| import Mathlib |
| import OAI.Computability.MaxCut.Estimates.Put |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace OptimalMaxCut.CounterMachine.Expr |
| open Polynomial |
|
|
| noncomputable def valuePolynomial : Expr → Polynomial ℕ |
| | .const n => C n |
| | .arg _ | .length => X |
| | .bit _ | .zero _ => 1 |
| | .add a b => valuePolynomial a+valuePolynomial b |
| | .sub a _ => valuePolynomial a |
| | .mul a b => valuePolynomial a*valuePolynomial b |
| | .sum b f => valuePolynomial b*(valuePolynomial f).comp (X+valuePolynomial b) |
|
|
| theorem valuePolynomial_eval (e : Expr) (M : ℕ) : |
| e.valuePolynomial.eval M = e.valueBudget M := by |
| induction e generalizing M <;> |
| simp_all [valuePolynomial, valueBudget, Polynomial.eval_comp] |
|
|
| noncomputable def timePolynomial : Expr → Polynomial ℕ |
| | .const n => C (n+1) |
| | .arg _ => 5*X+2 |
| | .length => 5*X+6 |
| | .bit e => timePolynomial e+3*valuePolynomial e+X+7 |
| | .add a b | .sub a b => timePolynomial a+timePolynomial b+2*valuePolynomial b+1 |
| | .mul a b => timePolynomial a+timePolynomial b+ |
| valuePolynomial b*(5*valuePolynomial a+3)+1+2*valuePolynomial a+1+ |
| 2*(valuePolynomial a*valuePolynomial b)+1 |
| | .zero a => timePolynomial a+2*valuePolynomial a+2 |
| | .sum b f => timePolynomial b+valuePolynomial b* |
| ((timePolynomial f).comp (X+valuePolynomial b)+ |
| 2*(valuePolynomial f).comp (X+valuePolynomial b)+3)+2*valuePolynomial b+2 |
|
|
| theorem timePolynomial_eval (e : Expr) (M : ℕ) : |
| e.timePolynomial.eval M = e.timeBudget M := by |
| induction e generalizing M <;> |
| simp_all [timePolynomial, timeBudget, valuePolynomial_eval, Polynomial.eval_comp] |
|
|
| end OptimalMaxCut.CounterMachine.Expr |
|
|
| end |
|
|
| end OAI |
|
|
