ := (pow_succ' _ _).symm
have hmul := Nat.mul_le_mul_right (C+1) h67
change _ ≤ u^8*(C+1)
unfold run
split <;> dsimp only <;> nlinarith only [hp,hs,hr,hlarge,hm,hmul]
end LowSpaceSubsetSum.CompressionPreparedCells
end OAI
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