File size: 4,425 Bytes
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 | import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStreamContracts
namespace OAI
namespace LowSpaceSubsetSum.BudgetedStreamStep
universe u v s t w
variable {α : Type u} {β : Type v} {σ₁ : Type s} {σ₂ : Type t} {τ : Type w}
/-- All nested prepaid charges are covered by the actual query and copy ledger. -/
theorem allowance_potential (base front fill outer inner copy : ℕ)
(read₁ : σ₁ → Option (α × σ₁) × ℕ) (key₁ : α → ℤ × ℕ)
(read₂ : σ₂ → Option (β × σ₂) × ℕ) (key₂ : β → ℤ × ℕ)
(cells : β → ℕ) (report : α → β → τ → (τ × Bool) × ℕ)
(clone : σ₂ → σ₂ × ℕ) (B fuel₁ fuel₂ : ℕ) (st : StreamJoin.State σ₁ σ₂ τ) :
let M := base + front + fill + outer + 2 * inner + 12
allowance base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ st +
CountedStreamJoin.potential M 0 copy st ≤
CountedStreamJoin.potential M 0 copy
(CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ st).1 + base := by
let M := base + front + fill + outer + 2 * inner + 12
change allowance _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ + CountedStreamJoin.potential M 0 copy st ≤
CountedStreamJoin.potential M 0 copy
(CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ st).1 + base
have hbase : base ≤ M := by dsimp [M]; omega
by_cases hd : st.done = true
· simp only [allowance, CountedStreamJoin.step, hd, ↓reduceIte]
omega
· cases hx : (read₁ st.left).1 with
| none =>
simp only [allowance, CountedStreamJoin.step, hd, Bool.false_eq_true, ↓reduceIte, hx,
CountedStreamJoin.potential, Nat.mul_add, Nat.mul_one, Nat.zero_mul, Nat.add_zero]
omega
| some p =>
rcases p with ⟨x, x'⟩
cases hy : (read₂ st.right).1 with
| none =>
simp only [allowance, CountedStreamJoin.step, hd, Bool.false_eq_true, ↓reduceIte, hx, hy,
CountedStreamJoin.potential, Nat.mul_add, Nat.zero_mul, Nat.add_zero]
omega
| some p =>
rcases p with ⟨y, y'⟩
simp only [allowance, CountedStreamJoin.step, hd, Bool.false_eq_true, ↓reduceIte, hx, hy]
split
· simp only [CountedStreamJoin.potential, Nat.mul_add, Nat.zero_mul, Nat.add_zero]
omega
· split
· simp only [CountedStreamJoin.potential, Nat.mul_add, Nat.zero_mul, Nat.add_zero]
omega
· let out := (CountedTieGroup.run read₁ key₁ read₂ key₂ cells (key₁ x).1 report clone
B fuel₁ fuel₂ st.left st.right st.user).1
have hg := BudgetedTieGroup.allowance_le front fill outer inner copy read₁ key₁ read₂ key₂ cells
(key₁ x).1 report clone B fuel₁ fuel₂ st.left st.right st.user
change BudgetedTieGroup.allowance front fill outer inner copy read₁ key₁ read₂ key₂ cells
(key₁ x).1 report clone B fuel₁ fuel₂ st.left st.right st.user ≤
(front + fill + outer + 2 * inner + 12) *
(out.outer.queries + out.fillQueries + out.outer.state.queries + 1) +
copy * ((if out.snapshot then 1 else 0) + out.outer.state.restores) at hg
have hm := Nat.mul_le_mul_right
(out.outer.queries + out.fillQueries + out.outer.state.queries + 1)
(show front + fill + outer + 2 * inner + 12 ≤ M by dsimp [M]; omega)
change base + BudgetedTieGroup.allowance front fill outer inner copy read₁ key₁ read₂ key₂ cells
(key₁ x).1 report clone B fuel₁ fuel₂ st.left st.right st.user +
CountedStreamJoin.potential M 0 copy st ≤
CountedStreamJoin.potential M 0 copy
⟨out.outer.cursor, out.outer.state.last, out.outer.state.user,
st.queries + 2 + out.outer.queries + out.fillQueries + out.outer.state.queries,
st.reports + out.outer.state.reports,
st.copies + (if out.snapshot then 1 else 0) + out.outer.state.restores,
out.outer.halted⟩ + base
simp only [CountedStreamJoin.potential, Nat.mul_add, Nat.mul_one, Nat.zero_mul, Nat.add_zero] at hg hm ⊢
omega
end LowSpaceSubsetSum.BudgetedStreamStep
end OAI
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