| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Bridge |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Outer.LocalTemplate |
| open MinUncut.Inner MinUncut.FiniteProof |
| open MinUncutGames.Foundations.Hastad.SourceOccurrences |
| variable {N t : ℕ} |
|
|
| def equationNumber (E : Equation (Fin N)) : ℕ := |
| E.rhs.val+2*∑p : Fin 3, (E.names p).val*N^p.val |
|
|
| def firstNumber (U : Fin t → Equation (Fin N)) : ℕ := |
| ∑i : Fin t,equationNumber (U i)*(N^3*2)^i.val |
|
|
| def secondNumber (U : Fin t → Equation (Fin N)) (h : Fin t → Bool) (pos : Fin t → Fin 3) : ℕ := |
| ∑i : Fin t,(if h i then ((U i).names (pos i)).val else N+equationNumber (U i))*(N+N^3*2)^i.val |
|
|
| lemma equation_code (E : Equation (Fin N)) : ((equationEncoding N).code E).val=equationNumber E := rfl |
| lemma first_code (U : Fin t → Equation (Fin N)) : |
| ((questionEncoding N t).code (.inl U)).val=firstNumber U := rfl |
| lemma second_code (U : Fin t → Equation (Fin N)) (h : Fin t → Bool) (pos : Fin t → Fin 3) : |
| ((questionEncoding N t).code (.inr (secondQuestion U h pos))).val= |
| (N^3*2)^t+secondNumber U h pos := by |
| change (N^3*2)^t+(∑i : Fin t, ((secondQuestionEncoding N).code (slotQuestion (U i) (h i) (pos i))).val*(N+N^3*2)^i.val)=_ |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| cases h i <;> rfl |
| |
| def tableNumber (f : (Fin t → Triple) → Bool) : ℕ := |
| (((ambientEncoding t).function Encoding.bool).code f).val |
|
|
| lemma variable_number (v : FamilyVariable (Fin N) (Fin t)) : |
| (variableCode v).val= tableNumber (extendTable v.1 v.2.val)+ |
| 2^((2^3)^t)*((questionEncoding N t).code v.1).val := rfl |
|
|
| variable {h : Fin t → Bool} |
| def Query.leftNumber (T : Query h) (U : Fin t → Equation (Fin N)) (pos : Fin t → Fin 3) : ℕ := |
| tableNumber (T.leftTable pos (signature U))+ |
| 2^((2^3)^t)*(if T.first then firstNumber U else (N^3*2)^t+secondNumber U h pos) |
|
|
| def Query.rightNumber (T : Query h) (U : Fin t → Equation (Fin N)) (pos : Fin t → Fin 3) : ℕ := |
| tableNumber T.rightTable+2^((2^3)^t)*((N^3*2)^t+secondNumber U h pos) |
|
|
| lemma Query.left_number (T : Query h) (U : Fin t → Equation (Fin N)) (pos : Fin t → Fin 3) : |
| (variableCode ((T.realize U pos).signed familyBase).left).val=T.leftNumber U pos := by |
| rw [variable_number] |
| change tableNumber (extendTable _ (rep (familyBase _) (T.realize U pos).left.2).val)+_= _ |
| rw [T.left_extended] |
| unfold Query.leftNumber Query.realize |
| by_cases hf : T.first=true |
| · simp only [ite_eq_left hf,comparison,QueryDemand.signed,vertex,first_code] |
| · simp only [ite_eq_right hf,comparison,QueryDemand.signed,vertex,second_code] |
|
|
| lemma Query.right_number (T : Query h) (U : Fin t → Equation (Fin N)) (pos : Fin t → Fin 3) : |
| (variableCode ((T.realize U pos).signed familyBase).right).val=T.rightNumber U pos := by |
| rw [variable_number] |
| change tableNumber (extendTable _ (rep (familyBase _) (T.realize U pos).right.2).val)+_= _ |
| rw [T.right_extended] |
| unfold Query.rightNumber Query.realize |
| by_cases hf : T.first=true |
| · simp only [ite_eq_left hf,comparison,QueryDemand.signed,vertex,second_code] |
| · simp only [ite_eq_right hf,comparison,QueryDemand.signed,vertex,second_code] |
|
|
| end MinUncut.Outer.LocalTemplate |
|
|
| end |
| end |
|
|
| end OAI |
|
|
