openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /CanonicalDictionaryNames.lean
| import OAI.Computability.PerfectCompleteness.Machines.CanonicalDictionaryMachine | |
| import OAI.Computability.PerfectCompleteness.Machines.CanonicalKeyShapeMachineLemmas | |
| namespace OAI | |
| namespace PerfectCompleteness.CanonicalDictionaryNames | |
| open CanonicalKeys Turing | |
| open UniqueGamesTheorem.Foundations.Complexity MachineComposition | |
| noncomputable section | |
| variable {n : Nat} | |
| theorem updatedTokens_eq (earlier : List (Key n)) (key : Key n) : | |
| CanonicalDictionaryMachine.updatedTokens (CanonicalVertexNames.tokens earlier) | |
| (CanonicalVertexNames.payload key) = | |
| CanonicalVertexNames.tokens (earlier ++ [key]) := | |
| (CanonicalOnlineNames.tokens_append_current earlier key).symm | |
| theorem keyWord_eq (key : Key n) : | |
| (CanonicalVertexNames.payload key).reverse = | |
| (CanonicalKeyEncoding.bits key ++ [true]).reverse := rfl | |
| theorem index_eq_onlineID (earlier : List (Key n)) (key : Key n) : | |
| CanonicalDictionaryMachine.index (CanonicalVertexNames.tokens earlier) | |
| (CanonicalVertexNames.payload key) = CanonicalOnlineNames.onlineID earlier key := by | |
| unfold CanonicalDictionaryMachine.index CanonicalOnlineNames.onlineID | |
| rw [updatedTokens_eq] | |
| theorem index_eq_finalName (earlier suffix : List (Key n)) (key : Key n) : | |
| CanonicalDictionaryMachine.index (CanonicalVertexNames.tokens earlier) | |
| (CanonicalVertexNames.payload key) = | |
| (CanonicalVertexNames.name (earlier ++ key :: suffix) key (by simp)).val := by | |
| rw [index_eq_onlineID, CanonicalOnlineNames.onlineID_eq_name_val earlier suffix key] | |
| def steps (earlier : List (Key n)) (key : Key n) : Nat := | |
| CanonicalDictionaryMachine.steps (CanonicalVertexNames.tokens earlier) | |
| (CanonicalVertexNames.payload key) | |
| theorem steps_le (earlier : List (Key n)) (key : Key n) : | |
| steps earlier key ≤ | |
| 3 * (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key]))).length ^ 2 + | |
| 16 * (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key]))).length + | |
| 12 := by | |
| simpa only [steps, updatedTokens_eq] using | |
| CanonicalDictionaryMachine.steps_le (CanonicalVertexNames.tokens earlier) | |
| (CanonicalVertexNames.payload key) | |
| def streamBound (n bound count : Nat) : Nat := | |
| (2 * CanonicalVertexNames.payloadBound n bound + 2) * count | |
| theorem steps_le_metadata (earlier : List (Key n)) (key : Key n) (bound : Nat) | |
| (hretained : ∀ k ∈ earlier ++ [key], k.retained.length ≤ n) | |
| (hIDs : ∀ k ∈ earlier ++ [key], ∀ entry ∈ k.retained, | |
| KeyMetadataEncoding.entryID entry.2 ≤ bound) : | |
| steps earlier key ≤ | |
| 3 * streamBound n bound (earlier.length + 1) ^ 2 + | |
| 16 * streamBound n bound (earlier.length + 1) + 12 := by | |
| have hstream : | |
| (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key]))).length ≤ | |
| streamBound n bound (earlier.length + 1) := by | |
| simpa only [streamBound, List.length_append, List.length_singleton] using | |
| CanonicalVertexNames.stream_length_le (earlier ++ [key]) bound hretained hIDs | |
| have hsquare := Nat.mul_le_mul hstream hstream | |
| apply (steps_le earlier key).trans | |
| nlinarith | |
| variable {K Λ A : Type} [DecidableEq K] | |
| theorem dictionaryTrace (tape : Fin 8 → K) (distinct : Function.Injective tape) | |
| (labels : CanonicalDictionaryMachine.Label → Λ) (done rejected : Option Λ) | |
| (program : Λ → TM2.Stmt (CanonicalDictionaryMachine.Alphabet (K := K)) Λ | |
| (CanonicalDictionaryMachine.State A)) | |
| (atLabels : ∀ l, program (labels l) = | |
| CanonicalDictionaryMachine.instruction tape labels done rejected l) | |
| (base : K → List Bool) (ambient : A) (earlier : List (Key n)) (key : Key n) | |
| (keyWord : base (tape 0) = (CanonicalVertexNames.payload key).reverse) | |
| (dictionaryWord : base (tape 1) = | |
| BinaryNameSearch.stream (CanonicalVertexNames.tokens earlier)) | |
| (empty : ∀ i : Fin 8, 2 ≤ i.val → i.val ≤ 6 → base (tape i) = []) : | |
| (advance (TM2.step program))^[steps earlier key] | |
| (some ⟨some (labels CanonicalDictionaryMachine.main), | |
| CanonicalDictionaryMachine.clean ambient, base⟩) = | |
| some ⟨done, CanonicalDictionaryMachine.clean ambient, | |
| Function.update (Function.update base (tape 1) | |
| (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key])))) (tape 7) | |
| ((encodeWord (CanonicalOnlineNames.onlineID earlier key)).reverse ++ base (tape 7))⟩ := by | |
| simpa only [steps, updatedTokens_eq, index_eq_onlineID] using | |
| CanonicalDictionaryMachine.dictionaryTrace tape distinct labels done rejected program | |
| atLabels base ambient (CanonicalVertexNames.tokens earlier) (CanonicalVertexNames.payload key) | |
| (CanonicalVertexNames.payload_canonical key) (CanonicalVertexNames.tokens_canonical earlier) | |
| keyWord dictionaryWord empty | |
| def dictionaryInTime (tape : Fin 8 → K) (distinct : Function.Injective tape) | |
| (labels : CanonicalDictionaryMachine.Label → Λ) (done rejected : Option Λ) | |
| (program : Λ → TM2.Stmt (CanonicalDictionaryMachine.Alphabet (K := K)) Λ | |
| (CanonicalDictionaryMachine.State A)) | |
| (atLabels : ∀ l, program (labels l) = | |
| CanonicalDictionaryMachine.instruction tape labels done rejected l) | |
| (base : K → List Bool) (ambient : A) (earlier : List (Key n)) (key : Key n) | |
| (keyWord : base (tape 0) = (CanonicalVertexNames.payload key).reverse) | |
| (dictionaryWord : base (tape 1) = | |
| BinaryNameSearch.stream (CanonicalVertexNames.tokens earlier)) | |
| (empty : ∀ i : Fin 8, 2 ≤ i.val → i.val ≤ 6 → base (tape i) = []) : | |
| StateTransition.EvalsToInTime (TM2.step program) | |
| ⟨some (labels CanonicalDictionaryMachine.main), CanonicalDictionaryMachine.clean ambient, | |
| base⟩ | |
| (some ⟨done, CanonicalDictionaryMachine.clean ambient, | |
| Function.update (Function.update base (tape 1) | |
| (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key])))) (tape 7) | |
| ((encodeWord (CanonicalOnlineNames.onlineID earlier key)).reverse ++ base (tape 7))⟩) | |
| (3 * (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key]))).length ^ 2 + | |
| 16 * (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key]))).length + | |
| 12) where | |
| steps := steps earlier key | |
| evals_in_steps := dictionaryTrace tape distinct labels done rejected program atLabels | |
| base ambient earlier key keyWord dictionaryWord empty | |
| steps_le_m := steps_le earlier key | |
| theorem dictionaryTrace_finalName (tape : Fin 8 → K) (distinct : Function.Injective tape) | |
| (labels : CanonicalDictionaryMachine.Label → Λ) (done rejected : Option Λ) | |
| (program : Λ → TM2.Stmt (CanonicalDictionaryMachine.Alphabet (K := K)) Λ | |
| (CanonicalDictionaryMachine.State A)) | |
| (atLabels : ∀ l, program (labels l) = | |
| CanonicalDictionaryMachine.instruction tape labels done rejected l) | |
| (base : K → List Bool) (ambient : A) (earlier suffix : List (Key n)) (key : Key n) | |
| (keyWord : base (tape 0) = (CanonicalVertexNames.payload key).reverse) | |
| (dictionaryWord : base (tape 1) = | |
| BinaryNameSearch.stream (CanonicalVertexNames.tokens earlier)) | |
| (empty : ∀ i : Fin 8, 2 ≤ i.val → i.val ≤ 6 → base (tape i) = []) : | |
| (advance (TM2.step program))^[steps earlier key] | |
| (some ⟨some (labels CanonicalDictionaryMachine.main), | |
| CanonicalDictionaryMachine.clean ambient, base⟩) = | |
| some ⟨done, CanonicalDictionaryMachine.clean ambient, | |
| Function.update (Function.update base (tape 1) | |
| (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key])))) (tape 7) | |
| ((encodeWord (CanonicalVertexNames.name | |
| (earlier ++ key :: suffix) key (by simp)).val).reverse ++ base (tape 7))⟩ := by | |
| simpa only [CanonicalOnlineNames.onlineID_eq_name_val earlier suffix key] using | |
| dictionaryTrace tape distinct labels done rejected program atLabels | |
| base ambient earlier key keyWord dictionaryWord empty | |
| def dictionaryInTime_metadata (tape : Fin 8 → K) (distinct : Function.Injective tape) | |
| (labels : CanonicalDictionaryMachine.Label → Λ) (done rejected : Option Λ) | |
| (program : Λ → TM2.Stmt (CanonicalDictionaryMachine.Alphabet (K := K)) Λ | |
| (CanonicalDictionaryMachine.State A)) | |
| (atLabels : ∀ l, program (labels l) = | |
| CanonicalDictionaryMachine.instruction tape labels done rejected l) | |
| (base : K → List Bool) (ambient : A) (earlier : List (Key n)) (key : Key n) | |
| (keyWord : base (tape 0) = (CanonicalVertexNames.payload key).reverse) | |
| (dictionaryWord : base (tape 1) = | |
| BinaryNameSearch.stream (CanonicalVertexNames.tokens earlier)) | |
| (empty : ∀ i : Fin 8, 2 ≤ i.val → i.val ≤ 6 → base (tape i) = []) | |
| (bound : Nat) | |
| (hretained : ∀ k ∈ earlier ++ [key], k.retained.length ≤ n) | |
| (hIDs : ∀ k ∈ earlier ++ [key], ∀ entry ∈ k.retained, | |
| KeyMetadataEncoding.entryID entry.2 ≤ bound) : | |
| StateTransition.EvalsToInTime (TM2.step program) | |
| ⟨some (labels CanonicalDictionaryMachine.main), CanonicalDictionaryMachine.clean ambient, | |
| base⟩ | |
| (some ⟨done, CanonicalDictionaryMachine.clean ambient, | |
| Function.update (Function.update base (tape 1) | |
| (BinaryNameSearch.stream (CanonicalVertexNames.tokens (earlier ++ [key])))) (tape 7) | |
| ((encodeWord (CanonicalOnlineNames.onlineID earlier key)).reverse ++ base (tape 7))⟩) | |
| (3 * streamBound n bound (earlier.length + 1) ^ 2 + | |
| 16 * streamBound n bound (earlier.length + 1) + 12) where | |
| steps := steps earlier key | |
| evals_in_steps := dictionaryTrace tape distinct labels done rejected program atLabels | |
| base ambient earlier key keyWord dictionaryWord empty | |
| steps_le_m := steps_le_metadata earlier key bound hretained hIDs | |
| end | |
| end PerfectCompleteness.CanonicalDictionaryNames | |
| end OAI | |