| import Mathlib |
| import OAI.Computability.MaxCut.Arithmetic.Prime |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace OptimalMaxCut.RawGrid |
| open scoped BigOperators |
| open Finset CounterMachine.Expr |
| attribute [local instance] Classical.propDecidable |
|
|
| def param (q K : ℕ) (s : List Bool) : ℕ := Unweighted.searchPrime (K*(vertices s*2^q+1)^2) |
| def size (q K : ℕ) (s : List Bool) : ℕ := vertices s*2^q*(param q K s)^6 |
| def scale (q K : ℕ) (s : List Bool) : ℕ := (param q K s)^12 |
| def coordinate (p i k : ℕ) : ℕ := (i % p^6)/p^k % p |
| def pairing (p i j : ℕ) : ℕ := (∑ k : Fin 6, coordinate p i k.val * coordinate p j k.val)%p |
| noncomputable def adjacency (q : ℕ) (t : ℚ) (ht : t ∈ Set.Icc (-1:ℚ) 1) (K L : ℕ) |
| (s : List Bool) (i j : ℕ) : ℕ := |
| let p := param q L s |
| if i/p^6=j/p^6 then 0 else |
| ∑ r ∈ range (p*weights q t ht K s (i/p^6) (j/p^6)/gridSize q K s), |
| if r%p=pairing p i j then 1 else 0 |
|
|
| def frameValue (n i : ℕ) : ℕ := |
| if i<2*n.size then if i%2=0 then 1 else n/2^(i/2)%2 else 0 |
| def length (q L : ℕ) (s : List Bool) : ℕ := |
| (2*(size q L s).size+1)+(2*(scale q L s).size+1)+(size q L s)^2 |
| noncomputable def entry (q : ℕ) (t : ℚ) (ht : t ∈ Set.Icc (-1:ℚ) 1) (K L : ℕ) |
| (s : List Bool) (i : ℕ) : ℕ := |
| let n := size q L s |
| let d := 2*n.size+1 |
| let e := 2*(scale q L s).size+1 |
| if i<d then frameValue n i else if i<d+e then frameValue (scale q L s) (i-d) else |
| adjacency q t ht K L s ((i-d-e)/n) ((i-d-e)%n) |
|
|
| namespace Rep |
| open CounterMachine.Expr.Represented |
| variable {f g h : List Bool → (ℕ → ℕ) → ℕ} |
| theorem param (q K : ℕ) : Represented (fun s _ => RawGrid.param q K s) := |
| searchPrime ((Represented.const K).mul (((vertices.mul (Represented.const (2^q))).add (Represented.const 1)).pow 2)) |
| theorem size (q K : ℕ) : Represented (fun s _ => RawGrid.size q K s) := |
| (vertices.mul (Represented.const (2^q))).mul ((param q K).pow 6) |
| theorem scale (q K : ℕ) : Represented (fun s _ => RawGrid.scale q K s) := (param q K).pow 12 |
| theorem coordinate (k : ℕ) (hp : Represented f) (hi : Represented g) : |
| Represented (fun s a => RawGrid.coordinate (f s a) (g s a) k) := |
| ((hi.mod (hp.pow 6)).div (hp.pow k)).mod hp |
| theorem pairing (hp : Represented f) (hi : Represented g) (hj : Represented h) : |
| Represented (fun s a => RawGrid.pairing (f s a) (g s a) (h s a)) := |
| (Represented.finiteSum (fun k : Fin 6 => (coordinate k.val hp hi).mul (coordinate k.val hp hj))).mod hp |
| theorem adjacency (q : ℕ) (t : ℚ) (ht : t ∈ Set.Icc (-1:ℚ) 1) (K L : ℕ) |
| (hi : Represented f) (hj : Represented g) : |
| Represented (fun s a => RawGrid.adjacency q t ht K L s (f s a) (g s a)) := by |
| let hp := param q L |
| let hu := hi.div (hp.pow 6) |
| let hv := hj.div (hp.pow 6) |
| apply iteEq hu hv (Represented.const 0) |
| apply bounded ((hp.mul (weights q t ht K hu hv)).div (gridSize q K)) |
| exact eq ((Represented.arg 0).mod hp) (pairing hp (hi.rename Nat.succ) (hj.rename Nat.succ)) |
| theorem frameValue (hn : Represented f) (hi : Represented g) : |
| Represented (fun s a => RawGrid.frameValue (f s a) (g s a)) := framedBit hn hi |
| theorem length (q L : ℕ) : Represented (fun s _ => RawGrid.length q L s) := |
| ((framedLen (size q L)).add (framedLen (scale q L))).add ((size q L).pow 2) |
| theorem entry (q : ℕ) (t : ℚ) (ht : t ∈ Set.Icc (-1:ℚ) 1) (K L : ℕ) |
| (hi : Represented f) : Represented (fun s a => RawGrid.entry q t ht K L s (f s a)) := by |
| let hn := size q L |
| let hd := framedLen hn |
| let he := framedLen (scale q L) |
| apply iteLt hi hd (frameValue hn hi) |
| apply iteLt hi (hd.add he) (frameValue (scale q L) (hi.sub hd)) |
| exact adjacency q t ht K L (((hi.sub hd).sub he).div hn) (((hi.sub hd).sub he).mod hn) |
| end Rep |
| end OptimalMaxCut.RawGrid |
|
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| end |
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| end OAI |
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