| import Mathlib |
| import OAI.Computability.VertexCover.Machines.ExpanderStep |
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| namespace OAI |
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| namespace VertexCover.Machine.ExpandMachine |
| open UniqueGames.Foundations.PCP |
|
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| @[simp] theorem lookup_fst_val {q v : ℕ} (T : ExpanderTables.Table v q) (x : Fin v × Fin q) : |
| (ExpanderTables.lookup T x).1.val = (lookup q (T.rows.toList.map Fin.val,(x.1.val,x.2.val))).1 := |
| (congrArg Prod.fst (lookup_erase T x)).symm |
| @[simp] theorem lookup_snd_val {q v : ℕ} (T : ExpanderTables.Table v q) (x : Fin v × Fin q) : |
| (ExpanderTables.lookup T x).2.val = (lookup q (T.rows.toList.map Fin.val,(x.1.val,x.2.val))).2 := |
| (congrArg Prod.snd (lookup_erase T x)).symm |
|
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| @[simp] theorem index_val {v q : ℕ} (x : Fin v × Fin q) : |
| (ExpanderTables.rowIndex v q x).val=x.2.val+q*x.1.val := rfl |
| @[simp] theorem index_inv_fst {v q : ℕ} (i : Fin (v*q)) : |
| ((ExpanderTables.rowIndex v q).symm i).1.val=i.val/q := rfl |
| @[simp] theorem index_inv_snd {v q : ℕ} (i : Fin (v*q)) : |
| ((ExpanderTables.rowIndex v q).symm i).2.val=i.val%q := rfl |
|
|
| theorem stepRow_lookup {v d : ℕ} (T : ExpanderTables.Table v (d*d)) |
| (H : ExpanderTables.Table ((d*d)*(d*d)) d) (i : Fin ((v*((d*d)*(d*d)))*(d*d))) : |
| stepRow d (H.rows.toList.map Fin.val) (erase ⟨v,T⟩,i.val) = |
| (ExpanderTables.rowIndex _ _ (ExpanderTables.stepLookup T H |
| ((ExpanderTables.rowIndex _ _).symm i))).val := by |
| simp only [ExpanderTables.stepLookup,index_val,index_inv_fst,index_inv_snd, |
| lookup_fst_val,lookup_snd_val] |
| rfl |
|
|
| theorem stepRow_source {v d : ℕ} (T : ExpanderTables.Table v (d*d)) |
| (H : ExpanderTables.Table ((d*d)*(d*d)) d) (i : Fin ((v*((d*d)*(d*d)))*(d*d))) : |
| stepRow d (H.rows.toList.map Fin.val) (erase ⟨v,T⟩,i.val) = |
| (ExpanderTables.step T H).rows[i].val := by |
| rw [stepRow_lookup,← ExpanderTables.lookup_step] |
| simp only [ExpanderTables.lookup,Equiv.apply_symm_apply,ExpanderTables.reverseIndex] |
|
|
| def step {d : ℕ} (H : ExpanderTables.Table ((d*d)*(d*d)) d) (T : Input (d*d)) : Input (d*d) := |
| ⟨T.1*((d*d)*(d*d)),ExpanderTables.step T.2 H⟩ |
|
|
| theorem stepData_source {d : ℕ} (H : ExpanderTables.Table ((d*d)*(d*d)) d) (T : Input (d*d)) : |
| stepData d (H.rows.toList.map Fin.val) (erase T)=erase (step H T) := by |
| apply Prod.ext |
| · rfl |
| · apply List.ext_getElem |
| · simp [stepData,erase,step] |
| · intro i h₁ h₂ |
| simp only [stepData,List.getElem_map,List.getElem_range,erase,step,Vector.getElem_toList] |
| exact stepRow_source T.2 H ⟨i, by simpa [stepData,erase] using h₁⟩ |
|
|
| noncomputable def stepPoly {d : ℕ} (H : ExpanderTables.Table ((d*d)*(d*d)) d) : |
| Poly (code (q := d*d)) code (step H) := |
| (stepDataPoly d (H.rows.toList.map Fin.val)).encodeCongr erase (fun _ => rfl) (fun T => by |
| change dataCode _ = dataCode _ |
| rw [stepData_source]) |
|
|
| end VertexCover.Machine.ExpandMachine |
| end |
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| end |
| end |
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| end |
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| end OAI |
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