openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /BoundedPowerProgram.lean
| import OAI.Combinatorics.ContingencyTables.Machines.BinaryPowerProgram | |
| namespace OAI | |
| /-! # Reusing a unary allowance across a binary transition table -/ | |
| namespace ContingencyTables.BoundedPowerPrograms | |
| open MatchingFPRAS MatchingFPRAS.TreeTyped | |
| def powerInput (z : List Bool×ℕ) : ℕ×ℕ := (2,z.1.length-z.2) | |
| def powerInputRealizer : Realizer powerInput := | |
| pair (constant 2) (composition (pair (composition first listLength) second) subtract) | |
| lemma polynomial_powerInput : PolynomialTime powerInputRealizer := | |
| polynomial_pair (polynomial_constant 2) (polynomial_composition (polynomial_pair | |
| (polynomial_composition polynomial_first polynomial_listLength) polynomial_second) | |
| polynomial_subtract) | |
| def power (z : List Bool×ℕ) : ℕ := BinaryPowerPrograms.power (powerInput z) | |
| def powerRealizer : Realizer power := | |
| composition powerInputRealizer BinaryPowerPrograms.powerRealizer | |
| /-- A single allowance list can be shared by all entries in the finite | |
| matrix. Each binary power then has polynomial cost in the supplied data. -/ | |
| theorem polynomial_power : PolynomialTime powerRealizer := by | |
| apply work_composition polynomial_powerInput BinaryPowerPrograms.work_power (Polynomial.X+9) | |
| intro z | |
| have hl: z.1.length ≤ weight z := (list_length_le_weight z.1).trans (weight_fst z) | |
| have hw : weight (2:ℕ)≤9 := by decide | |
| simp only [powerInput,Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_ofNat] | |
| omega | |
| theorem power_eq (xs : List Bool) (gap : ℕ) : power (xs,gap)=2^(xs.length-gap) := | |
| BinaryPowerPrograms.power_eq 2 (xs.length-gap) | |
| end ContingencyTables.BoundedPowerPrograms | |
| end OAI | |