| import OAI.Combinatorics.EditApproximation.Binary.FractionAllocation |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryFixedDenominator |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryCeilingWork |
| import OAI.Combinatorics.EditApproximation.Coarse.CoarseDenominatorArithmetic |
| import OAI.Combinatorics.EditApproximation.Probability.FiniteEnumerationWork |
| import OAI.Combinatorics.EditApproximation.Definitions |
| |
| |
| namespace OAI |
| |
| namespace EditApproximation |
| |
| |
| |
| theorem bitPowerAllocation_le_work (base : List Bool) (n : ℕ) : |
| bitPowerAllocation base n ≤ (bitPowerWithWork base n).2 := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| have hm := bitMulAllocation_le_work (bitPowerWithWork base n).1 base |
| have ht := trimBitWordAllocation_le_work |
| (bitMulWithWork (bitPowerWithWork base n).1 base).1 |
| simp only [bitPowerAllocation, bitPowerWithWork] |
| omega |
| |
| def coarseDenominatorAllocation (allowanceDen q L branching : List Bool) (D : ℕ) : ℕ := |
| let first := bitMulWithWork L branching |
| let base := bitMulWithWork first.1 q |
| let trimmed := trimBitWordWithWork base.1 |
| let power := bitPowerWithWork trimmed.1 D |
| let initial := bitMulWithWork allowanceDen q |
| let product := bitMulWithWork initial.1 power.1 |
| bitMulAllocation L branching + bitMulAllocation first.1 q + trimBitWordAllocation base.1 + |
| bitPowerAllocation trimmed.1 D + bitMulAllocation allowanceDen q + |
| bitMulAllocation initial.1 power.1 + trimBitWordAllocation product.1 |
|
|
| theorem coarseDenominatorAllocation_le_work (allowanceDen q L branching : List Bool) (D : ℕ) : |
| coarseDenominatorAllocation allowanceDen q L branching D ≤ |
| (coarseDenominatorWithWork allowanceDen q L branching D).2 := by |
| have h1 := bitMulAllocation_le_work L branching |
| have h2 := bitMulAllocation_le_work (bitMulWithWork L branching).1 q |
| have h3 := trimBitWordAllocation_le_work (bitMulWithWork (bitMulWithWork L branching).1 q).1 |
| have h4 := bitPowerAllocation_le_work |
| (trimBitWordWithWork (bitMulWithWork (bitMulWithWork L branching).1 q).1).1 D |
| have h5 := bitMulAllocation_le_work allowanceDen q |
| have h6 := bitMulAllocation_le_work (bitMulWithWork allowanceDen q).1 |
| (bitPowerWithWork (trimBitWordWithWork (bitMulWithWork (bitMulWithWork L branching).1 q).1).1 D).1 |
| have h7 := trimBitWordAllocation_le_work |
| (bitMulWithWork (bitMulWithWork allowanceDen q).1 |
| (bitPowerWithWork (trimBitWordWithWork (bitMulWithWork (bitMulWithWork L branching).1 q).1).1 D).1).1 |
| unfold coarseDenominatorAllocation coarseDenominatorWithWork |
| dsimp only |
| omega |
| |
| |
| |
| theorem bitCeilDivAllocation_le_work (divisor bits : List Bool) : |
| bitCeilDivAllocation divisor bits ≤ (bitCeilDivWithWork divisor bits).2 := by |
| have hd := bitDivModAllocation_le_work divisor bits |
| have ha := bitAddAllocation_le_work (bitDivModWithWork divisor bits).1 [true] false |
| unfold bitCeilDivAllocation bitCeilDivWithWork |
| dsimp only |
| split_ifs <;> dsimp only <;> omega |
|
|
| /-- Map allocation includes one result spine cell for every callback. -/ |
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|
| theorem arithmeticMapAllocation_le_work {α β : Type*} (f : α → β × ℕ) |
| (allocation : α → ℕ) (values : List α) |
| (h : ∀ a ∈ values, allocation a ≤ (f a).2) : |
| arithmeticMapAllocation allocation values ≤ (arithmeticMapWithWork f values).2 := by |
| induction values with |
| | nil => rfl |
| | cons a rest ih => |
| have hh := h a List.mem_cons_self |
| have ht := ih (fun b hb => h b (List.mem_cons_of_mem _ hb)) |
| simp only [arithmeticMapAllocation, arithmeticMapWithWork] |
| omega |
| |
| namespace BinaryFraction |
| |
| |
| |
| theorem invAllocation_le_work (a : BinaryFraction) : |
| invAllocation a ≤ a.invWithWork.2 := by |
| unfold invAllocation invWithWork |
| dsimp only |
| split_ifs <;> dsimp only |
| · exact Nat.succ_le_succ (Nat.zero_le _) |
| · exact Nat.zero_le _ |
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|
|
| theorem divAllocation_le_work (a b : BinaryFraction) : |
| divAllocation a b ≤ (divWithWork a b).2 := by |
| have hi := invAllocation_le_work b |
| have hm := canonicalMulAllocation_le_work a b.invWithWork.1 |
| unfold divAllocation divWithWork |
| dsimp only |
| omega |
| |
| |
| |
| theorem fixedDenominatorAllocation_le_work (a : BinaryFraction) (denominator : List Bool) |
| (hden : 0 < bitWordValue denominator) : |
| fixedDenominatorAllocation a denominator hden ≤ (fixedDenominatorWithWork a denominator hden).2 := by |
| have hm := bitMulAllocation_le_work a.numerator.bits denominator |
| have hd := bitDivModAllocation_le_work a.denominator |
| (bitMulWithWork a.numerator.bits denominator).1 |
| have hc := canonicalizeAllocation_le_work |
| (⟨⟨a.numerator.negative, |
| (bitDivModWithWork a.denominator (bitMulWithWork a.numerator.bits denominator).1).1⟩, |
| denominator, hden⟩ : BinaryFraction) |
| unfold fixedDenominatorAllocation fixedDenominatorWithWork |
| dsimp only |
| omega |
| |
| |
| |
| theorem naturalCeilingAllocation_le_work (a : BinaryFraction) : |
| naturalCeilingAllocation a ≤ a.naturalCeilingWithWork.2 := by |
| have h := bitCeilDivAllocation_le_work a.denominator a.numerator.bits |
| unfold naturalCeilingAllocation naturalCeilingWithWork |
| split_ifs <;> dsimp only <;> omega |
|
|
| end BinaryFraction |
| end EditApproximation |
|
|
| end OAI |
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|
