openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CanonicalUpperDirectionSplit.lean
| import OAI.Computability.PerfectCompleteness.Foundations.CanonicalDirections | |
| import OAI.Computability.PerfectCompleteness.Foundations.QuarterBalanceLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.TupleIndexEmitterLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CanonicalUpperDirectionSplit | |
| noncomputable section | |
| open scoped Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {n : Nat} | |
| abbrev Complement (rows : Nat → Nat) (upper : Fin n) := | |
| FiniteProductSplit.Others | |
| (Ω := fun level : Fin n => PrefixTests.LevelDirection rows level) upper | |
| def splitEquiv (rows : Nat → Nat) (upper : Fin n) : | |
| CanonicalDirections.Tuple rows n ≃ | |
| PrefixTests.LevelDirection rows upper × Complement rows upper := | |
| Equiv.piSplitAt upper (fun level => PrefixTests.LevelDirection rows level) | |
| def complement (rows : Nat → Nat) (upper : Fin n) | |
| (directions : CanonicalDirections.Tuple rows n) : Complement rows upper := | |
| (splitEquiv rows upper directions).2 | |
| def assemble (rows : Nat → Nat) (upper : Fin n) | |
| (direction : PrefixTests.LevelDirection rows upper) (rest : Complement rows upper) : | |
| CanonicalDirections.Tuple rows n := | |
| (splitEquiv rows upper).symm (direction, rest) | |
| @[simp] theorem split_selected (rows : Nat → Nat) (upper : Fin n) | |
| (directions : CanonicalDirections.Tuple rows n) : | |
| (splitEquiv rows upper directions).1 = directions upper := rfl | |
| @[simp] theorem complement_apply (rows : Nat → Nat) (upper : Fin n) | |
| (directions : CanonicalDirections.Tuple rows n) (i : {i : Fin n // i ≠ upper}) : | |
| complement rows upper directions i = directions i.val := rfl | |
| @[simp] theorem assemble_at_upper (rows : Nat → Nat) (upper : Fin n) | |
| (direction : PrefixTests.LevelDirection rows upper) (rest : Complement rows upper) : | |
| assemble rows upper direction rest upper = direction := by | |
| exact congrArg Prod.fst ((splitEquiv rows upper).apply_symm_apply (direction, rest)) | |
| @[simp] theorem assemble_at_other (rows : Nat → Nat) (upper : Fin n) | |
| (direction : PrefixTests.LevelDirection rows upper) (rest : Complement rows upper) | |
| (i : Fin n) (hne : i ≠ upper) : | |
| assemble rows upper direction rest i = rest ⟨i, hne⟩ := by | |
| exact congrArg | |
| (fun z : PrefixTests.LevelDirection rows upper × Complement rows upper => z.2 ⟨i, hne⟩) | |
| ((splitEquiv rows upper).apply_symm_apply (direction, rest)) | |
| theorem assemble_split (rows : Nat → Nat) (upper : Fin n) | |
| (directions : CanonicalDirections.Tuple rows n) : | |
| assemble rows upper (directions upper) (complement rows upper directions) = directions := | |
| (splitEquiv rows upper).symm_apply_apply directions | |
| theorem lower_preserved (rows : Nat → Nat) (upper lower : Fin n) (hne : lower ≠ upper) | |
| (direction : PrefixTests.LevelDirection rows upper) | |
| (directions : CanonicalDirections.Tuple rows n) : | |
| assemble rows upper direction (complement rows upper directions) lower = directions lower := by | |
| rw [assemble_at_other rows upper direction (complement rows upper directions) lower hne] | |
| rfl | |
| def complementLaw (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) : | |
| FiniteDistribution (Complement rows upper) := | |
| FiniteProductSplit.otherLaw (CanonicalDirections.levelLaw rows hrows) upper | |
| theorem split_law (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) : | |
| (CanonicalDirections.law rows n hrows).pushforward (splitEquiv rows upper) = | |
| (CanonicalDirections.levelLaw rows hrows upper).product (complementLaw rows hrows upper) := | |
| FiniteProductSplit.split_law (CanonicalDirections.levelLaw rows hrows) upper | |
| theorem assemble_law (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) : | |
| ((CanonicalDirections.levelLaw rows hrows upper).product | |
| (complementLaw rows hrows upper)).pushforward | |
| (fun z => assemble rows upper z.1 z.2) = CanonicalDirections.law rows n hrows := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution (PrefixTests.LevelDirection rows upper × Complement rows upper) => | |
| μ.pushforward (splitEquiv rows upper).symm) (split_law rows hrows upper) | |
| rw [FiniteDistribution.pushforward_comp] at h | |
| have hid : (fun directions : CanonicalDirections.Tuple rows n => | |
| (splitEquiv rows upper).symm (splitEquiv rows upper directions)) = id := by | |
| funext directions | |
| exact (splitEquiv rows upper).symm_apply_apply directions | |
| rw [hid, FiniteDistribution.pushforward_id] at h | |
| exact h.symm | |
| theorem selected_law (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) : | |
| (CanonicalDirections.law rows n hrows).pushforward (fun directions => directions upper) = | |
| CanonicalDirections.levelLaw rows hrows upper := | |
| FiniteProduct.eval_pushforward (CanonicalDirections.levelLaw rows hrows) upper | |
| theorem probability_split (rows : Nat → Nat) (hrows : ∀ k, 0 < rows (k + 1)) | |
| (upper : Fin n) (event : CanonicalDirections.Tuple rows n → Bool) : | |
| (CanonicalDirections.law rows n hrows).probability event = | |
| ((CanonicalDirections.levelLaw rows hrows upper).product | |
| (complementLaw rows hrows upper)).probability | |
| (fun z => event (assemble rows upper z.1 z.2)) := by | |
| rw [← assemble_law rows hrows upper, FiniteDistribution.probability_pushforward] | |
| theorem probability_complement_first (rows : Nat → Nat) | |
| (hrows : ∀ k, 0 < rows (k + 1)) (upper : Fin n) | |
| (event : CanonicalDirections.Tuple rows n → Bool) : | |
| (CanonicalDirections.law rows n hrows).probability event = | |
| (complementLaw rows hrows upper).expectation (fun rest => | |
| (CanonicalDirections.levelLaw rows hrows upper).probability | |
| (fun direction => event (assemble rows upper direction rest))) := by | |
| rw [probability_split rows hrows upper] | |
| exact UsefulAdvice.product_probability_advice_first _ _ _ | |
| theorem lower_choice_preserved {branch : Nat → Nat} (rows : Nat → Nat) | |
| (leaf : RecursiveSpaces.Slots branch n) (upper lower : Fin n) (hne : lower ≠ upper) | |
| (direction : PrefixTests.LevelDirection rows upper) | |
| (directions : CanonicalDirections.Tuple rows n) : | |
| PrefixTests.choiceAt rows leaf lower | |
| (assemble rows upper direction (complement rows upper directions) lower) = | |
| PrefixTests.choiceAt rows leaf lower (directions lower) := by | |
| rw [lower_preserved rows upper lower hne direction directions] | |
| end | |
| end PerfectCompleteness.CanonicalUpperDirectionSplit | |
| end | |
| end OAI | |