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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStoppedLoop
import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStreamPotential
namespace OAI
namespace LowSpaceSubsetSum.BudgetedJoinExecution
universe u v s t w
variable {α : Type u} {β : Type v} {σ₁ : Type s} {σ₂ : Type t} {τ : Type w}
abbrev step (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) :=
BudgetedStreamStep.execute base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂
abbrev run (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) :=
BudgetedStoppedLoop.run
(step base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂)
theorem reference_eq (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 fuel : ℕ) (st : StreamJoin.State σ₁ σ₂ τ) :
BudgetedStoppedJoin.reference
(fun s => (CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ s).1) fuel st =
(CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1 := by
induction fuel generalizing st with
| zero => rfl
| succ fuel ih =>
simp only [BudgetedStoppedJoin.reference, CountedStreamJoin.runAcc]
split <;> simp only [ih]
theorem run_value (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 fuel : ℕ)
(st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) (out : StreamJoin.StoppedResult σ₁ σ₂ τ) (left : ℕ)
(h : (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone
B fuel₁ fuel₂ fuel st budget).1 = some (out, left)) :
out = (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1 ∧
left ≤ budget := by
simp only [run, BudgetedStoppedLoop.run_eq] at h
have hv := BudgetedStoppedJoin.execute_value
(step base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂)
(fun s => (CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ s).1)
(BudgetedStreamStep.execute_value base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂)
fuel st budget out left h
rw [reference_eq] at hv
exact hv
theorem run_work
(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)
(R₁ L₁ R₂ L₂ S C base front fill outer inner copy : ℕ)
(hr₁ : ∀ s, (read₁ s).2 ≤ R₁) (hk₁ : ∀ x, (key₁ x).2 ≤ L₁)
(hr₂ : ∀ s, (read₂ s).2 ≤ R₂) (hk₂ : ∀ y, (key₂ y).2 ≤ L₂)
(hs : ∀ y, cells y ≤ S) (hc : ∀ x y a, (report x y a).2 ≤ C)
(hbase : R₁ + L₁ + R₂ + L₂ + 16 ≤ base)
(hfront : R₁ + L₁ + R₂ + L₂ + 16 ≤ front)
(hfill : R₂ + L₂ + S + 8 ≤ fill) (houter : R₁ + L₁ + 8 ≤ outer)
(hinner : max R₂ 4 + L₂ + C + 8 ≤ inner) (hcopy : ∀ s, (clone s).2 ≤ copy)
(B fuel₁ fuel₂ fuel : ℕ) (st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ) :
(run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone
B fuel₁ fuel₂ fuel st budget).2 ≤ budget + 1 := by
simp only [run, BudgetedStoppedLoop.run_eq]
apply BudgetedStoppedJoin.execute_work
· exact BudgetedStreamStep.execute_work read₁ key₁ read₂ key₂ cells report clone
R₁ L₁ R₂ L₂ S C base front fill outer inner copy hr₁ hk₁ hr₂ hk₂ hs hc hbase hfront hfill houter hinner hcopy
B fuel₁ fuel₂
· intro s b out left h
exact ⟨(BudgetedStreamStep.execute_value base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone
B fuel₁ fuel₂ s b out left h).2,
BudgetedStreamStep.execute_spent read₁ key₁ read₂ key₂ cells report clone
R₁ L₁ R₂ L₂ S C base front fill outer inner copy hr₁ hk₁ hr₂ hk₂ hs hc hbase hfront hfill houter hinner hcopy
B fuel₁ fuel₂ s b out left h⟩
theorem run_complete (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 fuel : ℕ)
(st : StreamJoin.State σ₁ σ₂ τ) (budget : ℕ)
(hb : let out := (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1
(base + front + fill + outer + 2 * inner + 12) * out.state.queries + copy * out.state.copies +
(base + 4) * out.steps + 4 ≤ budget) :
∃ left, (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone
B fuel₁ fuel₂ fuel st budget).1 =
some ((CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel st).1, left) := by
let pure := fun s => (CountedStreamJoin.step read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ s).1
let bill := BudgetedStreamStep.allowance base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂
let M := base + front + fill + outer + 2 * inner + 12
have hp := BudgetedStoppedJoin.allowance_potential pure bill (CountedStreamJoin.potential M 0 copy) base
(BudgetedStreamStep.allowance_potential base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂)
fuel st
rw [reference_eq] at hp
simp only [CountedStreamJoin.potential, Nat.zero_mul, Nat.add_zero] at hp
have ha : BudgetedStoppedJoin.allowance pure bill fuel st ≤ budget := by
dsimp only [M] at hp
omega
have h := BudgetedStoppedJoin.execute_complete
(step base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂) pure bill
(BudgetedStreamStep.execute_complete base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂)
(BudgetedStreamStep.execute_remaining base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂)
fuel st budget ha
simpa only [run, BudgetedStoppedLoop.run_eq, pure, reference_eq] using h
/-- From fresh cursors, every control iteration performs at least one query. -/
theorem run_complete_queries (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) (hclone : ∀ s, (clone s).1 = s)
(B fuel₁ fuel₂ fuel : ℕ) (s₁ : σ₁) (s₂ : σ₂) (a : τ) (budget : ℕ)
(hb : let out := (CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel
(StreamJoin.start s₁ s₂ a)).1
(2 * base + front + fill + outer + 2 * inner + 16) * out.state.queries +
copy * out.state.copies + 4 ≤ budget) :
∃ left, (run base front fill outer inner copy read₁ key₁ read₂ key₂ cells report clone
B fuel₁ fuel₂ fuel (StreamJoin.start s₁ s₂ a) budget).1 =
some ((CountedStreamJoin.runAcc read₁ key₁ read₂ key₂ cells report clone B fuel₁ fuel₂ fuel
(StreamJoin.start s₁ s₂ a)).1, left) := by
have hv := CountedStreamJoin.runAcc_value read₁ key₁ read₂ key₂ cells report clone hclone
B fuel₁ fuel₂ fuel (StreamJoin.start s₁ s₂ a)
have hq := StreamJoin.stopped_steps_le_queries (fun s => (read₁ s).1) (fun x => (key₁ x).1)
(fun s => (read₂ s).1) (fun x => (key₂ x).1) (fun x y a => (report x y a).1)
B fuel₁ fuel₂ fuel s₁ s₂ a
rw [← StreamJoin.stopped_eq] at hq
change (StreamJoin.stoppedAcc _ _ _ _ _ _ _ _ _ _).steps ≤
(StreamJoin.stoppedAcc _ _ _ _ _ _ _ _ _ _).state.queries at hq
rw [← hv] at hq
have hm := Nat.mul_le_mul_left (base + 4) hq
apply run_complete
have he : 2 * base + front + fill + outer + 2 * inner + 16 =
(base + front + fill + outer + 2 * inner + 12) + (base + 4) := by omega
dsimp only at hb
rw [he, Nat.add_mul] at hb
omega
end LowSpaceSubsetSum.BudgetedJoinExecution
end OAI
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