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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.PreparedCompressionPass
namespace OAI
/-! # A prepaid compression pass including all four initializations -/
namespace LowSpaceSubsetSum.BudgetedPreparedCompressionPass
open BudgetedSeparatorFlow
open PreparedCompressionPass (Arrays)
variable {n : ℕ}
def charge (I : Instance n) (A : Arrays) : ℕ := inputScale I ^ 6 * PreparedCompressionPass.size A + 1
def run (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P) (budget : ℕ) :=
bind (BudgetedSeparatorCall.run (charge I A) (fun _ => PreparedCompressionPass.prepare A P t z hP) budget)
(fun D remaining => BudgetedCompressionPass.run I C B D remaining)
def allowance (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P) : ℕ :=
charge I A + BudgetedCompressionPass.allowance I C B (PreparedCompressionPass.prepare A P t z hP).1
theorem value (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P)
(budget : ℕ) (v : Bool) (remaining : ℕ)
(h : (run I C B A P t z hP budget).1 = some (v, remaining)) :
v = BudgetedCompressionPass.reference I C B (PreparedCompressionPass.prepare A P t z hP).1 ∧
remaining ≤ budget := by
obtain ⟨D, first, hf, hp⟩ := (bind_eq_some _ _ _ _).mp h
obtain ⟨rfl, hfirst, hcharge⟩ := BudgetedSeparatorCall.value _ _ _ _ _ hf
have hv := BudgetedCompressionPass.value I C B _ first v remaining hp
exact ⟨hv.1, by omega⟩
theorem account (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P)
(budget : ℕ) (hn : 10000 ≤ n)
(hfirst : ∀ i, (A.first i).size ≤ 2 ^ n) (hsecond : ∀ i, (A.second i).size ≤ 2 ^ n)
(hpass : ∀ b, Account b (BudgetedCompressionPass.run I C B
(PreparedCompressionPass.prepare A P t z hP).1 b)) :
Account budget (run I C B A P t z hP budget) := by
have hc : (PreparedCompressionPass.prepare A P t z hP).2 + 1 ≤ charge I A :=
Nat.add_le_add_right (PreparedCompressionPass.prepare_work I hn A P t z hP hfirst hsecond) 1
apply bind_account
· exact ⟨BudgetedSeparatorCall.work _ _ _ hc,
fun D remaining h => BudgetedSeparatorCall.spent _ _ _ hc D remaining h⟩
· intro D remaining h
obtain ⟨rfl, _, _⟩ := BudgetedSeparatorCall.value _ _ _ _ _ h
exact hpass remaining
theorem complete (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P)
(hn : 10000 ≤ n) (budget : ℕ) (hb : allowance I C B A P t z hP ≤ budget) :
∃ v remaining, (run I C B A P t z hP budget).1 = some (v, remaining) := by
have hc : charge I A ≤ budget := by unfold allowance at hb; omega
obtain ⟨v, remaining, hp⟩ := BudgetedCompressionPass.complete I C B
(PreparedCompressionPass.prepare A P t z hP).1 hn (budget - charge I A)
(by unfold allowance at hb; omega)
refine ⟨v, remaining, bind_complete _ _ _ _ _ _ ?_ hp⟩
exact congrArg Prod.fst (BudgetedSeparatorCall.complete _ _ _ hc)
theorem remaining (I : Instance n) (C B : ℕ) (A : Arrays) (P t z : ℤ) (hP : 0 < P)
(hn : 10000 ≤ n) (budget : ℕ) (v : Bool) (rest : ℕ)
(h : (run I C B A P t z hP budget).1 = some (v, rest)) :
budget ≤ rest + allowance I C B A P t z hP := by
obtain ⟨D, first, hf, hp⟩ := (bind_eq_some _ _ _ _).mp h
obtain ⟨rfl, hfirst, hcharge⟩ := BudgetedSeparatorCall.value _ _ _ _ _ hf
have hr := BudgetedCompressionPass.remaining I C B _ hn first v rest hp
unfold allowance
omega
theorem source_value (D : CompressionData n) (B : ℕ) (e : Equiv.Perm (Fin n))
(q : CompressionSetup.PrimeSpace n) (z : Fin q.val) (he : D.compactTests e = true)
(budget : ℕ) (v : Bool) (rest : ℕ)
(h : (run D.input (compressionDictionaryCapacity n) B (PreparedCompressionPass.source D e)
q.val D.input.target z.val (by exact_mod_cast (primeRange_prime q.property).pos) budget).1 =
some (v, rest)) : v = D.residuePass B e q z ∧ rest ≤ budget := by
have hv := value D.input (compressionDictionaryCapacity n) B (PreparedCompressionPass.source D e)
q.val D.input.target z.val _ budget v rest h
rw [PreparedCompressionPass.source_value] at hv
exact ⟨hv.1.trans (D.pass_reference B e q z he), hv.2⟩
theorem source_account (D : CompressionData n) (e : Equiv.Perm (Fin n))
(q : CompressionSetup.PrimeSpace n) (z : Fin q.val) (he : D.compactTests e = true)
(budget : ℕ) (hn : 10000 ≤ n) :
Account budget (run D.input (compressionDictionaryCapacity n) (representationBufferCapacity n)
(PreparedCompressionPass.source D e) q.val D.input.target z.val
(by exact_mod_cast (primeRange_prime q.property).pos) budget) := by
have hs := PreparedCompressionPass.source_sizes D e he hn
apply account _ _ _ _ _ _ _ _ _ hn hs.1 hs.2
intro b
rw [PreparedCompressionPass.source_value]
exact D.pass_account e he q z b hn
end LowSpaceSubsetSum.BudgetedPreparedCompressionPass
end OAI
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