| import Mathlib |
| import OAI.Computability.MinUncut.Games.QuestionArithmetic |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Costed.SourceWords |
| open MinUncut.Inner MinUncut.Outer MinUncut.Outer.LocalTemplate |
| open MinUncutGames.Reduction MinUncut.SourceBridge |
|
|
| variable {t : ℕ} |
|
|
| def tupleDigit (j : ℕ) : AExpr := .mod (.div (.reg 0) ((AExpr.reg 2).pow j)) (.reg 2) |
| def slotExpr (j p : ℕ) : AExpr := .at (.add (.add (.const 3) (.mul (.const 4) (tupleDigit j))) (.const p)) |
| def inputTuple (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) : List ℕ := |
| (finFunctionFinEquiv u).val::SourceEncoding.inputWords input |
|
|
| lemma tupleDigit_eval (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) (j : Fin t) : |
| (tupleDigit j.val).eval (inputTuple input u)=(u j).val := by |
| have hh := congrArg Fin.val (congrFun (finFunctionFinEquiv.symm_apply_apply u) j) |
| change (finFunctionFinEquiv u).val/input.equations.length^j.val%input.equations.length=(u j).val at hh |
| simpa only [tupleDigit,AExpr.eval,AExpr.eval_pow,inputTuple,SourceEncoding.inputWords, |
| List.cons_append,List.nil_append,List.drop_succ_cons,List.drop_zero,List.headI_cons] using hh |
|
|
| lemma slotExpr_name (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) (j : Fin t) (p : Fin 3) : |
| (slotExpr j.val p.val).eval (inputTuple input u)=((equations input (u j)).names p).val := by |
| simp only [slotExpr,AExpr.eval,tupleDigit_eval] |
| have he : 3+4*(u j).val+p.val=(2+4*(u j).val+p.val)+1 := by omega |
| rw [he] |
| exact input_name input (u j) p |
|
|
| lemma slotExpr_rhs (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) (j : Fin t) : |
| (slotExpr j.val 3).eval (inputTuple input u)=(equations input (u j)).rhs.val := by |
| simp only [slotExpr,AExpr.eval,tupleDigit_eval] |
| have he : 3+4*(u j).val+3=(2+4*(u j).val+3)+1 := by omega |
| rw [he] |
| exact input_rhs input (u j) |
|
|
| def equationExpr (j : ℕ) : AExpr := .add (slotExpr j 3) (.mul (.const 2) |
| (AExpr.sum (List.ofFn (fun p : Fin 3=>.mul (slotExpr j p.val) ((AExpr.reg 1).pow p.val))))) |
|
|
| def firstExpr (t : ℕ) : AExpr := AExpr.sum (List.ofFn (fun i : Fin t=> |
| .mul (equationExpr i.val) ((AExpr.mul ((AExpr.reg 1).pow 3) (.const 2)).pow i.val))) |
|
|
| def secondExpr {t : ℕ} (h : Fin t → Bool) (pos : Fin t → Fin 3) : AExpr := |
| AExpr.sum (List.ofFn (fun i : Fin t=> |
| .mul (if h i then slotExpr i.val (pos i).val else .add (.reg 1) (equationExpr i.val)) |
| ((AExpr.add (.reg 1) (.mul ((AExpr.reg 1).pow 3) (.const 2))).pow i.val))) |
|
|
| lemma reg1_eval (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) : |
| (AExpr.reg 1).eval (inputTuple input u)=input.«variables» := rfl |
|
|
| lemma inputTuple_reg1 (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) : |
| ((inputTuple input u).drop 1).headI=input.«variables» := rfl |
|
|
| lemma equationExpr_eval (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) (j : Fin t) : |
| (equationExpr j.val).eval (inputTuple input u)=equationNumber (equations input (u j)) := by |
| simp only [equationExpr,AExpr.eval,AExpr.eval_sum,List.map_ofFn,Function.comp_def,List.sum_ofFn, |
| slotExpr_rhs,slotExpr_name,AExpr.eval_pow,inputTuple_reg1,equationNumber] |
|
|
| lemma firstExpr_eval (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) : |
| (firstExpr t).eval (inputTuple input u)=firstNumber (fun j=>equations input (u j)) := by |
| simp only [firstExpr,AExpr.eval_sum,List.map_ofFn,Function.comp_def,List.sum_ofFn,AExpr.eval, |
| equationExpr_eval,AExpr.eval_pow,inputTuple_reg1,firstNumber] |
|
|
| lemma secondExpr_eval (input : SourceEncoding.Input) (u : Fin t → Fin input.equations.length) |
| (h : Fin t → Bool) (pos : Fin t → Fin 3) : |
| (secondExpr h pos).eval (inputTuple input u)=secondNumber (fun j=>equations input (u j)) h pos := by |
| simp only [secondExpr,AExpr.eval_sum,List.map_ofFn,Function.comp_def,List.sum_ofFn,AExpr.eval,AExpr.eval_pow, |
| inputTuple_reg1,secondNumber] |
| apply Finset.sum_congr rfl |
| intro j _ |
| cases h j <;> simp only [Bool.false_eq_true,ite_false,ite_true,AExpr.eval,equationExpr_eval,slotExpr_name,inputTuple_reg1] |
| |
| end MinUncut.Costed.SourceWords |
| |
| end |
| end |
| |
| end OAI |
| |
